18 1. Preparatory material

non-zero variance, then there exist scalars a, b such that a + bX has mean

zero and variance one.

Exercise 1.1.7. We say that a real number M(X) is a median of a real-

valued random variable X if P(X M(X)), P(X M(X)) ≤ 1/2.

(i) Show that a median always exists, and if X is absolutely continuous

with strictly positive density function, then the median is unique.

(ii) If X has finite second moment, show that

M(X) = E(X) +

O(Var(X)1/2)

for any median M(X).

Exercise 1.1.8 (Jensen’s inequality). Let F : R → R be a convex function

(thus F ((1 − t)x + ty) ≤ (1 − t)F (x) + tF (y) for all x, y ∈ R and 0 ≤ t ≤ 1),

and let X be a bounded real-valued random variable. Show that EF (X) ≥

F (EX). (Hint: Bound F from below using a tangent line at EX.) Extend

this inequality to the case when X takes values in Rn (and F has Rn as its

domain.)

Exercise 1.1.9 (Paley-Zygmund inequality). Let X be a positive random

variable with finite variance. Show that

P(X ≥ λE(X)) ≥ (1 −

λ)2

(EX)2

EX2

for any 0 λ 1.

If X is sub-Gaussian (or has sub-exponential tails with exponent a 1),

then from dominated convergence we have the Taylor expansion

(1.27)

EetX

= 1 +

∞

k=1

tk

k!

EXk

for any real or complex t, thus relating the exponential and Fourier moments

with the

kth

moments.

1.1.3. Independence. When studying the behaviour of a single random

variable X, the distribution μX captures all the probabilistic information

one wants to know about X. The following exercise is one way of making

this statement rigorous:

Exercise 1.1.10. Let X, X be random variables (on sample spaces Ω, Ω ,

respectively) taking values in a range R, such that X ≡ X . Show that after

extending the spaces Ω, Ω , the two random variables X, X are isomorphic,

in the sense that there exists a probability space isomorphism π : Ω → Ω

(i.e., an invertible extension map whose inverse is also an extension map)

such that X = X ◦ π.