Let X1 ~ Binomial(m,p), X2 ~ Poisson(a) and X1 X2. Define X = X1 + X2, then X is said to follow the Charlier series (CS) distribution, denoted by X ~ CS(m, p, 1). (a) Show that the pmf of X is min(m,r) Pr(X = x) = k=0 I (%)*(1 p). 1e-1 (2 k)!' for x = 0,1,...,0, where m is a known positive integer, pe (0,1) and > 0. (b) Show that E(X) = mp + lu and Var(X) = mp(1 p) + 1 = H - mp (c) Let {Xn}n=1 lid CS(m,p, 1), where m and p are assumed to be known. Based on the central limit theorem, construct an approx- imate 100(1 - a)% CI for the mean p. Let X1 ~ Binomial(m,p), X2 ~ Poisson(a) and X1 X2. Define X = X1 + X2, then X is said to follow the Charlier series (CS) distribution, denoted by X ~ CS(m, p, 1). (a) Show that the pmf of X is min(m,r) Pr(X = x) = k=0 I (%)*(1 p). 1e-1 (2 k)!' for x = 0,1,...,0, where m is a known positive integer, pe (0,1) and > 0. (b) Show that E(X) = mp + lu and Var(X) = mp(1 p) + 1 = H - mp (c) Let {Xn}n=1 lid CS(m,p, 1), where m and p are assumed to be known. Based on the central limit theorem, construct an approx- imate 100(1 - a)% CI for the mean p