Page 80 - Elements of Distribution Theory

P1: JZP
052184472Xc02 CUNY148/Severini May 24, 2005 2:29

66 Conditional Distributions and Expectation

2.15 Let X and Y denote random vectors with ranges X and Y, respectively. Show that, if X and Y
are independent, then
E[g(X)|Y] = E[g(X)]

for any function g:X → R such that E[|g(X)|] < ∞.
Does the converse to this result hold? That is, suppose that
E[g(X)|Y] = E[g(X)]

for any function g:X → R such that E[|g(X)|] < ∞. Does it follow that X and Y are indepen-
dent?
2.16 Prove Theorem 2.4.
2.17 Let X, Y and Z denote real-valued random variables such that (X, Y) and Z are independent.
Assume that E(|Y|) < ∞. Does it follow that
E(Y|X, Z) = E(Y|X)?

2.18 Let X denote a nonnegative, real-valued random variable. The expected residual life function
of X is given by

R(x) = E(X − x|X ≥ x), x > 0.
Let F denote the distribution function of X.
(a) Find an expression for R in terms of the integral

∞
F(t) dt.
x
(b) Find an expression for F in terms of R.
(c) Let X 1 and X 2 denote nonnegative, real-valued random variables with distribution functions
F 1 and F 2 and expected residual life functions R 1 and R 2 .If
R 1 (x) = R 2 (x), x > 0
does it follow that
F 1 (x) = F 2 (x), −∞ < x < ∞?
2
2.19 Let L 2 denote the linear space of random variables X such that E(X ) < ∞,as described in
Exercises 1.28 and 1.29. Let X 1 , X 2 denote elements of L 2 ;we say that X 1 and X 2 are orthogonal,
written X 1 ⊥ X 2 ,ifE[X 1 X 2 ] = 0.
Let Z denote a given element of L 2 and let L 2 (Z) denote the elements of L 2 that are functions
of Z.For a given random variable Y ∈ L 2 , let P Z Y denote the projection of Y onto L 2 (Z), defined
to be the element of L 2 (Z) such that Y − P Z Y is orthogonal to all elements of L 2 (Z). Show that
P Z Y = E(Y|Z).
2.20 Let X 1 , X 2 , andZ denote independent, real-valued random variables. Assume that
Pr(Z = 0) = 1 − Pr(Z = 1) = α

for some 0 <α < 1. Define

X 1 if Z = 0
Y = .
X 2 if Z = 1
(a) Suppose that E(X 1 ) and E(X 2 )exist. Does it follow that E(Y)exists?
(b) Assume that E(|X 1 |) < ∞ and E(|X 2 |) < ∞. Find E(Y|X 1 ).