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2. Expectations of Functions of Random Variables 71

where as usual I(X > x) stands for the indicator function of the event X > x. If
X happens to be one, then I(X > 0) = 1, but I(X > x) = 0 for all x = 1, 2, ... .
Hence, the rhs of (2.2.12) is also one. In other words when X = 1, the two
sides of (2.2.12) agree. Now if X happens to be two, then I(X > 0) = I(X > 1)
= 1, but I(X > x) = 0 for all x = 2, 3, ... . Hence, the rhs of (2.2.12) is also then
two. In other words when X = 2, the two sides of (2.2.12) agree. The reader
should check that (2.2.12) holds whatever be the value of X observed from
the set {1, 2, ...}.
For any fixed x ∈ {0, 1, 2, ...} the random variable I(X > x) takes only one
of the values 0 or 1. Hence from the Definition 2.2.1 of the expectation or using
the Example 2.2.3 which follows shortly, we can claim that E[I(X > x)] = P(X
> x) and this equals 1 – F(x).
Now the desired result will follow provided that the expectation operation
“E” can be taken inside the summation in order to write x)] =
Since I(X > x) ≥ 0 w.p.1, in-
deed by applying the Monotone Convergence Theorem (refer to the Exercise
2.2.24) one can justify exchanging the expectation and the infinite summation.
The proof is now complete. !
The Exercise 2.2.21 provides an alternate sufficient condition for the con-
clusion of the Theorem 2.2.5 to hold. The Exercises 2.2.22-2.2.23 give inter-
esting applications of the Theorems 2.2.5-2.2.6.
At this point we consider some of the standard distributions from Section
1.7 and derive the expressions for their means and variances. The exact calcu-
lation of the mean and variance needs special care and attention to details. In
what follows, we initially emphasize this.

2.2.1 The Bernoulli Distribution
From (1.7.1) recall that a Bernoulli(p) random variable X takes the values 1
and 0 with probability p and 1 – p respectively. Then according to the Defini-
tions 2.2.1-2.2.3 and using (2.2.9), we have

Example 2.2.3 (Exercise 1.7.22 Continued) Consider an arbitrary ran-
dom variable Y which may be discrete or continuous. Let A be an arbitrary
event (Borel set) defined through the random variable Y. For example, the
event A may stand for the set where Y ≥ 2 or |Y| > 4 ∪ |Y| ≤ 1/2 or one of