12. Let X has a Poisson distribution such that 3P{X =1}: P{X :2]: nd P(X = 4) ['2 13. In the decade of the 198031 the number of X of cases of diphtheria. reported each year in the United States followed a Poisson distribution with mean 2.5. What is the probability that {a} No cases were reported in a given year? {b} Exactly three cases were reported? (c) At least three cases were reported? {d} At most three cases were reported? [3 14. Let X equal the number of ips of a fair coin that are required to observe headstails on consecutive ips. Given that the m.g.f of X is 82: 2 (er 212 Find the values of the mean and the variance. rm}