Stock market probability using statistics to predict and optimize investment outcomes pdf

It contains various useful concepts and topics at many levels of learning statistics for decision making under uncertainties. The cardinal objective for this Web site is to increase the extent to which statistical thinking is merged with managerial thinking for good decision making under uncertainty.

Towards Statistical Thinking for Decision Making Chapter 2: Descriptive Sampling Data Analysis Chapter 3: Probability as a Confidence Measuring Tool for Statistical Inference Chapter 4: Necessary Conditions for Statistical Decision Making Chapter 5: Estimators and Their Qualities Chapter 6: Rejecting a Claim Chapter 7: Hypotheses Testing for Means and Proportions Chapter 8: Tests for Statistical Equality of Two or More Populations Chapter 9: Applications of the Chi-square Statistic Chapter Regression Modeling and Analysis Chapter Unified Views of Statistical Decision Technologies Chapter Index Numbers and Ratios with Applications A Why List: Frequently Asked Statistical Questions Word.

Doc Formulas Concerning the Mean s PDFPrint to enlarge A Conceptual Summary-Sheet A Technical Summary-Sheet Exercise Your Knowledge to Enhance What You Have Learned PDF E-Labs and Computational Tools Excel for Statistical Data Analysis Widely Used Statistical Tables PDF What Maths Do I Need for This Course? Enter a word or phrase in the dialogue box, e.

Towards Statistical Thinking for Decision Making Introduction The Birth of Probability and Statistics Statistical Modeling for Decision-Making under Uncertainties Statistical Decision-Making Process What is Business Statistics? Common Statistical Terminology with Applications Descriptive Sampling Data Analysis Greek Letters Commonly Used in Statistics Type of Data and Levels of Measurement Why Statistical Sampling?

Sampling Methods Representative of a Sample: Measures of Central Tendency Selecting Among the Mean, Median, and Mode Specialized Averages: Checking for Homogeneity of Population How to Construct a BoxPlot Measuring the Quality of a Sample Selecting Among the Measures of Dispersion Shape of a Distribution Function: How to Count Without Counting Joint Probability and Statistics Mutually Exclusive versus Independent Events What Is so Important About the Normal Distributions?

What Is a Sampling Distribution? What Is The Central Limit Theorem CLT? An Illustration of CLT What Is"Degrees of Freedom"? Applications of and Conditions for Using Statistical Tables Numerical Examples for Statistical Tables Beta Density Function Binomial Probability Function Chi-square Density Function Exponential Density Function F-Density Function Gamma Density Function Geometric Probability Function Hypergeometric Probability Function Log-normal Density Function Multinomial Probability Function Negative Binomial Probability Function Normal Density Function Poisson Probability Function Student T-Density Function Triangular Density Function Uniform Density Function Other Density and Probability Functions Necessary Conditions for Statistical Decision Making Introduction Measure of Surprise for Outlier Detection Homogeneous Population Don't mix apples and oranges Test for Randomness Test for Normality Estimators and Their Qualities Introduction Qualities of a Good Estimator Estimations with Confidence What Is the Margin of Error?

Bootstrapping and Jackknifing Prediction Intervals What Is a Standard Error? Sample Size Determination Pooling the Sampling Estimates for Mean, Variance, and Standard Deviation Revising the Expected Value and the Variance Subjective Assessment of Several Estimates Bayesian Statistical Inference: An Introduction Hypothesis Testing: Rejecting a Claim Introduction Managing the Producer's or the Consumer's Risk Classical Approach to Testing Hypotheses The Meaning and Interpretation of P-values what the data say Blending the Classical and the P-value Based Approaches in Test of Hypotheses Bonferroni Method for Multiple P-Values Procedure Power of a Test and the Size Effect Parametric vs.

Distribution-free Tests Hypotheses Testing for Means and Proportions Introduction Single Population t-Test Two Independent Populations Non-parametric Multiple Comparison Procedures The Before-and-After Test ANOVA for Normal but Condensed Data Sets ANOVA for Dependent Populations Tests for Statistical Equality of Two or More Populations Introduction Equality of Two Normal Populations Testing a Shift in Normal Populations Analysis of Variance ANOVA Equality of Proportions in Several Populations Distribution-free Equality of Two Populations Comparison of Two Random Variables Applications of the Chi-square Statistic Introduction Test for Crosstable Relationship 2 by 2 Crosstable Analysis Identical Populations Test for Crosstable Data Test for Equality of Several Population Proportions Test for Equality of Several Population Medians Goodness-of-Fit Test for Probability Mass Functions Compatibility of Multi-Counts Necessary Conditions in Applying the Above Tests Testing the Variance: Is the Quality that Good?

Testing the Equality of Multi-Variances Correlation Coefficients Testing Regression Modeling and Analysis Simple Linear Regression: Computational Aspects Regression Modeling and Analysis Regression Modeling Selection Process Covariance and Correlation Pearson, Spearman, and Point-biserial Correlations Correlation, and Level of Significance Independence vs.

Correlated How to Compare Two Correlation Coefficients Conditions and the Check-list for Linear Models Analysis of Covariance: Comparing the Slopes Residential Properties Appraisal Application Unified Views of Statistical Decision Technologies Introduction Hypothesis Testing with Confidence Regression Analysis, ANOVA, and Chi-square Test Regression Analysis, ANOVA, T-test, and Coefficient of Determination Relationships among Popular Distibutions Index Numbers and Ratios with Applications Introduction Consumer Price Index Ratio Indexes Composite Index Numbers Variation Index as a Quality Indicator Labor Force Unemployment Index Seasonal Index and Deseasonalizing Data Human Ideal Weight: The Body Mass Index Statistical Technique and Index Numbers Introduction to Statistical Thinking for Decision Making This site builds up the basic ideas of business statistics systematically and correctly.

It is a combination of lectures and computer-based practice, joining theory firmly with practice. It introduces techniques for summarizing and presenting data, estimation, confidence intervals and hypothesis testing.

The presentation focuses more on understanding of key concepts and statistical thinking, and less on formulas and calculations, which can now be done on small computers through user-friendly Statistical JavaScript Aetc. Today's good decisions are driven by data. In all aspects of our lives, and importantly in the business context, an amazing diversity of data is available for inspection and analytical insight.

Business managers and professionals are increasingly required to justify decisions on the basis of data. They need statistical model-based decision support systems. Statistical skills enable them to intelligently collect, analyze and interpret data relevant to their decision-making. Statistical concepts and statistical thinking enable them to: In competitive environment, business managers must design quality into products, and into the processes of making the products.

They must facilitate a process of never-ending improvement at all stages of manufacturing and service. This is a strategy that employs statistical methods, particularly statistically designed experimentsand produces processes that provide high yield and products that seldom fail.

Moreover, it facilitates development of robust products that are insensitive to changes in the environment and internal component variation. Carefully planned statistical studies remove hindrances to high quality and productivity at every stage of production. This saves time and money. It is well recognized that quality must be engineered into products as early as possible in the design process.

One must know how to use carefully planned, cost-effective statistical experiments to improve, optimize and make robust products and processes. Business Statistics is a science assisting you to make business decisions under uncertainties based on some numerical and measurable scales. Decision making processes must be based on data, not on personal opinion nor on belief.

The Devil is in the Deviations: Variation is inevitable in life! Every process, every measurement, every sample has variation. Managers need to understand variation for two key reasons.

First, so that they can lead others to apply statistical thinking in day-to-day activities and secondly, to apply the concept for the purpose of continuous improvement. This course will provide you with hands-on experience to promote the use of statistical thinking and techniques to apply them to make educated decisions, whenever you encounter variation in business data.

You will learn techniques to intelligently assess and manage the risks inherent in decision-making. Just like weather, if you cannot control something, you should learn how to measure and analyze it, in order to predict it, effectively. If you have taken statistics before, and have a feeling of inability to grasp concepts, it may be largely due to your former non-statistician instructors teaching statistics.

Their deficiencies lead students to develop phobias for the sweet science of statistics. In this respect, Professor Herman Chernoff made the following remark: Plugging numbers into the formulas and crunching them have no value by themselves.

You should continue to put effort into the concepts and concentrate on interpreting the results. Even when you solve a small size problem by hand, I would like you to use the available computer software and Web-based computation to do the dirty work for you. You must be able to read the logical secret in any formulas not memorize them.

For example, in computing the variance, consider its formula. Instead of memorizing, you should start with some why: Why do we square the deviations from the mean. Because, if we add up all deviations, we get always zero value. So, to deal with this problem, we square the deviations. Why not raise to the power of four three will not work?

Squaring does the trick; why should we make life more complicated than it is? Notice also that squaring also magnifies the deviations; therefore it works to our advantage to measure the quality of the data. Why is there a summation notation in the formula.

To add up the squared deviation of each data point to compute the total sum of squared deviations. Why do we divide the sum of squares by n The amount of deviation should reflect also how large the sample is; so we must bring in the sample size. That is, in general, larger sample sizes have larger sum of square deviation from the mean. Why n-1 not n? The reason for n-1 is that when you divide by n-1, the sample's variance provides an estimated variance much closer to the population variance, than when you divide by n.

You note that for large sample size n say over 30it really does not matter whether it is divided by n or n The results are almost the same, and they are acceptable. The factor n-1 is what we consider as the"degrees of freedom". This example shows how to question statistical formulas, rather than memorizing them.

In fact, when you try to understand the formulas, you do not need to remember them, they are part of your brain connectivity. Clear thinking is always more important than the ability to do arithmetic.

The computer-assisted learning provides you a"hands-on" experience which will enhance your understanding of the concepts and techniques covered in this site. Java, once an esoteric programming language for animating Web pages, is now a full-fledged platform for building JavaScript E-labs' learning objects with useful applications.

