Problems: Assignment 2
Basic Questions
are iid real random variables with
density f . Let 
be the X 's arranged
in increasing order, that is, 
is the minimum of 
and 
while 
is the maximum.
.
. Prove that 
is independent of 
.
.
.
are iid exponential. Let 
for i=1, 2 and 3.
.
.
.
. Prove from the
definition of density that the density of X is 
.
). 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.
be the bivariate normal density with mean 0,
unit variances and correlation 
and let 
be the standard
bivariate normal density. Let 
.
Show that p has normal margins but is not bivariate normal.
and 
. Let Z=X+Y.
Find the distribution of Z given X and that of X given Z.