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.
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.
Question 31 on page 216.
Question 33 on page 216.
Question 46 on page 216.
Question 55 on page 217.
Bonus Questions: if you do these you need do only questions
5 and 6 above for full marks on the assignment.
Suppose X and Y have joint density . Prove from the
definition of density that the density of X is .
Let 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.
Generalize the construction to show that there rv's X and Y
such that X and Y are each standard normal, X and Y are
uncorrelated but X and Y are not independent.