Q 2. (a) The probability mass function of Binomial random variable X is given by
f (x) = (n/x) θ^x (1 - θ)^(n-x) ; x = 0, 1, 2, …, n
for 0 ≤ θ ≤ 1. Show that its moment generating function is given by MX (t) = (θe^t + (1 - θ))^n. Hence show that its mean is E (X) = nθ and its variance is Var (X) = nθ (1 - θ).
(b) The probability mass function of Poisson random variable X is given by
f (x) = (e^-λ λ^x) / x! ; x = 0, 1, 2, …
for λ > 0.
(i) Show that its moment generating function is given by MX (t) = e^(λ(e^t - 1)). Hence show that its mean is E (X) = λ and its variance is Var (X) = λ.
(ii) Using the moment generating function given in part (a), show that the Binomial random variable X ~ Bin (n, θ) has a limiting distribution that is Poisson when λ = nθ and n → ∞. You may use the following result:
Let a1, a2, … be a sequence of numbers such that lim n → ∞ an = a. Then lim n → ∞ (1 + an/n)^n = e^a.
(iii) Consider a “standardized” Poisson random variable
Y = (X - λ) / √λ
in which the random variable X ~ Poisson (λ). Using the moment generating function given in part (i), show that as λ → ∞ the random variable Y has a limiting distribution that is standard normal.