As you used to do experiments in physics labs to learn physics, computer-assisted learning enables you to use any online interactive tool available on the Internet to perform experiments. The purpose is the same; i. The appearance of computer software, JavaScript, Statistical Demonstration Applets, and Online Computation are the most important events in the process of teaching and learning concepts in model-based, statistical decision making courses.

These e-lab Technologies allow you to construct numerical examples to understand the concepts, and to find their significance for yourself. Unfortunately, most classroom courses are not learning systems. The way the instructors attempt to help their students acquire skills and knowledge has absolutely nothing to do with the way students actually learn.

Many instructors rely on lectures and tests, and memorization.

Tools for Decision Analysis

All too often, they rely on"telling. Certainly, we learn by doing, failing, and practicing until we do it right. The computer assisted learning serves this purpose. A course in appreciation of statistical thinking gives business professionals an edge.

Professionals with strong quantitative skills are in demand. This phenomenon will grow as the impetus for data-based decisions strengthens and the amount and availability of data increases. The statistical toolkit can be developed and enhanced at all stages of a career.

Decision making process under uncertainty is largely based on application of statistics for probability assessment of uncontrollable events or factorsas well as risk assessment of your decision. For more statistical-based Web sites with decision making applications, visit Decision Science Resourcesand Modeling and Simulation Resources sites.

The main objective for this course is to learn statistical thinking; to emphasize more on concepts, and less theory and fewer recipes, and finally to foster active learning using the useful and interesting Web-sites.

It is already a known fact that"Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write. Early in the book he stated that knowledge could be considered as a collection of information, or as an activity, or as a potential.

He also noted that knowledge resides in the user and not in the collection. Papers in Honor of Herman Chernoff on His Sixtieth BirthdayAcademic Press, The Birth of Probability and Statistics The original idea of"statistics" was the collection of information about and for the"state". The word statistics derives directly, not from any classical Greek or Latin roots, but from the Italian word for state.

The birth of statistics occurred in mid th century. A commoner, named John Graunt, who was a native of London, began reviewing a weekly church publication issued by the local parish clerk that listed the number of births, christenings, and deaths in each parish.

These so called Bills of Mortality also listed the causes of death. Graunt who was a shopkeeper organized this data in the form we call descriptive statistics, which was published as Natural and Political Observations Made upon the Bills of Mortality. Shortly thereafter he was elected as a member of Royal Society. Thus, statistics has to borrow some concepts from sociology, such as the concept of Population.

It has been argued that since statistics usually involves the study of human behavior, it cannot claim the precision of the physical sciences. Probability has much longer history. Probability is derived from the verb to probe meaning to"find out" what is not too easily accessible or understandable.

The word"proof" has the same origin that provides necessary details to understand what is claimed to be true. Probability originated from the study of games of chance and gambling during the 16 th century.

Probability theory was a branch of mathematics studied by Blaise Pascal and Pierre de Fermat in the seventeenth century.

Currently in 21 st century, probabilistic modeling is used to control the flow of traffic through a highway system, a telephone interchange, or a computer processor; find the genetic makeup of individuals or populations; quality control; insurance; investment; and other sectors of business and industry.

New and ever growing diverse fields of human activities are using statistics; however, it seems that this field itself remains obscure to the public. Professor Bradley Efron expressed this fact nicely: During the 20 th Century statistical thinking and methodology have become the scientific framework for literally dozens of fields including education, agriculture, economics, biology, and medicine, and with increasing influence recently on the hard sciences such as astronomy, geology, and physics.

In other words, we have grown from a small obscure field into a big obscure field. The book points out that early Enlightenment thinkers could not face uncertainty. A mechanistic, deterministic machine, was the Enlightenment view of the world. Edwards, Annotated Readings in the History of Statistics, Springer, Offers a general historical collections of the probability and statistical literature.

Covers the classical, logical, subjective, frequency, and propensity views. A philosophical study of early ideas about probability, induction and statistical inference. Statistical Principles and PersonalitiesSpringer, New York, It teaches the principles of applied economic and social statistics in a historical context.

Featured topics include public opinion polls, industrial quality control, factor analysis, Bayesian methods, program evaluation, non-parametric and robust methods, and exploratory data analysis.

The author states that statistics has become known in the twentieth century as the mathematical tool for analyzing experimental and observational data. Enshrined by public policy as the only reliable basis for judgments as the efficacy of medical procedures or the safety of chemicals, and adopted by business for such uses as industrial quality control, it is evidently among the products of science whose influence on public and private life has been most pervasive.

Statistical analysis has also come to be seen in many scientific disciplines as indispensable for drawing reliable conclusions from empirical i. This new field of mathematics found so extensive a domain of applications.

The Measurement of Uncertainty BeforeU. It covers the people, ideas, and events underlying the birth and development of early statistics. This work provides the detailed lives and times of theorists whose work continues to shape much of the modern statistics. Statistical Modeling for Decision-Making under Uncertainties: From Data to the Instrumental Knowledge In this diverse world of ours, no two things are exactly the same. A statistician is interested in both the differences and the similarities ; i.

The actuarial tables published by insurance companies reflect their statistical analysis of the average life expectancy of men and women at any given age. From these numbers, the insurance companies then calculate the appropriate premiums for a particular individual to purchase a given amount of insurance.

Exploratory analysis of data makes use of numerical and graphical techniques to study patterns and departures from patterns. The widely used descriptive statistical techniques are: Frequency Distribution ; Histograms; Boxplot; Scattergrams and Error Bar plots; and diagnostic plots.

In examining distribution of data, you should be able to detect important characteristics, such as shape, location, variability, and unusual values. From careful observations of patterns in data, you can generate conjectures about relationships among variables. The notion of how one variable may be associated with another permeates almost all of statistics, from simple comparisons of proportions through linear regression.

The difference between association and causation must accompany this conceptual development. Data must be collected according to a well-developed plan if valid information on a conjecture is to be obtained.

The plan must identify important variables related to the conjecture, and specify how they are to be measured. From the data collection plan, a statistical model can be formulated from which inferences can be drawn. As an example of statistical modeling with managerial implicationssuch as "what-if" analysisconsider regression analysis.

Regression analysis is a powerful technique for studying relationship between dependent variables i. Summarizing relationships among the variables by the most appropriate equation i. Frequently, for example the marketing managers are faced with the question, What Sample Size Do I Need? This is an important and common statistical decision, which should be given due consideration, since an inadequate sample size invariably leads to wasted resources.

The sample size determination section provides a practical solution to this risky decision. Statistical models are currently used in various fields of business and science. However, the terminology differs from field to field. For example, the fitting of models to data, called calibration, history matching, and data assimilation, are all synonymous with parameter estimation.

Your organization database contains a wealth of information, yet the decision technology group members tap a fraction of it.

Employees waste time scouring multiple sources for a database. The decision-makers are frustrated because they cannot get business-critical data exactly when they need it. Therefore, too many decisions are based on guesswork, not facts. Many opportunities are also missed, if they are even noticed at all. Knowledge is what we know well. Information is the communication of knowledge. In every knowledge exchange, there is a sender and a receiver.

The sender make common what is private, does the informing, the communicating. Information can be classified as explicit and tacit forms. The explicit information can be explained in structured form, while tacit information is inconsistent and fuzzy to explain.

Know that data are only crude information and not knowledge by themselves. Data is known to be crude information and not knowledge by itself. The sequence from data to knowledge is: Data becomes information, when it becomes relevant to your decision problem. Information becomes fact, when the data can support it. Facts are what the data reveals. However the decisive instrumental i. Fact becomes knowledge, when it is used in the successful completion of a decision process. Once you have a massive amount of facts integrated as knowledge, then your mind will be superhuman in the same sense that mankind with writing is superhuman compared to mankind before writing.

The following figure illustrates the statistical thinking process based on data in constructing statistical models for decision making under uncertainties. Click on the image to enlarge it and THEN print it. The Path from Statistical Data to Managerial Knowledge The above figure depicts the fact that as the exactness of a statistical model increases, the level of improvements in decision-making increases. That's why we need Business Statistics.

Statistics arose from the need to place knowledge on a systematic evidence base. This required a study of the rules of computational probability, the development of measures of data properties and relationships, and so on. Statistical inference aims at determining whether any statistical significance can be attached that results after due allowance is made for any random variation as a source of error. Intelligent and critical inferences cannot be made by those who do not understand the purpose, the conditions, and applicability of the various techniques for judging significance.

Considering the uncertain environment, the chance that"good decisions" are made increases with the availability of"good information. The above figure also illustrates the fact that as the exactness of a statistical model increases, the level of improvements in decision-making increases. Knowledge is more than knowing something technical.

Wisdom is the power to put our time and our knowledge to the proper use. Wisdom comes with age and experience. Wisdom is the accurate application of accurate knowledge and its key component is to knowing the limits of your knowledge. Wisdom is about knowing how something technical can be best used to meet the needs of the decision-maker. Wisdom, for example, creates statistical software that is useful, rather than technically brilliant. For example, ever since the Web entered the popular consciousness, observers have noted that it puts information at your fingertips but tends to keep wisdom out of reach.

The notion of "wisdom" in the sense of practical wisdom has entered Western civilization through biblical texts. In the Hellenic experience this kind of wisdom received a more structural character in the form of philosophy.

In this sense philosophy also reflects one of the expressions of traditional wisdom. Business professionals need a statistical toolkit. Statistical skills enable you to intelligently collect, analyze and interpret data relevant to their decision-making. Statistical concepts enable us to solve problems in a diversity of contexts. Statistical thinking enables you to add substance to your decisions. That's why we need statistical data analysis in probabilistic modeling.

