- "We see that the theory of probability is at bottom only common sense reduced to calculation: it makes us appreciate with exactitude what reasonable minds feel by a sort of instinct, often without being able to account for it... . It is remarkable that this science, which originated in the consideration of games of chance, should have become the most important object of human knowledge. ... The most important questions of life are. for the most part, really only problems of probability.” So said the famous French mathematician and astronomer (the “Newton of France”) Pierre Simon, Marquis de Laplace. Although many people might feel that the famous marquis, who was also one of the great contributors to the development of probability, might have exaggerated somewhat, it is nevertheless true that probability theory has become a tool of fundamental importance to nearly all scientists, engineers, medical practitioners, jurists, and industrialists. In fact, the enlightened individual had learned to ask not “Is it so?” but rather “What is the probability that it is so?”
- This book is intended as an elementary introduction to the mathematical theory of probability for students in mathematics, engineering, and the sciences (including the social sciences and management science) who possess the prerequisite knowledge of elementary calculus. It attempts to present not only the mathematics of probability theory, but also, through numerous examples, the many diverse possible applications of this subject.
- In Chapter 1 we present the basic principles of combinatorial analysis, which are most useful in computing probabilities.
- In Chapter 2 we consider the axioms of probability theory and show how they can be applied to compute various probabilities of interest. This chapter includes a proof of the important (and. unfortunately, often neglected) continuity property of probabilities, which is then used in the study of a “logical paradox.”
- Chapter 3 deals with the extremely important subjects of conditional probability and independence of events. By a series of examples we illustrate how conditional probabilities come into play not only when some partial information is available, but also as a tool to enable us to compute probabilities more easily, even when no partial information is present. This extremely important technique of obtaining probabilities by “conditioning” reappears in Chapter 7, where we use it to obtain expectations.
- In Chapters 4, 5 and 6 we introduce the concept of random variables. Discrete random variables are dealt with in Chapter 4, continuous random variables in Chapter 5, and jointly distributed random variables in Chapter 6. The important concepts of the expected value and the variance of a random variable are introduced in Chapters 4 and 5. These quantities are then determined for many of the common types of random variables.
- Additional properties of the expected value are considered in Chapter 7. Many examples illustrating the usefulness of the result that the expected value of a sum of random variables is equal to the sum of their expected values are presented. Sections on conditional expectation, including its use in prediction, and moment generating functions are contained in this chapter. In addition, the final section introduces the multivariate normal distribution and presents a simple proof concerning the joint distribution of the sample mean and sample variance of a sample from a normal distribution.
- In Chapter 8 we present the major theoretical results of probability theory. In particular, we prove the strong law of large numbers and the central limit theorem. Our proof of the strong law is a relatively simple one which assumes that the random variables have a finite fourth moment, and our proof of the central limit theorem assumes Levy's continuity theorem. Also in this chapter we present such probability inequalities as Markov’s inequality, Chebyshev’s inequality, and Chernoff bounds. The final section of Chapter 8 gives a bound on the error involved when a probability concerning a sum of independent Bernoulli random variables is approximated by the corresponding probability for a Poisson random variable having the same expected value.
- Chapter 9 presents some additional topics, such as Markov chains, the Poisson process, and an introduction to information and coding theory, and Chapter 10 considers simulation.
Prentice-Hall International, 1998. A gentle & Interesting Introduction
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