Problem 159E

Use the result in Exercise 3.158 to prove that, if W = aY + b, then E(W ) = aE(Y ) + b and V (W ) = a2 V (Y ).

Reference

If Y is a random variable with moment-generating function m(t) and if W is given by W = aY + b, show that the moment-generating function of W is etbm(at).

Solution :

Step 1 of 1:

Let Y denotes a random variable with moment-generating function m(t).

Our goal is:

If W=aY+b. We need to prove that E(W)=aE(Y)+b and V(W)=.

Now we have to prove that E(W)=aE(Y)+b and V(W)=.

If W=aY+b we know that the moment generating function.

The moment generating function is .

The expected value is the first derivative of the moment generating function evaluated at t=0.

E(W)=

E(W)=

E(W)=

E(W)=

E(W)=

E(W)=

E(W)=

The variance is the second derivatives of the moment generating function evaluated at t=0.

V(W)=

V(W)=

V(W)=

V(W)=

Therefore, E(W)=and V(W)=.