please answer both and show working

1. (25 points) Two fair 6 sided dice are rolled. What is the probability that (a) a ve turns up exactly once? (b) at least one die roll is a ve, given that the two dice land on different numbers? (c) the roll results in a sum of 5? (d) in 12 rolls of this pair of dice, the rolls result in a sum of 5 for exactly 8 of those rolls? (e) one needs exactly 5 rolls until the rst time the roll results in a sum of 5? 2. (25 points) Let X be a random variable with a Poisson distribution with parameter A I} 0. (a) Show that the moment generating function M X (t) = email1). (b) Using the moment generating function MX (t), compute the the expected value E[X] (c) Using the moment generating function M X (t), compute the variance var(X). (d) Suppose X1, . . . , X5 are independently and identically distributed Poisson random variables with parameter A I} 0. Let 85 : X1 + X; + . - . + X5. Find var(S5/5). (e) Let 5'5 as dened in part (e). Find the exact distribution of 35 and justify your answer. (Hint: Use moment generating function.)