2 We know that the moment generating function for WaXb is MW

2. We know that the moment generating function for W=aX+b is MW/9t)=e^tb Mx(ta), where Mx(t) denotes the moment-generating function for the random variable X. (a) Use MW(t) to prove that E(W)=aE(X)+b and Var(W)=a^2Var(X). (b) Suppose that E(x)=2 and E(Y^2)=8, find E[(2+4X)^2 and V ar (2 X+3).

Solution

b)Given that E(X)=2, E(X^2)=8

Then, E[(2+4X)^2]

=E[4+16X+16X^2]=16E(X)+16E(X^2)

=16*2+16*8=160

Var(2X+3)=4Var(X)

=4[E(X)^2-(E(X))^2]

=4[16-8]=4*8=32