The document analyzes two independent random variables, x and y, using their moment generating functions. It calculates the means, variances, and combines these to find the variance of a new variable z defined as z = x - 5y. The final variance of z is determined to be 13.

1. Suppose X and Y are two independent random variables with the following moment generating
functions: Mx(t)= e^(-t+(t^2)), My(t)= e^(2t+1/2(t^2))
Let Z = X
Solution
Given Mx(t) = e-t+t^2)
= 1-t+t^2)+(-t+t^2)^2/2!+....
= 1-t+t^2+t^2/2+t^3+t^4/2+...
Coefficient of t = -1
Mean of x = -1
E(X^2) = coeff of t^2 =3/2
Variance x = 3/2-1 = 1/2
My(t) = e^(2t+t^2/2)
= 1+2t+t^2/2 +4t^2+...
Mean of Y = 2
E(Y^2) = 9/2
Var(Y) = 0.5
Hence Var (z)
= Var(x-5y) = Var(x)+25Var(Y)
= 1/2 + 25(0.5)
=0.5+12.5 = 13.