Problem 3. Lionel Messi plays in n soccer games. X; = of goals that he scores in game , i = 1, ..., n. Assume that X1, .... Xn are mutually independent. (a) Explain why we could assume that X1, .... X, iid ~ Poisson(A) for some parameter A > 0. (b) Find the MLE for A. You can assume that the likelihood function L(X) satisfies standard conditions on (0, co). (c) Assuming that Messi scores 0.85 goals per game on average, that is, A = 0.85. Use the CLT to approximate the probability that he scores more than 90 goals in 100 games. (d) Assume that in his professional career, Messi played 6 games every month for consecutive 20 years. To become the best goal scorer of all time, Messi needs to score at least 1300 goals in his career. Use the CLT to approximate the probability that Messi finishes his career being the best goal scorer of all time . (e) Let Y ~ Poisson() be the number of goals Cristiano Ronaldo scores in a game. What is the distribution of X1 + Y (assuming X1 ~ Poisson(A) and that X, and Y are independent)?3a. Poission distribution is used to model the occurrence of rare events. You can write a few (no more than 3} sentences explaining why you can use Poisson distribution for the number of goals Messi scores in a certain football match. 312d. Both parts have the assumption that A = 0.35. In 3:11 Messi plays Ex 12): 20 = 1440 games in his career.