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1.2 Interpretations of Probability This section describes three common operational interpretations of probability. Although the interpretations may seem incompatible, it is fortunate that the calculus of probability (the subject matter of the first six chapters of this book) applies equally well no matter which interpretation one prefers.

1.3 Experiments and Events Probability will be the way that we quantify how likely something is to occur (in the sense of one of the interpretations in Sec. 1.2). In this section, we give examples of the types of situations in which probability will be used.

1.4 Set Theory This section develops the formal mathematical model for events, namely, the theory of sets. Several important concepts are introduced, namely, element, subset, empty set, intersection, union, complement, and disjoint sets

1.5 The Definition of Probability We begin with the mathematical definition of probability and then present some useful results that follow easily from the definition.

1.6 Finite Sample Spaces The simplest experiments in which to determine and derive probabilities are those that involve only finitely many possible outcomes. This section gives several examples to illustrate the important concepts from Sec. 1.5 in finite sample spaces.

1.7 Counting Methods In simple sample spaces, one way to calculate the probability of an event involves counting the number of outcomes in the event and the number of outcomes in the sample space. This section presents some common methods for counting the number of outcomes in a set. These methods rely on special structure that exists in many common experiments, namely, that each outcome consists of several parts and that it is relatively easy to count how many possibilities there are for each of the parts.

1.8 Combinatorial Methods Many problems of counting the number of outcomes in an event amount to counting how many subsets of a certain size are contained in a fixed set. This section gives examples of how to do such counting and where it can arise.

1.9 Multinomial Coefficients We learn how to count the number of ways to partition a finite set into more than two disjoint subsets. This generalizes the binomial coefficients from Sec. 1.8. The generalization is useful when outcomes consist of several parts selected from a fixed number of distinct types.

1.10 The Probability of a Union of Events The axioms of probability tell us directly how to find the probability of the union of disjoint events. Theorem 1.5.7 showed how to find the probability for the union of two arbitrary events. This theorem is generalized to the union of an arbitrary finite collection of events.

2.1 The Definition of Conditional Probability A major use of probability in statistical inference is the updating of probabilities when certain events are observed. The updated probability of event A after we learn that event B has occurred is the conditional probability of A given B.

2.2 Independent Events If learning that B has occurred does not change the probability of A, then we say that A and B are independent. There are many cases in which events A and B are not independent, but they would be independent if we learned that some other event C had occurred. In this case, A and B are conditionally independent given C.

2.3 Bayes’ Theorem Suppose that we are interested in which of several disjoint events B1,...,Bk will occur and that we will get to observe some other event A. If Pr(A|Bi) is available for each i, then Bayes’ theorem is a useful formula for computing the conditional probabilities of the Bi events given A

 2.4 The Gambler’s Ruin Problem Consider two gamblers with finite resources who repeatedly play the same game against each other. Using the tools of conditional probability, we can calculate the probability that each of the gamblers will eventually lose all of his money to the opponent.

 3.1 Random Variables and Discrete Distributions A random variable is a real-valued function defined on a sample space. Random variables are the main tools used for modeling unknown quantities in statistical analyses. For each random variable X and each set C of real numbers, we could calculate the probability that X takes its value in C. The collection of all of these probabilities is the distribution of X. There are two major classes of distributions and random variables: discrete (this section) and continuous (Sec. 3.2). Discrete distributions are those that assign positive probability to at most countably many different values. A discrete distribution can be characterized by its probability function (p.f.), which specifies the probability that the random variable takes each of the different possible values. A random variable with a discrete distribution will be called a discrete random variable.

3.2 Continuous Distributions Next, we focus on random variables that can assume every value in an interval (bounded or unbounded). If a random variable X has associated with it a function f such that the integral of f over each interval gives the probability that X is in the interval, then we call f the probability density function (p.d.f.) of X and we say that X has a continuous distribution.

3.3 The Cumulative Distribution Function Although a discrete distribution is characterized by its p.f. and a continuous distribution is characterized by its p.d.f., every distribution has a common characterization through its (cumulative) distribution function (c.d.f.). The inverse of the c.d.f. is called the quantile function, and it is useful for indicating where the probability is located in a distribution.

3.4 Bivariate Distributions We generalize the concept of distribution of a random variable to the joint distribution of two random variables. In doing so, we introduce the joint p.f. for two discrete random variables, the joint p.d.f. for two continuous random variables, and the joint c.d.f. for any two random variables. We also introduce a joint hybrid of p.f. and p.d.f. for the case of one discrete random variable and one continuous random variable.

3.5 Marginal Distributions Earlier in this chapter, we introduced distributions for random variables, and in Sec. 3.4 we discussed a generalization to joint distributions of two random variables simultaneously. Often, we start with a joint distribution of two random variables and we then want to find the distribution of just one of them. The distribution of one random variable X computed from a joint distribution is also called the marginal distribution of X. Each random variable will have a marginal c.d.f. as well as a marginal p.d.f. or p.f.We also introduce the concept of independent random variables, which is a natural generalization of independent events.

3.6 Conditional Distributions We generalize the concept of conditional probability to conditional distributions. Recall that distributions are just collections of probabilities of events determined by random variables. Conditional distributions will be the probabilities of events determined by some random variables conditional on events determined by other random variables. The idea is that there will typically be many random variables of interest in an applied problem. After we observe some of those random variables, we want to be able to adjust the probabilities associated with the ones that have not yet been observed. The conditional distribution of one random variable X given another Y will be the distribution that we would use for X after we learn the value of Y .

3.7 Multivariate Distributions In this section, we shall extend the results that were developed in Sections 3.4, 3.5, and 3.6 for two random variables X and Y to an arbitrary finite number n of random variables X1,...,Xn. In general, the joint distribution of more than two random variables is called a multivariate distribution. The theory of statistical inference (the subject of the part of this book beginning with Chapter 7) relies on mathematical models for observable data in which each observation is a random variable. For this reason, multivariate distributions arise naturally in the mathematical models for data. The most commonly used model will be one in which the individual data random variables are conditionally independent given one or two other random variables.

3.8 Functions of a Random Variable Often we find that after we compute the distribution of a random variable X, we really want the distribution of some function of X. For example, if X is the rate at which customers are served in a queue, then 1/X is the average waiting time. If we have the distribution of X, we should be able to determine the distribution of 1/X or of any other function of X. How to do that is the subject of this section.

3.9 Functions of Two or More Random Variables When we observe data consisting of the values of several random variables, we need to summarize the observed values in order to be able to focus on the information in the data. Summarizing consists of constructing one or a few functions of the random variables that capture the bulk of the information. In this section, we describe the techniques needed to determine the distribution of a function of two or more random variables.

 3.10 Markov Chains A popular model for systems that change over time in a random manner is the Markov chain model. A Markov chain is a sequence of random variables, one for each time. At each time, the corresponding random variable gives the state of the system. Also, the conditional distribution of each future state given the past states and the present state depends only on the present state.













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