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.