Theory of Statistics - George Mason University

1.1 Some Important Probability Facts 29
Proof. This is a simple application of Fubini’s theorem, using the CDF F of
X:
E(X) =
=
=
=
=
∞
0
∞
0
∞
0
∞
0
∞
0
x dF(x)
dt dF(x)
]0,x[
]t,∞[
dF(x)dt
(1 − F(t))dt
Pr(X > t)dt
Theorem 1.15 leads in general to the following useful property for any
given random variable X for which E(X) exists:
E(X) =
∞
0
0
(1 − F(t))dt − F(t)dt. (1.44)
−∞
It is an exercise to write a proof of this statement.
Another useful fact in applications involving the Bernoulli distribution
with parameter π is the relationship
E(X) = Pr(X = 1) = π.
Expected Value of the Indicator Function
We define the indicator function, IA(x), as
1 if x ∈ A
IA(x) =
0 otherwise.
(1.45)
(This is also sometimes called the “characteristic function”, but we use that
term to refer to something else.) If X is an integrable random variable over
A, then IA(X) is an integrable random variable, and
Pr(A) = E(IA(X)). (1.46)
It is an exercise to write a proof of this statement. When it is clear from the
context, we may omit the X, and merely write E(IA).
Theory of Statistics c○2000–2013 James E. Gentle