Introduction to Probability and Stochastic Processes with Applications

CHAPTER 2

RANDOM VARIABLES AND THEIR DISTRIBUTIONS

In a random experiment, frequently there has been greater interest for certain numerical values that can be deduced from the results of the random experiment than the experiment itself. Suppose, for example, that a fair coin is tossed consecutively six times and that we want to know the number of heads obtained. In this case, the sample space is equal to:

If we define X := “number of heads obtained”, then we have that X is a mapping of Ω to {1,2, ··· ,6}. Suppose, for example, we have X((H, H, T, H, H, T)) = 4. The mapping X is an example of a random variable. That is, a random variable is a function defined on a sample space. We will explain this concept in this chapter.

2.1   DEFINITIONS AND PROPERTIES

Definition 2.1 (Random Variable) Let(Ω, , P) be a probability space. A (real) random variable is a mapping such that, for all A B, , where B is the Borel σ-algebra over .

Note 2.1 Let (Ω, , P) be an arbitrary probability space. Given ...