A coin is tossed once and heads turn up with probability p. Let X and Y be the numbers of heads and tails, respectively.
(a) Show that X and Y are not independent.
Suppose now that the coin is tossed a random number N of times, where N has the Poisson distribution with parameter λ. Let X and Y be the numbers of heads and tails, respectively.
(b) Find the joint PMF of X and Y.
(c) Find the marginal PMFs and identify the distributions for X and Y.
(d) Are X and Y independent?