Suppose X and Y are two independent random variables with th

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

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.