Probability Models and Applications

The following sections are included:

• Dedication
• Preface
• Contents

Everyone encounters uncertainty. Unpredictable results are said to be “random” or to “happen by chance.” The news media often discuss the “odds” or probability of a certain occurrence: for example, a favorite team winning a crucial game, or mortgage rates declining. Such forecasts may help us in choosing actions. Thus a weather prediction of a “50% chance of rain” may suggest carrying an umbrella. Businesses and governments also take risks in the face of uncertainty. Scientists and engineers encounter phenomena whose results cannot be predicted. Out of the need to find an order or pattern in unpredictable phenomena and to measure risks precisely, mathematical models of uncertainty, called probability models, have been developed. The present book is intended to introduce modern mathematical probability theory and the various methods used to calculate odds or probabilities. Some widely used probability models are described, along with examples of how such models are developed and used in science, technology, business, and everyday life.

The following sections are included:

• Introduction
• A Brief Outline of Set Theory
• Formal Structure: Probabilities
• Probability Calculations Using Venn Diagrams
• Formal Proofs
• Fitting a Probability Model
• Notes and References
• Glossary and Summary
• Exercises
• Theoretical Exercises

In this chapter we discuss probability models whose sample spaces contain only a finite number of outcomes. When every outcome is equally probable (the uniform probability model), the probabilities of events can be determined by counting outcomes. In Section 2 we show how uniform probability models arise in statistical sampling problems. Rules of combinational analysis useful for counting outcomes for probability calculations are developed in Section 3.

In this chapter we introduce the important concept of conditional probability and illustrate how knowledge of conditional probabilities aids both in computing probabilities of complex results and in prediction. The closely related and very useful concept of probabilistic independence is then discussed. Finally, experiments whose outcomes are n-tuples (pairs, for example) resulting from recording the outcomes of n component experiments are introduced, and it is shown how these composite experiments can sometimes be modeled using the concept of probabilistic independence.

The following sections are included:

• Introduction
• Discrete and Continuous Random Variables
• Probability Mass Functions of Discrete Random Variables
• Probability Density Functions
• Cumulative Distribution Functions
• Transformations of a Random Variable
• Notes and References
• Glossary and Summary
• Exercises
• Theoretical Exercises

The following sections are included:

• Introduction
• Measures of Location for a Distribution : The Mean
• Other Measures of Location for a Distribution: Median, Quantile, Mode
• The Expected Value of a Function of a Random Variable
• Measures of Dispersion for a Distribution: The Variance and the Standard Deviation
• Higher Moments
• Some Probability Inequalities
• Notes: Estimation of the Mean and Variance
• Glossary and Summary
• Exercises
• Theoretical Exercises

The following sections are included:

• Introduction
• Mutually Independent Random Variables
• Sums of Independent Random Variables
• The Moment Generating Function
• Notes and References
• Glossary and Summary
• Exercises
• Theoretical Exercises

Certain basic probability distributions are applicable in a wide variety of contexts, and thus arise repeatedly in practice. In this and in the next two chapters, we describe some frequently encountered discrete and continuous distributions.

The following sections are included:

• Introduction
• The Normal Density and Its Properties
• Probability Calculations: The Standard Normal Distribution
• Fitting a Normal Distribution
• The Central Limit Theorem
• The Moment Generating Function and a Proof of the Central Limit Theorem
• Glossary and Summary
• Exercises
• Theoretical Exercises

In Chapter 8 we presented a variety of discrete distributions that often arise in applications. In the present chapter we discuss continuous distributions other than the normal distribution that are frequently used: namely, the exponential, gamma (and chi-squared), Weibull, uniform, beta, lognormal, and Cauchy distributions. We continue the practice, begun in Chapters 8 and 9, of omitting identifying subscripts on density functions and cumulative distribution functions when only one random variable is under discussion.

The following sections are included:

• Introduction
• Determining Bivariate Distributions
• Descriptive Indices of Bivariate Distributions
• Joint Moment Generating Function
• Conditional Distributions
• Notes and References: Multivariate Distributions
• Glossary and Summary
• Exercises
• Theoretical Exercises

The following sections are included:

• The Bivariate Normal Density
• Conditional Distributions
• Joint Moment Generating Function
• Probability Calculations
• Glossary and Summary
• Exercises
• Theoretical Exercises

In Chapter 5 we discussed transformations of a single random variable. In experiments where pairs (X, Y) of random variables are observed, we may be interested in two functions

A probability model for a random phenomenon may arise from a theoretical construct. The validity of this model (and thus of the theory) can then be tested by comparing probabilities based on the model with observed relative frequencies (Chapter 6). In comparing observed relative frequencies to theoretical probabilities, one does not expect perfect agreement to occur because the number of observations will usually not be large enough to force relative frequencies to stabilize. Discrepancies may exist by chance between the theoretical probabilities specified by the model and the observed relative frequencies, even if the model has been chosen correctly. Consequently, procedures are needed that can distinguish between discrepancies expected to occur by chance when the model holds, and discrepancies that indicate that the model is insufficiently accurate. Such procedures are called goodness- of-fit tests in the literature of statistics. One of the most frequently used goodness-of-fit tests, Pearson’s chi-squared goodness-of-fit test, is presented in Section 1…

The following sections are included:

• Appendix A The Gamma Function
• Appendix B Tables
• Bibliography
• Name Index
• Subject Index