Expected values & Moment-generating function (MGF)

Having a little trouble with this problem. Thank you in advance!

A random variable Y has a density function f(y) = e^y, y < 0.
1. Find E(e^(3Y/2))
2. Find MFG for Y.
3. Find V(Y).

Re: Expected values & Moment-generating function (MGF)

Pls don't double post.

Recall $E(x) = \int_{x_1}^{x_2} x f(x)~dx$

Re: Expected values & Moment-generating function (MGF)

Quote:

Originally Posted by charmagne

Having a little trouble with this problem. Thank you in advance!

A random variable Y has a density function f(y) = e^y, y < 0.
1. Find E(e^(3Y/2))
2. Find MFG for Y.
3. Find V(Y).

$E(h(Y))=\int_{-\infty}^0 h(y) f(y)\; dy$

so for 1.:

$E(e^{3Y/2})=\int_{-\infty}^0 e^{3y/2}\; e^y\; dy$

which is elementary.

Now look up the definition of the MGF to do 2., and for 3. you should know that:

$V(Y)=E(Y^2)-\left[ E(Y) \right]^2$

and both of the terms on the right can be obtained from the MGF.

CB