Page 93 - Probability and Statistical Inference

70 2. Expectations of Functions of Random Variables

(i) a ≤ E(X) ≤ b if the support χ of X is the interval [a, b];
(ii) E(X) ≤ E(Y) if X ≤ Y w.p.1.
Theorem 2.2.4 Let X be a random variable. Then, we have

where a and b are any two fixed real numbers.
We now consider another result which provides an interesting perspective
by expressing the mean of a continuous random variable X in terms of its tail
area probabilities when X is assumed non-negative.
Theorem 2.2.5 Let X be a non-negative continuous random variable with
its distribution function F(x). Suppose that . Then, we
have:

Proof We have assumed that X ≥ 0 w.p.1 and thus

The proof is now complete. !
We can write down a discrete analog of this result as well. Look at the
following result.
Theorem 2.2.6 Let X be a positive integer valued random variable with its
distribution function F(x). Then, we have

Proof Recall that F(x) = P(X ≤ x) so that 1 – F(x) = P(X > x) for x = 1, 2,
.... Let us first verify that