# 4.1 Random Variable

In this section we introduce discrete random variables and their probability distributions.

Given a chance experiment, the collection of possible outcomes is called the sample space, denoted as ** S**. A

**random variable**is a function (or a mapping) from the sample space

**into real numbers. Random variables are usually denoted as uppercase letters, such as X, Y, Z. We use the corresponding lowercase letters x, y, z to represent possible values that random variables may attain.**

*S*Example: Random Variable

- Consider the chance experiment of flipping a balanced coin twice, the sample space is
= {HH, HT, TT, TH}. Let the random variable X = # of tails. It is a mapping from the sample space*S*to integers 0, 1, and 2.*S*

- Five students are asked to report the number of siblings they have; their responses are summarized in the following table:
Name Mark John Rebecca Sarah Mary # of Siblings 0 1 2 2 3 Randomly pick one student and let random variable X = # of siblings the student has. Then X is a mapping from the sample space

= {Mark, John, Rebecca, Sarah, Mary} to the numbers 0, 1, 2 and 3.*S*

In general, a discrete random variable maps the sample space ** S** to numbers that can be listed or counted; a continuous random variable maps the sample space

**to an interval that is a subset of the entire real line. If you need to review discrete and continuous data, refer to variables and data in Chapter 1.3.**

*S*