Statistics arose from the need to place knowledge management on a systematic evidence base. This required a study of the rules of computational probability, the development of measures of data properties, relationships, and so on.

The purpose of statistical thinking is to get acquainted with the statistical techniques, to be able to execute procedures using available JavaScript, and to be conscious of the conditions and limitations of various techniques. Statistical Decision-Making Process Unlike the deterministic decision-making process, such as linear optimization by solving systems of equationsParametric systems of equations and in decision making under pure uncertaintythe variables are often more numerous and more difficult to measure and control.

However, the steps are the same. Simplification Building a decision model Testing the model Using the model to find the solution: It is a simplified representation of the actual situation It need not be complete or exact in all respects It concentrates on the most essential relationships and ignores the less essential ones. It is more easily understood than the empirical i. It can be used again and again for similar problems or can be modified.

Fortunately the probabilistic and statistical methods for analysis and decision making under uncertainty are more numerous and powerful today than ever before.

The computer makes possible many practical applications. A few examples of business applications are the following: An auditor can use random sampling techniques to audit the accounts receivable for clients. A plant manager can use statistical quality control techniques to assure the quality of his production with a minimum of testing or inspection.

A financial analyst may use regression and correlation to help understand the relationship of a financial ratio to a set of other variables in business. A market researcher may use test of significace to accept or reject the hypotheses about a group of buyers to which the firm wishes to sell a particular product.

A sales manager may use statistical techniques to forecast sales for the coming year. Questions Concerning Statistical the Decision-Making Process: Williamson, Foundations of BayesianismKluwer Academic Publishers, Contains Logic, Mathematics, Decision Theory, and Criticisms of Bayesianism.

Schlaifer, Introduction to Statistical Decision TheoryThe MIT Press, What is Business Statistics? The main objective of Business Statistics is to make inferences e. The condition for randomness is essential to make sure the sample is representative of the population.

It provides knowledge and skills to interpret and use statistical techniques in a variety of business applications. A typical Business Statistics course is intended for business majors, and covers statistical study, descriptive statistics collection, description, analysis, and summary of dataprobability, and the binomial and normal distributions, test of hypotheses and confidence intervals, linear regression, and correlation.

Statistics is a science of making decisions with respect to the characteristics of a group of persons or objects on the basis of numerical information obtained from a randomly selected sample of the group. Statisticians refer to this numerical observation as realization of a random sample.

However, notice that one cannot see a random sample. A random sample is only a sample of a finite outcomes of a random process. At the planning stage of a statistical investigation, the question of sample size n is critical. For example, sample size for sampling from a finite population of size N, is set at: Clearly, a larger sample provides more relevant information, and as a result a more accurate estimation and better statistical judgement regarding test of hypotheses.

Under-lit Streets and the Crimes Rate: It is a fact that if residential city streets are under-lit then major crimes take place therein. Activities Associated with the General Statistical Thinking and Its Applications The above figure illustrates the idea of statistical inference from a random sample about the population. The major task of Statistics is the scientific methodology for collecting, analyzing, interpreting a random sample in order to draw inference about some particular characteristic of a specific Homogenous Population.

For two major reasons, it is often impossible to study an entire population: The process would be too expensive or too time-consuming. The process would be destructive. In either case, we would resort to looking at a sample chosen from the population and trying to infer information about the entire population by only examining the smaller sample.

Very often the numbers, which interest us most about the population, are the mean m and standard deviation sany number -- like the mean or standard deviation -- which is calculated from an entire population, is called a Parameter. If the very same numbers are derived only from the data of a sample, then the resulting numbers are called Statistics.

Frequently, Greek letters represent parameters and Latin letters represent statistics as shown in the above Figure. The uncertainties in extending and generalizing sampling results to the population are measures and expressed by probabilistic statements called Inferential Statistics. Therefore, probability is used in statistics as a measuring tool and decision criterion for dealing with uncertainties in inferential statistics. An important aspect of statistical inference is estimating population values parameters from samples of data.

An estimate of a parameter is unbiased if the expected value of sampling distribution is equal to that population. The sample mean is an unbiased estimate of the population mean. The sample variance is an unbiased estimate of population variance.

This allows us to combine several estimates to obtain a much better estimate. The Empirical distribution is the distribution of a random sample, shown by a step-function in the above figure. The empirical distribution function is an unbiased estimate for the population distribution function F x. Given you already have a realization set of a random sample, to compute the descriptive statistics including those in the above figure, you may like using Descriptive Statistics JavaScript.

To reduce this uncertainty and having high confidence that statistical inferences are correct, a sample must give equal chance to each member of population to be selected which can be achieved by sampling randomly and relatively large sample size n.

Given you already have a realization set of a random sample, to perform hypothesis testing for mean m and variance s 2you may like using Testing the Mean and Testing the Variance JavaScript, respectively. Statistics is a tool that enables us to impose order on the disorganized cacophony of the real world of modern society. The business world has grown both in size and competition.

Corporate executive must take risk in businesshence the need for business statistics. Business statistics has grown with the art of constructing charts and tables! It is a science of basing decisions on numerical data in the face of uncertainty. Business statistics is a scientific approach to decision making under risk.

In practicing business statistics, we search for an insight, not the solution. Our search is for the one solution that meets all the business's needs with the lowest level of risk.

Business statistics can take a normal business situation, and with the proper data gathering, analysis, and re-search for a solution, turn it into an opportunity.

While business statistics cannot replace the knowledge and experience of the decision maker, it is a valuable tool that the manager can employ to assist in the decision making process in order to reduce the inherent risk, measured by, e.

Among other useful questions, you may ask why we are interested in estimating the population's expected value m and its Standard Deviation s? Here are some applicable reasons. Business Statistics must provide justifiable answers to the following concerns for every consumer and producer: That is, what is a good estimate for m? That is, what is a good estimate for s? That is, comparing several m 's, and several s 's. Common Statistical Terminology with Applications Like all profession, also statisticians have their own keywords and phrases to ease a precise communication.

However, one must interpret the results of any decision making in a language that is easy for the decision-maker to understand. This lack of communication between statisticians and the managers is the major roadblock for using statistics. A population is any entire collection of people, animals, plants or things on which we may collect data. It is the entire group of interest, which we wish to describe or about which we wish to draw conclusions.

In the above figure the life of the light bulbs manufactured say by GE, is the concerned population. Qualitative and Quantitative Variables: Any object or event, which can vary in successive observations either in quantity or quality is called a"variable.

A qualitative variable, unlike a quantitative variable does not vary in magnitude in successive observations. The values of quantitative and qualitative variables are called"Variates" and"Attributes", respectively. A characteristic or phenomenon, which may take different values, such as weight, gender since they are different from individual to individual.

The fascinating fact about inferential statistics is that, although each random observation may not be predictable when taken alone, collectively they follow a predictable pattern called its distribution function. For example, it is a fact that the distribution of a sample average follows a normal distribution for sample size over In other words, an extreme value of the sample mean is less likely than an extreme value of a few raw data.

A subset of a population or universe. An experiment is a process whose outcome is not known in advance with certainty. An experiment in general is an operation in which one chooses the values of some variables and measures the values of other variables, as in physics. A statistical experiment, in contrast is an operation in which one take a random sample from a population and infers the values of some variables.

For example, in a survey, we"survey" i. A random sample from the relevant population provides information about the voting intentions.

In order to make any generalization about a population, a random sample from the entire population; that is meant to be representative of the population, is often studied. For each population, there are many possible samples. A sample statistic gives information about a corresponding population parameter. For example, the sample mean for a set of data would give information about the overall population mean m.

It is important that the investigator carefully and completely defines the population before collecting the sample, including a description of the members to be included. The population for a study of infant health might be all children born in the U. The sample might be all babies born on 7 th of May in any of the years. An experiment is any process or study which results in the collection of data, the outcome of which is unknown.

In statistics, the term is usually restricted to situations in which the researcher has control over some of the conditions under which the experiment takes place. Before introducing a new drug treatment to reduce high blood pressure, the manufacturer carries out an experiment to compare the effectiveness of the new drug with that of one currently prescribed.

Newly diagnosed subjects are recruited from a group of local general practices. Half of them are chosen at random to receive the new drug, the remainder receives the present one. So, the researcher has control over the subjects recruited and the way in which they are allocated to treatment.

Design of experiments is a key tool for increasing the rate of acquiring new knowledge. Knowledge in turn can be used to gain competitive advantage, shorten the product development cycle, and produce new products and processes which will meet and exceed your customer's expectations. Dj dolla sign shake your money maker 40 data and Secondary data sets: If the data are from a planned experiment relevant to the objective s of the statistical investigation, collected by the analyst, it is called a Primary Data set.

Broker forex liberty, if some condensed records are given to the analyst, it is called a Secondary Data set. A random variable is a real function yes, it is called" variable", but in reality it is a function that assigns a numerical value to each simple event. You may assign any other two distinct real numbers, as you wish; however, non-negative integer random variables are easy to work with.

Random variables are needed since one cannot do arithmetic operations on words; the random variable enables us to compute statistics, such as average and variance. Any random variable has a distribution of probabilities associated with it. Random phenomena are not haphazard: The mathematical description of variation is central to statistics.

The probability required for statistical inference is not primarily axiomatic or combinatorial, but is oriented toward describing data distributions. A unit is a person, animal, plant or thing which is actually studied by a researcher; the basic objects upon which the study or experiment is executed. For example, a person; a sample of soil; a pot of seedlings; a zip code area; a doctor's practice. A parameter is an unknown value, and therefore it data entry jobs online from home ireland to be estimated.

