520 CHAPTER 8 Discrete Probability
U Discrete Random Variables
Many probabilistic experiments have outcomes that are associated with real numbers. For
example, a gambling game may involve spinning a wheel that is divided into segments
that offer various payoffs or penalties (see Figure 8.6). The outcomes (wheel segments)
of the experiment have real numbers (cash values) associated with them. In the case of
flipping a coin n times, each outcome (n-tuple of heads and tails) can be associated with
the total number of heads in that outcome. Of course, we are accustomed to using numbers,
typically integers, as labels. For example, we use 1 to mean heads and 6 to mean a die that
shows six spots. However, our interest in this section is to study the situation in which
outcomes are associated with numerical values that are not used simply as labels for the
outcome.$500 $1Figure 8.6 Payoffs associated with outcomes.8.7.1 Distributions of a Random Variable
A typical question involving numerical values is how much one should be willing to pay
to gamble on a game that offers various payoffs with various probabilities. The following
definition explains what it means to associate real numbers with outcomes.Definition 1. A discrete random variable X is a real-valued function with a domain
that is a discrete (finite or countably infinite) sample space 02 endowed with a probability
density function. In other words, X (a)) c R for each 0) E Q2.Example 1. A game consists of rolling a fair die. You win $3 times the value on the top
face after the roll if the top face is 2 or 5. The game pays $1 if the top face after the roll is
1, 3, 4, or 6. Define the random variable associated with this game.Solution. Let X be the random variable defined as X(l) = 1, X(2) = 6, X(3) = 1,X(4) = 1, X(5) = 15, and X(6) = I where 1, 2,..., 6 represent the value on the top face
after rolling a fair die. The probability density for the roll of the die is the uniform proba-
bility density on {1, 2, 3, 4, 5, 6}. 0