Page 79 - Elements of Distribution Theory

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2.7 Exercises 65

2.6 Prove Corollary 2.1.
2.7 Prove Theorem 2.2.
2.8 Let (X, Y) denote a random vector with an absolutely continuous distribution with density
function p and let p X and p Y denote the marginal densities of X and Y, respectively. Let
p X|Y (·|y) denote the density of the conditional distribution of X given Y = y and let p Y|X (·|x)
denote the density of the conditional distribution of Y given X = x. Show that
p Y|X (y|x)p X (x)
p X|Y (x|y) =
p Y (y)
provided that p Y (y) > 0. This is Bayes Theorem for density functions.
2.9 Consider a sequence of random variables X 1 , X 2 ,..., X n which each take the values 0 and 1.
Assume that

Pr(X j = 1) = 1 − Pr(X j = 0) = φ, j = 1,..., n
where 0 <φ < 1 and that
Pr(X j = 1|X j−1 = 1) = λ, j = 2,..., n.
(a) Find Pr(X j = 0|X j−1 = 1), Pr(X j = 1|X j−1 = 0), Pr(X j = 0|X j−1 = 0).
(b) Find the requirements on λ so that this describes a valid probability distribution for
X 1 ,..., X n .
2.10 Let X and Y denote real-valued random variables such that the distribution of (X, Y)is absolutely
continuouswithdensityfunction p andlet p X denotethemarginaldensityfunctionof X.Suppose
that there exists a point x 0 such that p X (x 0 ) > 0, p X is continuous at x 0 , and for almost all y,
p(·, y)is continuous at x 0 . Let A denote a subset of R.For each > 0, let

d( ) = Pr(Y ∈ A|x 0 ≤ X ≤ x 0 + ].
Show that
Pr[Y ∈ A|X = x 0 ] = lim d( ).
→0
2.11 Let (X, Y) denote a random vector with the distribution described in Exercise 2.4. Find the den-
sity function of the conditional distribution of X given Y = y and of the conditional distribution
of Y given X = x.
2.12 Let X denote a real-valued random variable with range X, such that E(|X|) < ∞. Let A 1 ,..., A n
denote disjoint subsets of X. Show that

E(X) = n E(X|X ∈ A j )Pr(X ∈ A j ).
j=1
2.13 Let X denote a real-valued random variable with an absolutely continuous distribution with
distribution function F and density p.For c ≥ 0, find an expression for Pr(X > 0||X|= c).
2.14 Let X, Y, and Z denote random variables, possibly vector-valued. Let X denote the range of X
and let Y denote the range of Y. X and Y are said to be conditionally independent given Z if,
for any A ⊂ X and B ⊂ Y,

Pr(X ∈ A, Y ∈ B|Z) = Pr(X ∈ A|Z)Pr(Y ∈ B|Z)
with probability 1.
(a) Suppose that X and Y are conditionally independent given Z and that Y and Z are inde-
pendent. Does it follow that X and Z are independent?
(b) Suppose that X and Y are conditionally independent given Z and that X and Z are condi-
tionally independent given Y. Does it follow that X and Y are independent?