Page 246 - Elements of Distribution Theory

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232 Distribution Theory for Functions of Random Variables

7.9 Let X 1 and X 2 denote independent random variables, each with a uniform distribution on (0, 1).
Find the density function of Y = log(X 1 /X 2 ).
7.10 Let X and Y denote independent random variables such that X has a standard normal distribution
and Y has a standard exponential distribution. Find the density function of X + Y.
7.11 Suppose X = (X 1 , X 2 ) has density function
x , x 1 > 1, x 2 > 1.
p(x 1 , x 2 ) = x −2 −2
1 2
Find the density function of X 1 X 2 .
7.12 Let X 1 ,..., X n denote independent, identically distributed random variables, each of which is
n
uniformly distributed on the interval (0, 1). Find the density function of T = j=1 X j .
7.13 Let X 1 , X 2 ,..., X n denote independent, identically distributed random variables, each with an
2
absolutely continuous distribution with density function 1/x , x > 1, and assume that n ≥ 3.
Let
Y j = X j X n , j = 1,..., n − 1.
Find the density function of (Y 1 ,..., Y n−1 ).
7.14 Let X be a real-valued random variable with an absolutely continuous distribution with density
function p. Find the density function of Y =|X|.
7.15 Let X denote a real-valued random variable with a t-distribution with ν degrees of freedom.
2
Find the density function of Y = X .
7.16 Let X and Y denote independent discrete random variables, each with density function p(·; θ)
where 0 <θ < 1. For each of the choices of p(·; θ)given below, find the conditional distribution
of X given S = s where S = X + Y.
j
(a) p( j; θ) = (1 − θ)θ , j = 0,...
j
(b) p( j; θ) = (1 − θ)[− log(1 − θ)] /j!, j = 0,...
(c) p( j; θ) = θ j+1 /[ j(− log(1 − θ))], j = 0,...
Suppose that S = 3is observed. For each of the three distributions above, give the conditional
probabilities of the pairs (0, 3), (1, 2), (2, 1), (3, 0) for (X, Y).
7.17 Let X and Y denote independent random variables, each with an absolutely continuous distri-
bution with density function
α
, x > 1
x α+1
where α> 1. Let S = XY and T = X/Y. Find E(X|S) and E(T |S).
7.18 Let X denote a nonnegative random variable with an absolutely continuous distribution. Let F
and p denote the distribution function and density function, respectively, of the distribution.
The hazard function of the distribution is defined as
p(x)
h(x) = , x > 0.
1 − F(x)
Let X 1 ,..., X n denote independent, identically distributed random variables, each with the
same distribution as X, and let

Y = min(X 1 ,..., X n ).
Find the hazard function of Y.
7.19 Let X 1 , X 2 denote independent random variables, each with a standard exponential distribution.
Find E(X 1 + X 2 |X 1 − X 2 ) and E(X 1 − X 2 |X 1 + X 2 ).
7.20 Let X 1 ,..., X n denote independent random variables such that X j has a normal distribution
with mean µ j and standard deviation σ j . Find the distribution of ¯ X.