Moment Generating Function Method. (a) Show that if X1, . . .,Xn are independent Poisson distributed with parameters (i) respectively, then Y = n i=1

Moment Generating Function Method.

(a) Show that if X1, . . .,Xn are independent Poisson distributed with parameters (λi) respectively, then Y =

n i=1 Xi is Poisson with parameter

n i=1 λi.

(b) Show that if X1, . . .,Xn are independent Normally distributed with parameters (μi, σ2i

), then Y =

n i=1 Xi is Normal with mean

n i=1 μi and variance

n i=1 σ2i

.

(c) Deduce from part

(b) that if X1, . . . , Xn are IIN(μ, σ2), then ¯X ∼ N(μ, σ2/n).

(d) Show that if X1, . . .,Xn are independent χ2 distributed with parameters (ri) respectively, then Y =

n i=1 Xi is χ2 distributed with parameter

n i=1 ri.