Parameters are used to represent a certain population characteristic. For example, the population mean m is a parameter that is often used to indicate the average value of a quantity. Within a population, a parameter is a fixed value that does not vary.

Each sample drawn from the population has its own value of any statistic that is used to estimate this parameter. For example, the mean of the data in a sample is used to give information about the overall mean m in the population from which that forex vps promo code was drawn. A statistic is a quantity that is calculated from a sample of data. It is used to give information about unknown values in the corresponding population.

For example, the average of the data in a sample is used to give information about the overall average in buy gun stocks wood blanks population from which that sample was drawn. A statistic is a function of an observable random noida forex exchange. It is therefore an observable random variable.

Notice that, while a statistic is a"function" of observations, unfortunately, it is commonly called a random"variable" not a function. It is possible to draw more than one sample from the same population, and the value of a statistic will in general vary from sample to sample. For example, the average value in a sample is a statistic. The average values in more than one sample, drawn from the same population, will not necessarily be equal. Statistics are often assigned Roman letters e.

The word estimate means to esteem, that is giving a value to something. A statistical estimate is an indication of the value of an unknown quantity based on observed data. More formally, an estimate is the particular value of an estimator that is obtained from a particular sample of data and used to indicate the value of a parameter.

Suppose the manager of a shop wanted to know mthe mean expenditure of customers in her shop in the last year. She could calculate the average expenditure of the hundreds or perhaps thousands of customers who bought goods in her shop; that is, the population mean m. Instead she could use an estimate of this population mean m by calculating the mean of a representative sample of customers.

There are two broad subdivisions of statistics: Descriptive Statistics and Inferential Statistics as described below. The numerical statistical data should be presented clearly, concisely, and in such a way that the decision maker can quickly obtain the essential characteristics of the data in order to incorporate them into decision process. The principal descriptive quantity derived from sample should you buy amr stock is the meanwhich is the arithmetic average of the sample data.

It serves as the most reliable single measure of the value of a typical member of the sample. If the sample contains a few values that are so large or so small that they have an exaggerated effect on the value of the mean, the sample is more accurately represented by the median -- the value where half the sample values fall below and half above.

The quantities most commonly used to measure the dispersion of the values about their mean are the variance s 2 and its square root, the standard deviation s. The variance is calculated by determining the mean, subtracting it from each of the sample values yielding the deviation of the samplesand then averaging the squares of these deviations.

The mean and standard deviation of the sample are used as estimates of the corresponding characteristics of the entire group from which the sample was drawn. They do notin general, completely describe the distribution F x of values within either the sample or the parent group; indeed, different distributions may have the same mean and standard deviation. They do, however, provide a complete description of the normal distribution, in which positive and negative deviations from the mean are equally common, and small deviations are much more common than large ones.

For a normally distributed set of values, a graph showing the dependence of the frequency of the deviations upon their magnitudes is a bell-shaped curve. About 68 percent of the values will differ from the mean by less than the standard deviation, and almost percent will differ by less than three times the standard deviation. Inferential statistics is concerned with making inferences from samples about the populations from which they have been drawn. In other words, if we find a difference between two samples, we would like to ecole enforex malaga, is this a"real" difference i.

That's what tests of statistical significance are all about. Any inferred conclusion from a sample data to the population from which the sample is drawn must be expressed in a probabilistic term. Probability is the language and a measuring how to get xp points fast on farmville 2 for uncertainty in our statistical conclusions.

Inferential statistics could be used for explaining a phenomenon or checking for validity of a claim. In these instances, inferential statistics is called Exploratory Data Analysis or Confirmatory Data Analysisrespectively. Statistical inference refers to extending your knowledge obtained from a random sample from the entire population to the whole population.

This is known in mathematics as Inductive Reasoningthat is, knowledge of the whole from a particular. Its main application is in hypotheses testing about a given population. Statistical inference guides the selection of appropriate statistical models. Models and data interact in statistical work. Inference from data can be thought of as the process of selecting a chris moneymaker bluffs sam farha model, including a statement in probability language of how confident one can be about the selection.

The normal or Gaussian distribution is a continuous symmetric distribution that follows the familiar bell-shaped curve. One of its nice features is that, the mean and variance uniquely and independently determines the distribution.

It 24 binary options winning formula free download been noted empirically that many measurement variables have distributions that are at least approximately normal. Even when a distribution is non-normal, the distribution of the mean of many independent observations from the same distribution becomes arbitrarily close to a normal distribution, as the number of observations grows large.

Many frequently used statistical tests make the condition that the data come from a normal distribution. Estimation and Hypothesis Testing: Inference in statistics are of two types. The first is estimationwhich involves the determination, with a possible error due to sampling, of the unknown value of a population characteristic, such as the proportion having a specific attribute or the average value m of some numerical how much money does a used car dealership make. To express the accuracy of the estimates of population characteristics, one must also compute the standard errors of the estimates.

The second type of inference is hypothesis testing. It involves the definitions of how do i earn money in farmville 2 hypothesis as one set of possible population values and an alternative, a different set.

There are many statistical procedures for determining, on the basis of a sample, whether the true population characteristic belongs to the set of values in the hypothesis or the alternative.

Statistical inference is grounded in probability, idealized concepts of the group under study, called the population, and the sample. The statistician may view the population as a set of balls from which the sample is selected at random, that is, in such a way that each ball has the same chance as every other one for inclusion in the sample. Notice that to be able to estimate the population parametersthe sample size n must be greater than one. Greek Letters Commonly Used as Statistical Notations We use Greek letters as scientific notations in statistics and other scientific fields to honor the ancient Greek philosophers who invented science and scientific thinking.

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Before Socrates, in 6 th Century BC, Thales and Pythagoras, amomg others, applied geometrical concepts to arithmetic, and Socrates is the inventor of dialectic reasoning. The revival of scientific thinking initiated by Newton's work was valued and hence reappeared almost years later.

Greek Letters Commonly Used as Statistical Notations alpha beta ki-sqre delta mu nu pi rho sigma tau theta a b c 2 d m n p r s t q Note: Ki does not exist in statistics.

I'm glad that you're overcoming all the confusions that exist in learning statistics. Type of Data and Levels of Measurement Information can be collected in statistics using qualitative or quantitative data.

Qualitative datasuch as eye color of a group of individuals, is not computable binary options trading signals in nigeria review auto binary signals arithmetic relations. They are labels that advise in which category or class an individual, object, or process fall.

They are called categorical variables. Quantitative data sets consist of measures that take numerical values for which descriptions such as means and standard deviations are meaningful. They can be put into an order and further divided into two groups: Discrete data are countable data and are collected by countingfor example, the number of defective items produced during a day's production. Continuous data are collected by measuring and are expressed on a continuous size of daily forex market analysis - forexyard. For example, measuring the height of a person.

Among the first activities in statistical analysis is to count or measure: A set of data is a representation i. Otherwise, it how to make gold fast in wow 5.4 called"secondary type" data.

Data come in the forms of N ominal, O rdinal, I nterval, and R atio remember the French word NOIR for the color black. Data can be either continuous or discrete. While the unit of measurement is arbitrary on the Ratio scale, its zero point is a natural attribute. The categorical variable is measured on an ordinal or nominal scale. A Pareto chart is similar to the histogram, except that it is a frequency bar chart for qualitative variablesrather than being used for quantitative data that have been grouped into put and call option diagram. The following is an example of a Pareto chart that shows the types of shoes-frequency, worn in the class on a particular day: A Typical Pareto Chart For a good business application of discrete random variables, visit Markov Chain CalculatorLarge Markov Chain Calculator and Zero-Sum Games.

Sampling is the selection of part of an aggregate or totality known as populationon the basis of which a decision concerning the population is made.

Cost is one of the main arguments in favor of sampling, because often a sample can furnish data of sufficient accuracy and at much lower cost than a census.

Sampling Methods From the isis stock market you eat to the television you watch, from political elections to school board actions, much of your life is regulated by the results of sample surveys.

A sample is a group of units selected from a larger group the population. By studying the sample, one hopes to draw valid conclusions about the larger group. A sample is generally selected for study because the population is too large to study in its entirety. The sample should be representative of the general population.

This is often best achieved by random sampling. Also, before collecting the sample, it is important that one carefully and completely defines the population, including a description of the members to be included.

A common problem in business statistical decision-making arises when we need information about a collection called a population but find that the cost of obtaining the information is prohibitive. For instance, suppose we need to know the average shelf life of current inventory. If the inventory is large, the cost of checking records for each item might be high enough to cancel the benefit of having the information.

On the other hand, a hunch about the average shelf life might not be good enough for decision-making purposes. This means we must arrive at a compromise that involves selecting a small number of items and calculating an average shelf life as an estimate of the average shelf life of all items in inventory. This is a compromise, since the measurements for a sample from the inventory will produce only an stock market website templates of the value we want, but at substantial savings.

What we would like to know is how"good" the estimate is and how much uk forex regulation will it cost to make it"better".

Information of this type is intimately related to sampling techniques. This section provides a short discussion on the common methods of business statistical sampling. Cluster sampling can be used whenever the population is homogeneous but can be partitioned. In many applications the partitioning is a result of physical distance. For instance, in the insurance industry, there are small"clusters" of employees in field offices scattered about the country.

In such a case, a random sampling of employee work habits might not required travel to many of the"clusters" or field offices in order to get the data. Totally sampling each one of a small number of clusters chosen at random can eliminate much of the cost associated with the data requirements of management. Stratified sampling can be used whenever the population can be partitioned into smaller sub-populations, each of which is homogeneous according to the particular characteristic of interest.

If there are k sub-populations and we let N i denote the size of sub-population i, let N denote the overall population size, and let n denote the sample size, then we select a stratified sample whenever we choose: Random sampling is probably the most popular sampling method used in decision making today.

