Unformatted text preview: , Y be nonnegative random variables. Show the following properties,
using just the above deﬁnition and basic properties of measures, but no other
facts from integration theory.
(a) If we have two nonnegative random variables with P(X = Y ) = 1, then
E[X ] = E[Y ].
(b) If Y is a nonnegative random variable and E[Y ] = 0, then P(Y = 0) = 1.
(c) If X (ω ) ≤ Y (ω ) for all ω ∈ Ω, then E[X ] ≤ E[Y ].
(d) (Monotone convergence theorem) Let Xn be an increasing sequence of
nonnegative random variables, whose limit is X . Show that limn→∞ E[Xn ] →
E[X ]. Hint: This is really easy: use continuity of measures on the sets
AXn .
All this looks pretty simple, so you may wonder why this is not done in most
textbooks. The answer is twofold: (i) developing some of the other properties,
2 such as linearity, is not as straightforward; (ii) the construction of the product
measure, when carried out rigorously is quite involved.
Exercise 6. Suppose that X is a nonnegative random variable and that E[esX ] <
∞ for all s ∈ (−∞, a], where a is a positive number. Let k be a positive integer.
(a) Show that E[X k ] < ∞.
(b) Show that E[X k esX ] < ∞, for every s < a.
(c) Suppose that h > 0. Show that (ehX − 1)/h ≤ X ehX .
(d) Use the DCT to argue that
�
ehX − 1 �
E[ehX ] − 1
E[X ] = E lim
= lim
.
h↓ 0
h↓ 0
h
h 3...
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This document was uploaded on 03/19/2014 for the course EECS 6.436 at MIT.
 Fall '08
 DavidGarnarnik

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