Page 47 - Elements of Distribution Theory

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1.8 Expectation 33

so that

F X (z) − F Y (z) = [1 − x/ ] dF X 0 (x) − [1 − y/ ] dF Y 0 (y).
0 0
Using integration-by-parts,
1 1
F X (z) − F Y (z) = [F X 0 (x) − F X 0 (0)] dx + [F Y 0 (y) − F Y 0 (0)] dy.
0 0
The result now follows as in the scalar random variable case.

Inequalities
The following theorems give some useful inequalities regarding expectations; the proofs of
Theorems 1.12–1.14 are left as exercises.

Theorem 1.12 (Cauchy-Schwarz inequality). Let X denote a random variable with range
X and let g 1 , g 2 denote real-valued functions on X. Then
2
2
2
E[|g 1 (X)g 2 (X)|] ≤ E[g 1 (X) ]E[g 2 (X) ]
2
with equality if and only if either E[g j (X) ] = 0 for some j = 1, 2 or
Pr[g 1 (X) = cg 2 (X)] = 1
for some real-valued nonzero constant c.

Theorem 1.13 (Jensen’s inequality). Let X denote a real-valued random variable with
range X and let g denote a real-valued convex function defined on some interval containing
X such that E(|X|) < ∞ and E[|g(X)|] < ∞. Then
g[E(X)] ≤ E[g(X)].

Theorem 1.14 (Markov’s inequality). Let X be a nonnegative, real-valued random vari-
able. Then, for all a > 0,
1
Pr(X ≥ a) ≤ E(X).
a

Theorem 1.15 (H¨older inequality). Let X denote a random variable with range X and let
g 1 , g 2 denote real-valued functions on X. Let p > 1 and q > 1 denote real numbers such
that 1/p + 1/q = 1. Then
p 1 q 1
E(|g 1 (X)g 2 (X)|) ≤ E(|g 1 (X)| ) E(|g 2 (X)| ) .
p
q
The proof is based on the following lemma.

Lemma 1.1. Let a, b,α,β denote positive real numbers such that α + β = 1.
α
β
(i) a b ≤ αa + βb.
(ii) If p > 1 and q > 1 satisfy 1/p + 1/q = 1, then
a p b q
ab ≤ + .
p q