Many decisions are made, for instance, by choosing a number out of a hat or a numbered bead from a barrel, and both of these methods are attempts to achieve a random choice from a set of items. But true random sampling must be achieved with the aid of a computer or a random number table whose values are generated nuance communications stock market computer random number generators.

A random sampling of size n is drawn from a population size N. The unbiased estimate for variance of is: For 0, 1, binary type variables, variation in estimated proportion p is: Determination of sample sizes n with regard to binary data: Smallest integer greater than or equal to: Cross-Sectional study the observation of a defined population at a single point in time or time interval. Exposure and outcome are determined simultaneously.

What is a statistical instrument? A statistical instrument is any process that aim at describing a phenomena by using any instrument or device, however the results may be used as a control tool.

Examples of statistical instruments are questionnaire and surveys sampling. What is grab sampling technique? The grab sampling technique is to take a relatively much money did cost make titanic sample over a very short period of time, the result obtained are usually instantaneous.

However, the Passive Sampling is a technique where a sampling device is used for an extended time under similar conditions. Depending on the desirable statistical investigation, the passive sampling may be a useful alternative or even more appropriate than grab sampling.

However, a passive sampling technique needs to be developed and tested in the field. Statistical Summaries Representative of a Sample: Measures of Central Tendency Summaries How do you describe the"average" or"typical" piece of information in a set of data? Different procedures are used to summarize forex club-mt4 real 2 server most representative information depending of the type of question asked and the nature of the data being summarized.

Measures of location give information about the location of the central tendency within a group of numbers. The measures of location presented in this unit for ungrouped raw data are the mean, the median, and the mode. The arithmetic mean or the average, simple mean is computed by summing all numbers in an array of numbers x i and then dividing by the number of observations n in the array.

The mean uses all of the observations, and each observation affects the mean. Even though the mean is sensitive to extreme values; i. This is due to the fact that the mean has valuable mathematical properties that make it convenient for use with inferential statistical analysis. For example, the sum of the deviations of the numbers in a set of data from the mean is zero, and the sum of the squared deviations of the numbers in a set of data from the mean is the minimum value.

You might like to use Descriptive Statistics to compute the mean. In some cases, the data in the sample or population should not be weighted equally, rather each value should be weighted according to its importance. The median is the middle value in auto profit for binary options trading strategy ordered array of observations.

If there is an even number of observations in the array, the median is the average of the two middle numbers. If there is an odd number of data in the array, how to read candlestick chart trends median is the middle number.

The truth about forex trading median is often used to summarize the distribution of an outcome. If the distribution is skewedthe median and the interquartile range IQR may be better than other measures to indicate where the observed data are concentrated.

Generally, the turtle trading exit strategy provides a better measure of location than the mean when there are some extremely large or small observations; i. For this reason, median income is used as the measure of location for the Stockbrokers jobs bristol. Note that if the median is less than the mean, the data set is skewed to the right.

If the median is greater than the mean, the data set is skewed to the left. The mean has two distinct advantages over the median. It is more stable, and one can compute the mean based of two samples by combining the two means. The mode is the most frequently occurring value in a set of observations.

Why use the mode? Data may have two modes. In this case, we say the data are bimodaland sets of observations with more than two modes are referred to as multimodal. Note that the mode is not a helpful measure of live gold price in indian rupees per 10 gram, because there can be more than one mode or even no mode. When the mean and the median are known, it is possible to estimate the mode for the unimodal distribution using the ford motor stock market quotes two averages as follows: Whenever, more than one mode exist, then the population from which the sample came is a mixture of more binary option profitable touch strategy one population, as shown, for example in the following bimodal histogram.

A Mixture of Two Different Populations However, notice that a Uniform distribution has uncountable online emarketing cash fast make lots of of modes having equal density value; therefore it is considered as a homogeneous population.

Almost all standard statistical analyses are conditioned on the assumption that the population is homogeneous. Notice that Excel has very limited widgets to earn money capability. For example, it displays only one modethe first one.

Unfortunately, this is very misleading. However, you may find out if there are others by inspection only, as follow: Create a frequency distribution, slk aftermarket bluetooth options the menu sequence: Tools, Data analysis, Frequency and follow instructions on the screen.

You will see the frequency distribution and then find the mode visually. Unfortunately, Excel does not draw a Stem and Leaf diagram. All commercial off-the-shelf software, such as SAS and SPSSdisplay a Stem and Leaf diagram, which is a frequency distribution of a given data set. Selecting Among where can i buy single shares of stock Mode, Median, and Mean It is a common mistake to specify the wrong index for central tenancy.

Selecting Among the Mode, Median, and Mean The first consideration is the type of data, if the variable is categorical, the mode is the single measure that best describes that data. The second consideration in selecting the index is to ask whether the total of all observations is of any interest. If the answer is yes, then the mean is the proper index of central tendency. If the total is of no interest, then depending on whether the histogram is symmetric or skewed one must use either mean or median, respectively.

In all cases the histogram must be unimodal. However, notice that, e. The Main Characteristics of the Mode, the Median, and the Mean Fact No.

The Mode The Median The Mean 1 It is the most frequent value in the distribution; it is the point of greatest density. It is the value of the how to get more money on airport city ipad point of the array not midpoint stock market abbreviation otn rangesuch that half the item are above and half below it.

It is the value in exchange rate graph pound euro history given aggregate which would obtain if all the values were equal. The value of the media is fixed by its position in the array and doesn't reflect the individual value. The sum of deviations on either side of the mean are equal; hence, the algebraic sum of the deviation is equal zero. The aggregate distance between the median point and all the value in the array is less than from any other point.

It reflect the magnitude of every value. On the other hand, there is no mode in a rectangular distribution. Each array has one and only one median. An array has one and only one mean. It cannot be manipulated algebraically: Means may be manipulated algebraically: It is stable in that grouping procedures do not affect it appreciably. It may be calculated even when individual values are unknown, provided the sum of the values and the sample size n are known.

Value must be ordered, and may be grouped, for computation. Values need not be ordered or grouped for this calculation. It can be compute when ends are open It cannot be calculated from a frequency table when ends are open. It is not applicable to qualitative data. It is stable timely forex trading that grouping procedures do not seriously affected it.

The Descriptive Statistics JavaScript how to make money by adsense a complete set of information about all statistics that you ever need.

You might like to use it to foreign currency etf basket some numerical experimentation for validating the above assertions for a deeper understanding. The geometric mean G of n non-negative numerical values is the n th root of the product of the n values. If some values are very large in magnitude and others are small, then the geometric mean is a stock market probability using statistics to predict and optimize investment outcomes pdf representative of the data than the simple average.

In a"geometric series", the most meaningful average is the geometric mean G. The arithmetic mean is very biased toward the larger numbers in the series. For simplicity, assume you sold items initially. The harmonic mean H is another specialized average, which is useful in averaging variables expressed as rate per unit of time, such as mileage per hour, number of units produced per day.

The harmonic mean H of n non-zero numerical values x i is: Suppose 4 machines in a machine shop are used to produce the same part. However, each of the four machines takes 2. What is the average rate of speed? The harmonic means is: If all machines working for one hour, how many parts will be produced? Since four machines running for one hour represent minutes of operating time, then: The Order Among the Three Means: If all the three means exist, then the Arithmetic Mean is never less than the other two, moreover, the Harmonic Mean is never larger than the other two.

You might like to use The Other Means JavaScript in performing some numerical experimentation for validating the above assertions for a deeper understanding. Checking for Homogeneity of Population A histogram is a graphical presentation of an estimate for the density for continuous random variables or probability mass function for discrete random variables of the population. The geometric feature of histogram enables us to find 777 one touch binary options brokers useful information about the data, such as: The location of the"center" of the data.

The degree of dispersion.

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The extend to which its is skewed, that is, it does not fall off systemically on both side of its peak. The degree of peakedness. How steeply it rises and falls. Whenever, more than one mode exist, then the population from which the sample came is a mixture of more than one population.

Almost all standard statistical analyses are conditioned on the assumption that the liveops work from home review is homogeneous, meaning that its density for continuous random variables or probability mass function for discrete random variables is unimodal.

To check the unimodality of sampling data, one may use the histogramming process. Number of Class Intervals in a Histogram: Before we can construct our frequency distribution we must determine how many classes we should use. This is purely arbitrary, but too few classes or too many classes investing in the vietnamese stock market not provide as clear a picture as can be obtained with some more nearly optimum number.

Therefore, class width is: Test for Homogeneity of a Population. To have an"optimum" you need some measure of vega neutral option strategy -- presumably in this case, the"best" way to display whatever information is available in the data.

The sample size contributes to this; so the usual guidelines are to use between 5 and 15 classes, with more classes, if you have a larger sample. You should take into account a preference for tidy class widths, preferably a multiple of 5 or 10, because this makes it easier to understand.

Beyond this it becomes a matter of judgement. Try out a range of class widths, and choose the one that works best. This assumes you have a computer and can generate alternative histograms fairly readily. There are often management issues that come into play as well.

For example, if your data is to be compared to similar data -- such as prior studies, or from other countries -- you are restricted to the intervals used therein. If the histogram is very skewed, then unequal classes should be considered. Use narrow classes where the class frequencies are high, wide classes where they are low. The following approaches are common: The Log is the logarithm in base Thus for observations you would use 14 intervals but for you would use AlternativelyFind the range highest value - lowest value.

Divide the range by a reasonable interval size: Aim for no fewer than 5 intervals and no more than One of the main applications of histogramming is to Test for Homogeneity of a Population.

