Question

(2) (a) Argue using the Central Limit Theorem that one may approxímate X ~ Poisson(n) by a normal law when the integer n is large. (b) Compare the true values of E(X3) and E(X4) with those based on the CLT approximation. Note: the requisite moments of normal and Poisson random variables can be extracted from their MGFs.

Answer 1

the probability of random variable of x following Poisson distribution is given by p(x) = e^-n n^x/x ! Hence 'n' is the parameter of the distribution the mean is given by mu_2 = elementof (x) = sigma^infinity_x = 0 x p (x) but p (x) = e^-n n^x/x mu_1 = sigma^infinity_x = 0 x e^-n n^x/x = 0 x e^-n n^0/0 + 1 x e^-n n^1/1 + 2 x e^-n n^2/2 + 3 e^-n n^3/3 +.+ 0 infinity = e^-n 1 + n/1 + n^2/2 + . infinity mu_1 = e^-n n e^n = 0 Similarly mu_2 = elementof (x)62 = sigma^infinity_x = 0 x^2 p (x) But p (x) = e^-n n^x/x mu_2 = sigma^infinity_x-0 x^2 e^-n n^x/x = 0 x e^-n n^o/o + x e^-n n^1/1 + 2^2 x e^-n n^2/2 +. Infinity = e^-n [n + 2 n^2 + 3n^3/2 + 4n^4/3 +. Infinity] = e^-n (n {1 + n + n^2/2 +.[1 + n^2 n^2/2] But e^n = 1 + n/1 + n^2/2 + n^3/3 6 +. infinity mu_2 = \r\n