Leonid Koralov - Theory of Probability and Random Processes

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Part 1
Probability Theory

Leonid Koralov and Yakov G. Sinai Universitext Theory of Probability and Random Processes Second 10.1007/978-3-540-68829-7_1 Springer-Verlag Berlin Heidelberg 2012

1. Random Variables and Their Distributions

Leonid Koralov 1 and Yakov G. Sinai 2

(1)

Department of Mathematics, University of Maryland, College Park, MD, USA

(2)

Department of Mathematics, Princeton University, New Jersey, Fine Hall, USA

Abstract

The first object encountered in probability theory is the space of elementary outcomes. It is simply a non-empty set, usually denoted by , whose elements are called elementary outcomes. Here are several simple examples.

1.1 Spaces of Elementary Outcomes, -Algebras, and Measures

The first object encountered in probability theory is the space of elementary outcomes. It is simply a non-empty set, usually denoted by , whose elements are called elementary outcomes. Here are several simple examples.

Example

Take a finite set and the set consisting of sequences of length , where for each . In applications, is a result of n statistical experiments, while is the result of the i -th experiment. It is clear that , where denotes the number of elements in the finite set . If , then each is a sequence of length n made of zeros and ones. Such a space can be used to model the result of n consecutive tosses of a coin. If , then can be viewed as the space of outcomes for n rolls of a die.

Example

A generalization of the previous example can be obtained as follows. Let X be a finite or countable set, and I be a finite set. Then is the space of all functions from I to X .

If and is a finite set, then each is a configuration of zeros and ones on a bounded subset of d -dimensional lattice. Such spaces appear in statistical physics, percolation theory, etc.

Example

Consider a lottery game where one tries to guess n distinct numbers and the order in which they will appear out of a pool of r numbers (with ). In order to model this game, define . Let consist of sequences of length n such that for . It is easy to show that .

Later in this section we shall define the notion of a probability measure, or simply probability. It is a function which ascribes real numbers between zero and one to certain (but not necessarily all!) subsets . If is interpreted as the space of possible outcomes of an experiment, then the probability of A may be interpreted as the likelihood that the outcome of the experiment belongs to A . Before we introduce the notion of probability we need to discuss the classes of sets on which it will be defined.

Definition 1.1

A collection of subsets of is called an algebra if it has the following three properties.

.

implies that .

implies that .

Example

Given a set of elementary outcomes , let

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