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) +
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
Exercise 1.1.9 (Paley-Zygmund inequality). Let X be a positive random
variable with finite variance. Show that
P(X ≥ λE(X)) ≥ (1 −
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 +
for any real or complex t, thus relating the exponential and Fourier 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.