Probability: A Lively Introduction

In performing a chance experiment, one is often not interested in the particular outcome that occurs but in a specific numerical value associated with that outcome. Any function that assigns a real number to each outcome in the sample space of the experiment is called a random variable. Intuitively, a random variable takes on its value by chance. The observed value, or realization, of a random variable is completely determined by the realized outcome of the chance experiment and consequently probabilities can be assigned to the possible values of the random variable.

The first purpose of this chapter is to familiarize you with the concept of a random variable and with characteristics such as the expected value and the variance of a random variable. In addition, we give rules for the expected value and the variance of a sum of random variables, including the square-root rule. These rules are easiest explained and understood in the context of discrete random variables. Such random variables can take on only a finite or countably infinite number of values (the so-called continuous random variables that can take on a continuum of values are treated in the next chapter). The second purpose of the chapter is to introduce you to important discrete random variables such as the binomial, Poisson, hypergeometric, geometric, and negative binomial random variables among others. A comprehensive discussion of these random variables is given, together with appealing examples. Much attention is given to the Poisson distribution, which is the most important discrete distribution. Unlike most other introductory texts, we consider at length the practically useful Poisson heuristic for weakly dependent trials.

Concept of a Random Variable

The concept of a random variable is always difficult for the beginner. Intuitively, a random variable is a function that takes on its value by chance. A random variable is not a variable in the traditional sense of the word and actually it is a little misleading to call it a variable. Formally, a random variable is defined as a real-valued function on the sample space of a chance experiment. A random variable X assigns a numerical value X(ω) to each element ω of the sample space.