Discrete Mathematics for Computer Science

524 CHAPTER 8 Discrete Probability

is given by

b(0; 10, 25/1000) = C(10, 0) • (25/1000)0 .(1 - 25/1000)1°

= (975/1000)'°

S0.7763

Being able to approximate one distribution with another can be useful, especially if the

approximating distribution is easier to compute.

8.74 Expectation of a Random Variable

Suppose we choose a card from a shuffled deck. If we draw a spade, we must pay a $10

penalty, but if we draw a card from any other suit, we receive $5. How do we estimate our

winnings in playing the game a large number n of times? If we believe that a spade will be

drawn approximately n/4 times, then we estimate our winnings as

( ($)3)+ (_$10)(n) = ($1.25)n

4

and we estimate our average winning per draw to be $1.25.

In mathematical terms, the dollar amount associated with each card defines a real-

valued random variable X on the sample space Q2 of cards, each of which is assigned

a probability of 1/52. The range of X consists of just two values: $5, with probability

3/4 of occurring, and -$10, with probability 1/4 of occurring. The sum of these payoffs,

weighted by their probabilities, is

($5)(~ + (-$10) (1) = $1.25

This quantity is called the expectation of X. When we interpret probabilities as estimates

of the frequency of occurrence of real-world events, then the expectation of X estimates

the average value of X when the underlying experiment is repeated many times. In general,

the expectation of X does not tell us what we expect to happen on any particular trial of the

experiment. Only under very special circumstances, such as the ones described in the Law

of Averages given at the end of Section 8.9, does the expectation tell us what to expect.

Definition 5. The expectation (also called mean or expected value) E(X) of a random

variable X is defined by

E(X) = E x • px(x)

where the sum is taken over all x in the range Q x of X and px(x) = P(X = x), the

probability that x occurs.

Notation. When only one random variable is under discussion, the value E(X) is often

denoted by it.

Example 4. A loaded die pays $1 for an even number of spots occurring on the top face

after a roll, and charges $1, $3, or $5 for showing one, three, or five spots after a roll,

respectively. Suppose the probability p(w) of getting

(0 = 1, 2, 3, 4, 5,6