1. ## Moment-Generating Function Questions

1) Let Y have pdf:
fy(y) = y, between 0 and 1 (inclusive), 2-y, between 1 and 2 (inclusive), and 0 everywhere else.
Find My(t)

2) Fined E(Y^4) if Y is an exponential random variable with fy(y) = xe^(-xy), y > 0

3) Find the variance of Y if My(t) = e^-2t/(1-t^2)

4) Calculate P(X less than or equal to 2) if Mx(t) = (1/4 + 3/4e^t)^5

2. 1) Let Y have pdf:
fy(y) = y, between 0 and 1 (inclusive), 2-y, between 1 and 2 (inclusive), and 0 everywhere else.
Find My(t)

$M_Y(t)=E(e^{Yt})= \int_0^1ye^{yt}dy +\int_1^2(2-y)e^{yt}dy$

NOW, you do the PARTs

3. For 2), do I find E(Y^4) by doing integral of Y^4 * fy(y)?? I forget what it means when you have to find expectation of something to a certain power.

And for 3), I'm assuming I have to find expectation first, but how do I use the moment-generating function to do that?

4. $E(Y^4)=\int_{-\infty}^{\infty} y^4f_Y(y)dy$

Find the variance of Y if My(t) = e^-2t/(1-t^2).....

Differentiate once and plug in 0 for t, that gives you $E(Y)$
(THE first moment)
Differentiate a second time and plug in 0 for t, that gives you $E(Y^2)$
(The second moment)
That's because this function generates all the moments.