5. Diamond distribution. Consider the random variables X and Y with the joint pdf where c is a constant. (a) Find c. |xy(x,y) = { c, if |x|+|y| 1, 0, otherwise, (b) Find fx(x) and fxy(xy). (c) Are X and Y independent random variables? Justify your answer. (d) Define the random variable Z = (|X|+|Y]). Find the pdf fz(z). 6. Coin with random bias. You are given a coin but are not told what its bias (probability of heads) is. You are told instead that the bias is the outcome of a random variable P~ U[0, 1]. To get more infromation about the coin bias, you flip it independently 10 times. Let X be the number of heads you get. Thus X ~ Binom(10, P). Assuming that X = 9, find and sketch the a posteriori probability of P, i.e., fp|x(p|9). ~ 7. First available teller. Consider a bank with two tellers. The service times for the tellers are independent exponentially distributed random variables X ~ Exp(1) and X2 Exp(2), respectively. You arrive at the bank and find that both tellers are busy but that nobody else is waiting to be served. You are served by the first available teller once he/she is free. (a) What is the probability that you are served by the first teller? (Hint: Recall that the exponential distribution has the memoryless property.) (b) Let the random variable Y denote your waiting time. Find the pdf of Y.