Problems: Assignment 2
Suppose X and Y have joint density . Prove from the
definition of density that the density of X is
.
Suppose X is Poisson(). After observing X a coin landing
Heads with probability p is tossed X times. Let Y be the number of
Heads and Z be the number of Tails. Find the joint and marginal distributions
of Y and Z.
Let be the bivariate normal density with mean 0,
unit variances and correlation
and let
be the standard
bivariate normal density. Let
.
Warning: This is probably hard. Don't waste too much time on
it. Suppose X and Y are independent and
random variables. Show that
is a
random variable.
Suppose X and Y are independent with
and
. Let Z=X+Y.
Find the distribution of Z given X and that of X given Z.