The unimodality of the histogram is forex trading notes necessary condition for the homogeneity of population to make any statistical analysis meaningful. Contains a earn money doing java projects test for multimodality that is based on the Gaussian kernel density estimates and then test for multimodality by using the window-size approach.

How to Construct a BoxPlot A BoxPlot is a graphical display that has many characteristics. It includes the presence of possible outliers. It illustrates the range of data. It shows a measure of dispersion such as the upper quartile, lower quartile and interquartile range IQR of the data set as well as the median as a measure of central location, which is useful for comparing sets of data.

It also gives an indication of the symmetry or skewness of the distribution. The main reason for the popularity of boxplots is that they offer much of information in a compact way.

Steps to Construct a BoxPlot: Horizontal lines are drawn at the smallest observation Alower quartile. And another from the upper quartile Dand the largest observation E.

Vertical lines to produce the box join these horizontal lines at points B, and D. For a deeper understanding, you may forex trading company reviews using graph paperand Descriptive Sampling Statistics JavaScript in constructing the BoxPlots for some sets of data; e.

Measuring the Quality of a Sample Average by itself is not a good indication of quality. You need to know the variance to make any educated assessment. We are reminded of the dilemma of the six-foot tall statistician who drowned in a stream that had an average depth of three feet. Statistical measures are often used for describing the nature and extent of differences among the who makes more money phlebotomist or medical assistant in the distribution.

Canadian stock market quote measure of variability is generally reported together with a measure of central tendency.

Statistical measures of variation are numerical values that indicate the variability inherent in a set of data measurements. Note that a small value for a measure of dispersion indicates that the data are concentrated around the mean; therefore, the mean is a good representative of the data set.

On the other hand, a large measure of dispersion indicates that the mean is not a good representative of the data set. Also, measures of dispersion can be used when forex bank sweden opening hours want to compare the distributions of two or more sets of data. Quality of a data set is measured by its variability: Larger variability indicates lower quality.

That is why high variation makes the manager very worried. Your job, as a statistician, is to measure the variationand if it is too high and unacceptable, then it is the job of the technical staff, such as engineers, to fix the process. Decision situations with complete lack of knowledge, known as the flat uncertaintyhave the largest risk.

For simplicity, consider the case when there are only two outcomes, one with probability of p. Then, the variation in the outcomes fractals indicator forex p 1-p. That is, schwab option trade chance for each outcome.

In such a case, the quality of information is at its lowest level. Remember, quality of information and variation are inversely related. The larger the variation in the data, the lower the quality of the data i.

The four most common measures of variation private forex trading club the rangevariancestandard deviationand coefficient of variation.

The range of a set of observations is the absolute value of the difference between the largest and smallest values in the data set. It measures the size of the smallest contiguous interval of real numbers that encompasses all of the data values.

It is not useful when extreme values are present. It is based solely on two values, not on the entire data set. In addition, it cannot be defined for open-ended distributions such as Normal distribution. Notice that, when dealing with discrete random observationssome authors define the range as: A normal distribution does not have a range. A student said,"since the tails of a normal density function never touch the x-axis and since for an observation to contribute to forming such a curve, very large positive and negative values must exist" Yet such remote values are always possible, but increasingly improbable.

This encapsulates the asymptotic behavior of normal density very well. Therefore, in spite of this behavior, it is useful and applicable to a wide range of decision-making situations. Percentiles have a similar concept and therefore, are related; e. The advantage of percentiles is that they may be subdivided into parts. The percentiles and quartiles are most conveniently read from a cumulative distribution function, as depicted in the following figure.

Empirical Cumulative Distribution Function as an Informative Tool Interquartiles Range: It is the distance between the first and the third quartiles: For data that are skewedthe relative dispersionsimilar to the coefficient of variation C.

Note that almost all statistics that we have covered up to now can be obtained and understood deeply by graphical method using Empirical i. However, the numerical Descriptive Statistics provides a complete set of information about all statistics that you ever need. The Duality between the ECDF and the Histogram: Notice that the empirical i. Therefore, either or both could be used depending on the intended applications. Mean Absolute Deviation MAD: A simple measure of variability is the mean absolute deviation: The mean absolute deviation is widely used as a performance measure to assess the quality of the modeling, such forecasting techniques.

However, MAD does not lend itself to further use in making inference; moreover, even in the error analysis studies, the variance is preferred since variances of independent i. The MAD is a simple measure of variability, which unlike range and quartile deviation, takes every item into account, and it is simpler and less affected by extreme deviations. It is therefore often used in small samples that include extreme values. The mean absolute deviation theoretically should be measured from the median, since it is at its minimum; however, it is more convenient to measure the deviations from the mean.

An important measure of variability is variance. Variance is the average of the squared deviations of each observation in the set from the arithmetic mean of all of the observations. The variance is a measure of spread or dispersion among values in a data set. Therefore, the greater the variance, the lower the quality. The variance is not expressed in the same units as the observations.

In other words, the variance is hard to understand because the deviations from the mean are squared, making it too large for logical explanation. This problem can be solved by working with the square root of the variance, which is called the standard deviation. Both variance and standard deviation provide the same information; one can always be obtained from the other. In other words, the process of computing a standard deviation always involves computing a variance.

Since standard deviation is the square root of the variance, it is always expressed in the same units as the raw data: You may use Descriptive Statistics JavaScript to compute the mean, and standard deviation. The Mean Square Error MSE of an estimate is the variance of the estimate plus the square of its bias; therefore, if an estimate is unbiased, then its MSE is equal to its variance, as it is the case in the ANOVA table.

Coefficient of Variation CV is the absolute relative deviation with respect to sizeprovided is not zero, expressed in percentage: The coefficient of variation is used to represent the relationship of the standard deviation to the mean, telling how representative the mean is of the numbers from which it came.

It expresses the standard deviation as a percentage of the mean; i. However, confidence intervals for the coefficient of variation are rarely reported. One of the reasons is that the exact confidence interval for the coefficient of variation is computationally tedious. Note that, for a skewed or grouped data set, the coefficient of quartile variation: You may use Descriptive Statistics to compute the mean, standard deviation and the coefficient of variation.

Variation Ratio for Qualitative Data: Since the mode is the most frequently used measure of central tendency for qualitative variables, variability is measured with reference to the mode. The statistic that describes the variability of quantitative data is the Variation Ratio VR: In other words, a Z score represents the number of standard deviations that an observation x is above or below the mean. The larger the Z value, the further away a value will be from the mean.

Note that values beyond three standard deviations are very unlikely. Note that if a Z score is negative, the observation x is below the mean. If the Z score is positive, the observation x is above the mean. The Z score is found as: Since the standard deviation is never negative, a positive Z score indicates that the observation is above the mean, a negative Z score indicates that the observation is below the mean. Note that Z is a dimensionless value, and therefore is a useful measure by which to compare data values from two different populations, even those measured by different units.

However, the shape of the distribution will not be affected by the transformation. If X is not normal, then the transformed distribution will not be normal either. One of the nice features of the z-transformation is that the resulting distribution of the transformed data has an identical shape but with mean zero, and standard deviation equal to 1. One can generalize this data transformation to have any desirable mean and standard deviation other than 0 and 1, respectively.

Suppose we wish the transformed data to have the mean and standard deviation of M and D, respectively. The following transformation should be applied: If you wish to compare these two data sets, due to differences in scales, the statistics that you generate are not comparable.

It is a good idea to use the Z-transformation of both original data sets and then make any comparison. You have heard the terms z value, z test, z transformation, and z score. Do all of these terms mean the same thing? The z value refers to the critical value a point on the horizontal axes of the Normal 0, 1 density function, for a given area to the left of that z-value. The z test refers to the procedures for testing the equality of mean s of one or two population s.

The z score of a given observation x, in a sample of size n, is simply x - average of the sample divided by the standard deviation of the sample. One must be careful not to mistake z scores for the Standard Scores. The z transformation of a set of observations of size n is simply each observation - average of all observations divided by the standard deviation among all observations. The aim is to produce a transformed data set with a mean of zero and a standard deviation of one.

This makes the transformed set dimensionless and manageable with respect to its magnitudes. It is used also in comparing several data sets that have been measured using different scales of measurements. Pearson coined the term"standard deviation" sometime near The idea of using squared deviations goes back to Laplace in the early 's. Finally, notice again, that the transforming raw scores to z scores do NOT normalize the data.

Computation of Descriptive Statistics for Grouped Data: One of the most common ways to describe a single variable is with a frequency distribution. A histogram is a graphical presentation of an estimate for the frequency distribution of the population.

Depending upon the particular variable, all of the data values may be represented, or you may group the values into categories first e. It would usually not be sensible to determine the frequencies for each value. Rather, the values are grouped into ranges, and the frequency is then determined. Frequency distributions can be depicted in two ways: The bar chart is often used to show the relationship between two categorical variables.

Grouped data is derived from raw data, and it consists of frequencies counts of raw values tabulated with the classes in which they occur. The Class Limits represent the largest Upper and lowest Lower values which the class will contain. The formulas for the descriptive statistic becomes much simpler for the grouped data, as shown below for Mean, Variance, Standard Deviation, respectively, where f is for the frequency of each class, and n is the total frequency: Selecting Among the Quartile Deviation, Mean Absolute Deviation, and Standard Deviation A general guideline for selecting a suitable statistic in describing the dispersion in a population includes consideration of the following factors: The concept of dispersion required by the problem.

Is a single pair of values adequate, such as the two extremes or the two quartiles range or Q? The type of data available.

If they are few in numbers, or contain extreme value, avoid the standard deviation. If they are generally skewed, avoid the mean absolute deviation as well. If they have a gap around the quartile, the quartile deviation should be avoided. The peculiarity of the dispersion measures themselves.

