An Introduction to Probability and Statistics, 3rd Edition

2RANDOM VARIABLES AND THEIR PROBABILITY DISTRIBUTIONS

2.1 INTRODUCTION

In Chapter 1 we dealt essentially with random experiments which can be described by finite sample spaces. We studied the assignment and computation of probabilities of events. In practice, one observes a function defined on the space of outcomes. Thus, if a coin is tossed n times, one is not interested in knowing which of the 2ⁿ n-tuples in the sample space has occurred. Rather, one would like to know the number of heads in n tosses. In games of chance one is interested in the net gain or loss of a certain player. Actually, in Chapter 1 we were concerned with such functions without defining the term random variable. Here we study the notion of a random variable and examine some of its properties.

In Section 2.2 we define a random variable, while in Section 2.3 we study the notion of probability distribution of a random variable. Section 2.4 deals with some special types of random variables, and Section 2.5 considers functions of a random variable and their induced distributions.

The fundamental difference between a random variable and a real-valued function of a real variable is the associated notion of a probability distribution. Nevertheless our knowledge of advanced calculus or real analysis is the basic tool in the study of random variables and their probability distributions.

2.2 RANDOM VARIABLES

In Chapter 1 we studied properties of a set function P defined on a sample space (Ω, ). Since P is ...