Page 509 - Discrete Mathematics and Its Applications

488 7 / Discrete Probability

We can use Theorems 3 and 6 to derive an alternative formula for V(X) that provides some
insight into the meaning of the variance of a random variable.

2
COROLLARY 1 If X is a random variable on a sample space S and E(X) = μ, then V(X) = E((X − μ) ).

μ is the Greek letter mu. Proof: If X is a random variable with E(X) = μ, then

2
2
2
E((X − μ) ) = E(X − 2μX + μ ) expanding (X − μ) 2
2 2
= E(X ) − E(2μX) + E(μ ) by part (i) of Theorem 3
2
2
= E(X ) − 2μE(X) + E(μ ) by part (ii) of Theorem 3, noting that μ is a constant
2
2
2
2
= E(X ) − 2μE(X) + μ 2 as E(μ ) = μ , because μ is a constant
2
2
= E(X ) − 2μ + μ 2 because E(X) = μ
2
= E(X ) − μ 2 simplifying
= V(X) by Theorem 6 and noting that E(X) = μ.

This completes the proof.

Corollary 1 tells us that the variance of a random variable X is the expected value of the
square of the difference between X and its own expected value. This is commonly expressed
as saying that the variance of X is the mean of the square of its deviation. We also say that the
standard deviation of X is the square root of the mean of the square of its deviation (often read
as the “root mean square” of the deviation).
We now compute the variance of some random variables.

EXAMPLE 14 What is the variance of the random variable X with X(t) = 1 if a Bernoulli trial is a success
and X(t) = 0 if it is a failure, where p is the probability of success and q is the probability of
failure?

2
Solution: Because X takes only the values 0 and 1, it follows that X (t) = X(t). Hence,
2 2 2
V(X) = E(X ) − E(X) = p − p = p(1 − p) = pq. ▲

EXAMPLE 15 Variance of the Value of a Die What is the variance of the random variable X, where X is
the number that comes up when a fair die is rolled?
2
2
Solution: We have V(X) = E(X ) − E(X) . By Example 1 we know that E(X) = 7/2. To find
2
2
2
E(X ) note that X takes the values i , i = 1, 2,..., 6, each with probability 1/6. It follows
that
1 91
2 2 2 2 2 2 2
E(X ) = (1 + 2 + 3 + 4 + 5 + 6 ) = .
6 6