Problem 1.

Let

X,Y,Z

be independent random variables where

X,Y

are uniformly distributed

on

[0

,

1]

and

Z

is an exponential with mean 1.

(a) What is the conditional density of

X

given

XY

=

t

for

0

t

1

?

(b) What is the conditional density of

X

given

X

+

Z

=

t

for

t

0

?