Page 81 - Elements of Distribution Theory

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2.8 Suggestions for Further Reading 67

2.21 Let (X, Y) denote a two-dimensional random vector with an absolutely continuous distribution
with density function
1
p(x, y) = exp(−y), 0 < x < y < ∞.
y
r
Find E(X |Y) for r = 1, 2,....
2.22 For some n = 1, 2,..., let Y 1 , Y 2 ,..., Y n+1 denote independent, identically distributed, real-
valued random variables. Define

X j = Y j Y j+1 , j = 1,..., n.

(a) Are X 1 , X 2 ,..., X n independent?
(b) Are X 1 , X 2 ,..., X n exchangeable?
2.23 Let X and Y denote real-valued exchangeable random variables. Find Pr(X ≤ Y).
2.24 Prove Theorem 2.7.
2.25 Let X 0 , X 1 ,..., X n denote independent, identically distributed real-valued random variables.
For each definition of Y 1 ,..., Y n given below, state whether or not Y 1 ,..., Y n are exchangeable
and justify your answer.
(a) Y j = X j − X 0 , j = 1,..., n.
(b) Y j = X j − X j−1 , j = 1,..., n.
n
(c) Y j = X j − ¯ X, j = 1,..., n where ¯ X = j=1 X j /n.
(d) Y j = ( j/n)X j + (1 − j/n)X 0 , j = 1,..., n.
2.26 Let Y 1 , Y 2 ,... denote independent, identically distributed nonnegative random variables with
E(Y 1 ) = 1. For each n = 1, 2,..., let

X n = Y 1 ··· Y n .
Is {X 1 , X 2 ,...} a martingale?
2.27 Let {X 1 , X 2 ,...} denote a martingale. Show that

E(X 1 ) = E(X 2 ) = ··· .
Exercises 2.28 and 2.29 use the following definition. A sequence of real-valued random
variables {X 1 , X 2 ,...} such that E[|X n |] < ∞, n = 1, 2,..., is said to be a submartingale if,
for each n = 1, 2,...,

E[X n+1 |X 1 ,..., X n ] ≥ X n
with probability 1.
2.28 Show that if {X 1 , X 2 ,...} is a submartingale, then {X 1 , X 2 ,...} is a martingale if and only if

E(X 1 ) = E(X 2 ) = ··· .
2.29 Let {X 1 , X 2 ,...} denote a martingale. Show that {|X 1 |, |X 2 |,...} is a submartingale.

2.8 Suggestions for Further Reading

Conditional distributions and expectation are standard topics in probability theory. A mathematically
rigorous treatment of these topics requires measure-theoretic probability theory; see, for example,
Billingsley (1995, Chapter 6) and Port (1994, Chapter 14). For readers without the necessary back-
ground for these references, Parzen (1962, Chapter 2), Ross (1985, Chapter 3), Snell (1988, Chapter 4),