Page 45 - Elements of Distribution Theory

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1.8 Expectation 31

Theorem 1.10. Let X denote a random variable with range X.
(i) If g is a nonnegative real-valued function on X, then E[g(X)] ≥ 0 and E[g(X)] = 0
if and only if Pr[g(X) = 0] = 1.
(ii) If g is the constant function identically equal to c then E[g(X)] = c.
(iii) If g 1 , g 2 ,..., g m are real-valued functions on X such that E[|g j (X)|] < ∞,
j = 1,..., m, then
E[g 1 (X) +· · · + g m (X)] = E[g 1 (X)] + ··· + E[g m (X)].

(iv) Let g 1 , g 2 ,... denote an increasing sequence of nonnegative, real-valued functions
on X with limit g. Then

lim E[g n (X)] = E[g(X)].
n→∞
(v) Let g 1 , g 2 ,... denote a sequence of nonnegative, real-valued functions on X. Then

E[lim inf g n (X)] ≤ lim inf E[g n (X)].
n→∞ n→∞
(vi) Let g 1 , g 2 ,... denote a sequence of real-valued functions on X. Suppose there exist
real-valued functions g and G, defined on X, such that, with probability 1,
|g n (X)|≤ G(X), n = 1, 2,...,

and
lim g n (X) = g(X).
n→∞
If E[G(X)] < ∞, then
lim E[g n (X)] = E[g(X)].
n→∞

An important property of expectation is that the expectation operator E(·) completely
defines a probability distribution. A formal statement of this fact is given in the following
theorem.

Theorem 1.11. Let X and Y denote random variables.
E[g(X)] = E[g(Y)]
for all bounded, continuous, real-valued functions g, if and only if X and Y have the same
probability distribution.

Proof. If X and Y have the same distribution, then clearly E[g(X)] = E[g(Y)] for all
functions g for which the expectations exist. Since these expectations exist for bounded g,
the first part of the result follows.
Now suppose that E[g(X)] = E[g(Y)] for all bounded continuous g.We will show that
in this case X and Y have the same distribution. Note that we may assume that X and Y have
the same range, neglecting sets with probability 0, for if they do not, it is easy to construct
a function g for which the expected values differ.
The proof is based on the following idea. Note that the distribution function of a random
variable X can be written as the expected value of an indicator function:
Pr(X ≤ z) = E[I {X≤z} ].