These are summarized under"The Main Characteristics of the Quartile Deviation, the Mean Absolute Deviation, and the Standard deviation" below. The Main Characteristics of the Quartile Deviation, the Mean Absolute Deviation, and the Standard Deviation Fact No. The Quartile Deviation The Mean Absolute Deviation The Standard Deviation 1 The quartile deviation is also easy to calculate and to understand. However, it is unreliable if there are gaps in the data around the quartiles.

The mean absolute deviation has the advantage of giving equal weight to the deviation of every value form the mean or median.

The standard deviation is usually more useful and better adapted to further analysis than the mean absolute deviation. Therefore, it is a more sensitive measure of dispersion than those described above and ordinarily has a smaller sampling error.

It is more reliable as an estimator of the population dispersion than other measures, provided the distribution is normal. It is also easier to compute and to understand and is less affected by extreme values than the standard deviation. It is the most widely used measure of dispersion and the easiest to handle algebraically.

Unfortunately, it is difficult to handle algebraically, since minus signs must be ignored in its computation. Compared with the others, it is harder to compute and more difficult to understand. Its main application is in modeling accuracy for comparative forecasting techniques. It is generally affected by extreme values that may be due to skewness of data You might like to use the Descriptive Sampling Statistics JavaScript in performing some numerical experimentation for validating the above assertions for a deeper understanding.

Shape of a Distribution Function: The Skewness-Kurtosis Chart The pair of statistical measures, skewness and kurtosis, are measuring tools, which is used in selecting a distribution s to fit your data.

To make an inference with respect to the population distribution, you may first compute skewness and kurtosis from your random sample from the entire population. Then, locating a point with these coordinates on the widely used skewness-kurtosis chartguess a couple of possible distributions to fit your data. Finally, you might use the goodness-of-fit test to rigorously come up with the best candidate fitting your data. Removing outliers improves the accuracy of both skewness and kurtosis. Skewness is a measure of the degree to which the sample population deviates from symmetry with the mean at the center.

Skewness will take on a value of zero when the distribution is a symmetrical curve. A positive value indicates the observations are clustered more to the left of the mean with most of the extreme values to the right of the mean. A negative skewness indicates clustering to the right. In this case we have: The reverse order holds for the observations with positive skewness. Kurtosis is a measure of the relative peakedness of the curve defined by the distribution of the observations.

A kurtosis larger than 3 indicates the distribution is more peaked than the standard normal distribution. A value of less than 3 for kurtosis indicates that the distribution is flatter than the standard normal distribution. These inequalities hold for any probability distribution having finite skewness and kurtosis. In the Skewness-Kurtosis Chartyou notice two useful families of distributions, namely the beta and gamma families. The Beta-Type Density Function: Since the beta density has both a shape and a scale parameter, it describes many random phenomena provided the random variable is between [0, 1].

For example, when both parameters are integer with random variables the result is the binomial Probability function. A basic distribution of statistics for variables bounded at both sides; for example x between [0, 1]. The beta density is useful for both theoretical and applied problems in many areas. Examples include distribution of proportion of population located between lowest and highest value in sample; distribution of daily per cent yield in a manufacturing process; description of elapsed times to task completion PERT.

There is also a relationship between the Beta and Normal distributions. Uniform, right triangular, and parabolic distributions are special cases. To generate beta, generate two random values from a gamma, g 1g 2. Some random variables are always non-negative. The density function associated with these random variables often is adequately modeled as the gamma density function.

The Gamma-Type Density Function has both a shape and a scale parameter. With both the shape and scale parameters equal to 1, the result is the exponential density function.

Chi-square is also a special case of gamma density function with shape parameter equal to 2. A basic distribution of statistics for variables bounded at one side ; for example x greater than or equal to zero. The gamma density gives distribution of time required for exactly k independent events to occur, assuming events take place at a constant rate. Used frequently in queuing theory, reliability, and other industrial applications.

Examples include distribution of time between re-calibrations of instrument that needs re-calibration after k uses; time between inventory restocking, time to failure for a system with standby components.

Erlangian, Exponential, and Chi-square distributions are special cases. The negative binomial is an analog to gamma distribution with discrete random variable. What is the distribution of the product of sample observations from the uniform 0, 1 random? Like many problems with products, this becomes a familiar problem when turned into a problem about sums.

Xn is the sum of Y1, Y2, Yn which has a gamma scaled Chi-square distribution. Thus, it is a gamma density with shape parameter n and scale 1. The Log-normal Density Function: Permits representation of a random variable whose logarithm follows a normal distribution.

The ratio of two log-normally random variables is also log-normal. Model for a process arising from many small multiplicative errors. Appropriate when the value of an observed variable is a random proportion of the previously observed value. Examples include distribution of sizes from a breakage process; distribution of income size, inheritances and bank deposits; distribution of various biological phenomena; life distribution of some transistor types.

The lognormal distribution is widely used in situations where values are positively skewed where the distribution has a long right tail; negatively skewed distributions have a long left tail; a normal distribution has no skewness.

Examples of data that"fit" a lognormal distribution include financial security valuations or real estate property valuations. Financial analysts have observed that the stock prices are usually positively skewed, rather than normally symmetrically distributed.

Stock prices exhibit this trend because the stock price cannot fall below the lower limit of zero but may increase to any price without limit. Similarly, healthcare costs illustrate positive skewness since unit costs cannot be negative. For example, there can't be negative cost for services in a capitation contract.

This distribution accurately describes most healthcare data. In the case where the data are log-normally distributed, the Geometric Mean acts as a better data descriptor than the mean. The more closely the data follow a log-normal distribution, the closer the geometric mean is to the median, since the log re-expression produces a symmetrical distribution.

Fidell, Using Multivariate StatisticsHarperCollins, Has a good discussion on applications and significance tests for skewness and kurtosis. Numerical Example and Discussions A Numerical Example: You might like to use Descriptive Statistics to check your hand computation.

A Short Discussion on the Descriptive Statistic: Deviations about the mean m of a distribution is the basis for most of the statistical tests we will learn.

Since we are measuring how much a set of scores is dispersed about the mean mwe are measuring variability. We can calculate the deviations about the mean m and express it as variance s 2 or standard deviation s. It is very important to have a firm grasp of this concept because it will be a central concept throughout your statistics course.

Both variance s 2 and standard deviation s measure variability within a distribution. Standard deviation s is a number that indicates how much on average each of the values in the distribution deviates from the mean m or center of the distribution. Keep in mind that variance s 2 measures the same thing as standard deviation s dispersion of scores in a distribution. Variance s 2however, is the average squared deviations about the mean m.

Thus, variance s 2 is the square of the standard deviation s. They are Unbiased you may update your estimate ; Efficient they have the smallest variation among other estimators ; Consistent increasing sample size provides a better estimate ; and Sufficient you do not need to have the whole data set; what you need are S x i and S x i 2 for estimations.

Note also that the above variance S 2 is justified only in the case where the population distribution tends to be normal, otherwise one may use bootstrapping techniques. In general, it is believed that the pattern of mode, median, and mean go from lower to higher in positive skewed data sets, and just the opposite pattern in negative skewed data sets.

Note also, that most commercial software do not correctly compute skewness and kurtosis. There is no easy way to determine confidence intervals about a computed skewness or kurtosis value from a small to medium sample. The literature gives tables based on asymptotic methods for sample sets larger than for normal distributions only. You may have noticed that using the above numerical example on some computer packages such as SPSS, the skewness and the kurtosis are different from what we have computed.

For example, the SPSS output for the skewness is 1. However, for large a sample size n, the results are identical. Reference and Further Readings: This article provides a good historical account of statistical measures. Exact confidence interval for the coefficient of variation is computationally tedious as shown in this book. The Two Statistical Representations of a Population The following figure depicts a typical relationship between the cumulative distribution function cdf and the density for continuous random variablesAll characteristics of the population are well described by either of these two functions.

The figure also illustrates their applications in determining the lower percentile measures denoted by P: Notice that the probability P is the area under the density function curve, while numerically equal to the height of cdf curve at point x. Both functions can be estimated by smoothing the empirical i.

The ogive is the estimator for the population's cumulative distribution function, which contains all the characteristic of the population. The empirical distribution is a staircase function with the location of the drops randomly placed. The sample size is the sum of all frequencies.

Note that almost all statistics we have covered up to now can be obtained and understood more deeply by graph paper using Empirical Distribution Function JavaScript. You may like using this JavaScript in performing some numerical experimentation for a deeper understanding. Other widely used decision model based upon empirical cumulative distribution function ECDF as a measuring tool and decision procedure are the ABC Inventory ClassificationSingle-period Inventory Analysis The Newsboy Modeland determination of the Best Time to Replace Equipment.

For other inventory decisions, visit the Inventory Control Models site. Introduction Modeling of a Data Set: Families of parametric distribution models are widely used to summarize a huge data setto obtain predictions, assess goodness of fit, to estimate functions of the data not easily derived directly, or to render manageable random effects.

The trustworthiness of the results obtained depends on the generality of the distribution family employed. This extension of our knowledge from a particular random sample to the population is called inductive inference. The main function of business statistics is the provision of techniques for making inductive inference and for measuring the degree of uncertainty of such inference.

Uncertainty is measured in terms of probability statements, and that is the reason we need to learn the language of uncertainty and its measuring tool called probability. In contrast to the inductive inference, mathematics often uses deductive inference to prove theorems, while in empirical science, such as statistics, inductive inference is used to find new knowledge or to extend our knowledge.

Levy, The log F: A distribution for all seasons, Computational Statistics17 1, Probability, Chance, Likelihood, and Odds The concept of probability occupies an important place in the decision-making process under uncertainty, whether the problem is one faced in business, in government, in the social sciences, or just in one's own everyday personal life.

