5. (10 points) Let X be a Poisson random variable with parameter Y, that is, the...

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5. (10 points) Let X be a Poisson random variable with parameter Y, that is, the p.d.f. of X is given by f(xy) = ey x=0, 1, 2,... Suppose that the parameter Y itself is also a random variable, and it follows a gamma distribution with parameters a > 0 and > 0, that 1 is, f(y) = (a) Bay-e, y > 0. Find the marginal p.d.f. of X. (please simplify your final expression as much as you can.) 6. (10 points) In a series of gambling games, the probability that you win each game is p (p1/2). You plan to first play 3 games to see what happens. If you win at least 2 of the first 3 games, then you will keep playing until you loose. Assume that all games are independent. (a) Find the expected number of games that you will play. (b) Find the expected number of games that you will loose. 5. (10 points) Let X be a Poisson random variable with parameter Y, that is, the p.d.f. of X is given by f(xy) = ey x=0, 1, 2,... Suppose that the parameter Y itself is also a random variable, and it follows a gamma distribution with parameters a > 0 and > 0, that 1 is, f(y) = (a) Bay-e, y > 0. Find the marginal p.d.f. of X. (please simplify your final expression as much as you can.) 6. (10 points) In a series of gambling games, the probability that you win each game is p (p1/2). You plan to first play 3 games to see what happens. If you win at least 2 of the first 3 games, then you will keep playing until you loose. Assume that all games are independent. (a) Find the expected number of games that you will play. (b) Find the expected number of games that you will loose.