Thread: 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

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)



NOW, you do the PARTs

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?

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

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