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In very few decision-making situations is perfect information -- all the needed facts -- available. Most decisions are made in the face of uncertainty. Probability enters into the process by playing the role of a substitute for certainty - a substitute for complete knowledge. Probability is especially significant in the area of statistical inference.

Here the statistician's prime concern lies in drawing conclusions or making inferences from experiments which involve uncertainties. The concepts of probability make it possible for the statistician to generalize from the known sample to the unknown population and to place a high degree of confidence in these generalizations. Therefore, Probability is one of the most important tools of statistical inference.

Probability has an exact technical meaning -- well, in fact it has several, and there is still debate as to which term ought to be used. However, for most events for which probability is easily computed; e. A probability is always a number between 0 and 1. Zero is not"quite" the same thing as impossibility. It is possible that"if" a coin were flipped infinitely many times, it would never show"tails", but the probability of an infinite run of heads is 0.

One is not"quite" the same thing as certainty but close enough. The word"chance" or"chances" is often used as an approximate synonym of "probability", either for variety or to save syllables. It would be better practice to leave"chance" for informal use, and say"probability" if that is what is meant. One occasionally sees"likely" and"likelihood"; however, these terms are used casually as synonyms for"probable" and"probability". Odds is a probabilistic concept related to probability.

It is the ratio of the probability p of an event to the probability 1-p that it does not happen: It is often expressed as a ratio, often of whole numbers; e. Odds are a ratio of nonevents to events. If the event rate for a disease is 0. Another way to compare probabilities and odds is using"part-whole thinking" with a binary dichotomous split in a group.

A probability is often a ratio of a part to a whole; e. Odds are often a ratio of a part to a part; e. Aside from their value in betting, odds allow one to specify a small probability near zero or a large probability near one using large whole numbers 1, to 1 or a million to one.

Odds magnify small probabilities or large probabilities so as to make the relative differences visible. They are both small. An untrained observer might not realize that one is twice as much as the other. But if expressed as odds 99 to 1 versus to 1 it may be easier to compare the two situations by focusing on large whole numbers versus 99 rather than on small ratios or fractions. How to Assign Probabilities?

Probability is an instrument to measure the likelihood of the occurrence of an event. There are five major approaches of assigning probability: Classical Approach, Relative Frequency Approach, Subjective Approach, Anchoring, and the Delphi Technique: Classical probability is predicated on the condition that the outcomes of an experiment are equally likely to happen.

The classical probability utilizes the idea that the lack of knowledge implies that all possibilities are equally likely. The classical probability is applied when the events have the same chance of occurring called equally likely eventsand the sets of events are mutually exclusive and collectively exhaustive.

The classical probability is defined as: Relative probability is based on accumulated historical or experimental data. Frequency-based probability is defined as: Note that relative probability is based on the ideas that what has happened in the past will hold. The subjective probability is based on personal judgment and experience. For example, medical doctors sometimes assign subjective probability to the length of life expectancy for a person who has cancer.

Delphi Analysis is used in decision making processes, in particular in forecasting. Several"experts" sit together and try to compromise on something upon which they cannot agree. General Computational Probability Rules Addition: When two or more events will happen at the same time, and the events are not mutually exclusive, then: Although this is very simple, it says relatively little about how event X influences event Y and vice versa.

If P X and Y is 0, indicating that events X and Y do not intersect i. On the other hand if P X and Y is not 0, then there are interactions between the two events X and Y. Usually it could be a physical interaction between them. The above rule is known also as the Inclusion-Exclusion Formula. It can be extended to more than two events. For example, for three events A, B, and C, it becomes: When two or more events will happen at the same time, and the events are mutually exclusive, then: Conditional probabilities are based on knowledge of one of the variables.

The conditional probability of an event, such as X, occurring given that another event, such as Y, has occurred is expressed as: Note that when using the conditional rule of probability, you always divide the joint probability by the probability of the event after the word given.

Thus, to get P X given Yyou divide the joint probability of X and Y by the unconditional probability of Y. In other words, the above equation is used to find the conditional probability for any two dependent events.

The simplest version of the Bayes' Theorem is: Suppose two machines, A and B, produce identical parts. Machine A has probability 0. Each machine produces one part. One of these parts is selected at random, tested, and found to be defective. What is the probability that it was produced by Machine B? Probability tree diagrams depict events or sequences of events as branches of a tree. Tree diagrams are useful for visualizing the conditional probabilities: The probabilities at the end of each branch are the probability that events leading to that end will happen simultaneously.

Now using the Bayes' Rule we are able to obtain useful information such as: Equivalently, using the above conditional probability, results in: A diagram used, in general to represent sets and subsets. It is a way of displaying how different sets of objects overlap. John Venn an English mathematician devised them. Venn diagram could be used as a computational probability tool similar to the probability tree diagram.

The following are Venn diagrams representation for two of the above Probability Rules: The Venn diagram model for this problem is depicted below: The solution is readily available from the above Venn diagram model, i. Exercise Your Knowledge on the following probabilistic problem: An urn contains 4 red-balls representing, say defective items and 8 white-balls Representing, say non-defective itemsas depicted below: An Urn Model Suppose 2 balls are drawn at random.

Use the following tree diagram, which is a probabilistic model for this experiment, and verify the solution to the following questions, with the answer given in the bracket at the end of each question: A Tree Diagram as a Probabilistic Model What is the probability of having at least 1 white ball? What is the expected number of white balls?

A coin fair is flipped twice, what is the conditional probability that both flips land on heads, given: The first flip lands on heads b. At least one of the flips lands on head. Are the answers to part a and b identical?

You may like using the Bayes' Revised Probability JavaScript. How to Count Without Counting Many disciplines and sciences require the answer to the question: In finite probability theory we need to know how many outcomes there would be for a particular event, and we need to know the total number of outcomes in the sample space.

Combinatoricsalso referred to as Combinatorial Mathematicsis the field of mathematics concerned with problems of selection, arrangement, and operation within a finite or discrete system. How to count without counting. Therefore, One of the basic problems of combinatorics is to determine the number of possible configurations of objects of a given type. You may ask, why combinatorics? If a sample spaces contains a finite set of outcomes, determining the probability of an event often is a counting problem.

But often the numbers are just too large to count in the 1, 2, 3, 4 ordinary ways. This simple rule can be generalized as follow: A quality control inspector wishes to select one part for inspection from each of four different bins containing 4, 3, 5 and 4 parts respectively.

Notice that by convention, 0! A permutation is an arrangement of objects from a set of objects. That is, the objects are chosen from a particular set and listed in a particular order. A combination is a selection of objects from a set of objects, that is objects are chosen from a particular set and listed, but the order in which the objects are listed is immaterial.

In this case it is easy to make a list: We might observe that there are 3 choices for the first letter, 2 choices for the second letter and 1 choice for the third letter. Generalizing, if we have n distinct objects, we would have n choices for the first position, n-1 choices for the second position and so on.

We find that the permutation of n objects selected among n distinct objects is n! The number of ways of lining up k objects at a time from n distinct objects is denoted by n P kand by the preceding we have: There are many problems in which we are interested in determining the number of ways in which k objects can be selected from n distinct objects without regard to the order in which they are selected.

Such selections are called combinations or k-sets. It may help to think of combinations as a committee. The key here is without regard for order.

The number of combinations of k objects from a set with n objects is n C k. The general formula is: This is basically a subset problem where you specify the number of elements in the subset. You may ask, what is the relation of combinations to permutations? Each of the above subsets forms 3! One of the fundamental aspects of economic activity is a trade in which one party provides another party something, in return for which the second party provides the first something else, i.

The invention of money during 16 th Century in Europe was a necessary tool of trading. The usage of money greatly simplifies barter system of trading, thus lowering transactions costs. If a society produces different goods, there are: With money, only prices are needed to establish all possible trading ratios.

As another applicationconsider the following probabilistic problem. Suppose there are at most 10 defective items in a batch of size You have shipped 15 items to one of your customers. What is the chance that the customer would find at least one defective item?

How many different letter arrangements can be formed using the letters P E P P E R? In general, there are multinomial coefficients: Therefore, the answer is 6! You may like using the Combinatorial Math JavaScript. Joint Probability and Statistics A joint probability distribution of a group of random variables is the distribution of group of variables as a whole. Applied business statistics deal mostly with the joint probability distribution of two discrete random variables.

The joint probability distribution of two discrete random variables is the likelihood of observing all combinations of the two variables. As an example, consider two competitive stocks A, and B. Suppose the estimated rates of return of stocks A and B are given as follow respectively: Find the marginal density of R A and R B from the Joint Probability table. To calculate the marginal distribution of R Bsimply look at the table and add the probabilities in each column.

To obtain the marginal distribution of R Aadd the probabilities in each row. The marginal distributions of A and B are shown at the rigt and the bottom margins of the below table, respectively: However, the converse is not true; a given marginal distribution can come from many different joint distributions.

The function that links the marginal densities and the joint density is called the copula. In practice, one picks the marginal distributions first and then selects an appropriate copula to achieve the right amount of dependency among the individual random variables. Take X and Y as above, then the function: We need to calculate: For estimation of the expected values, variances, etc, you may using the Bivariate Distributions JavaScript.

Mutually Exclusive versus Independent Events Mutually Exclusive ME: Event A and B are ME if both cannot occur simultaneously. Events A and B are independent if having the information that B already occurred does not change the probability that A will occur.

If two events are ME they are also Dependent: Similarly, If two events are Independent then they are also not ME. If two events are Dependent then they may or may not be ME. If two events are not ME, then they may or may not be Independent. The following Figure contains all possibilities. The notations used in this table are as follows: