Question

Question 2
(a) Suppose X ∼ N(μ, σ) and Z ∼ N(0, 1). The moment generating
function (m.g.f) of X is given by e^ut+1/2t^2σ^2

(i) What is the m.g.f of Z. [2 Marks]
(ii) If Y = cZ +d, where c and d are constant, find the m.g.f of Y and
hence the distribution of Y. [4 Marks]
(b) Suppose a random variable X follows a geometric distribution with pmf
p(x) = p(1−p)^(x−1), x = 1, 2, 3, ..., find the m.g.f of X and hence use the
m.g.f to obtain the mean and variance of X. [4 + 3 + 5 Marks]

Answer 1