Springer Texts in Business and Economics

Moment Generating Function Method

a. If Xi,.., Xn are independent Poisson distributed with parameters (Xi) respec­tively, then from problem 2.14c, we have

MXi (t) = eAi(e-1) for i = 1,2,... ,n

n n

Y = Xi has My(t) = П MXi (t) since the X/s are independent. Hence

i=i i=i

n

Ai (e‘-l)

MY(t) = ei=1

n

which we recognize as a Poisson with parameter ^ Xi.

i=i

b. IfXi, ..,Xn are IIN (^i, a2), then from problem 2.14b, we have MXi(t) = ew‘+ 1ai2‘2 for i = 1,2,.., n

nn

Y = ^ Xi has MY(t) = ]""[ MXi (t) since the X/s are independent. Hence

i=i i=i

MY(t) = e 1=i

c. If Xi,.., Xn are IIN(|a,, a2),thenY = J2 Xi is N(np,,na2) from part b using

i=i

the equality of means and variances. Therefore, X = Y/nis N(p,, a2/n).

d. If Xi,.., Xn are independent x2 distributed with parameters (ri) respectively, then from problem 2.14f, we get

MXi(t) = (i - 2t)-ri/2 fori = i,2,..,n

nn

Y = ^ Xi has MY(t) = ]""[ MXi (t) since the X/s are independent. Hence,

i=i i=i

- ri/2

MY(t) = (i - 2t) i-i

n

which we recognize as x2 with degrees of freedom J2 ri.

i=i