Hi, I am at a loss for a problem...
If Y1 and Y2 are independent random variables, each having a normal distribution with mean 0 and variance 1, find the moment-generating function of U=Y1Y2. Use this moment-generating function to find E(U) and V(U). Check the result by evaluating E(U) and V(U) directly from the density functions for Y1 and Y2.
I try to do Mu(t) = E(e^tu) the normal way, but then I get: E(e^ty1y2) which doesn't help me in the least...
I know that E(U) = E(Y1)*E(Y2) = 0 and V(U) = 1 similarly, but I cannot find the MGF. Any pointers? Am I missing something obvious, or not so obvious?
But I admit that I don't understand the order of the questions since you're first asked to find the mgf and then the expectations and all the stuff. And finding that mgf is more difficult than finding the expectation if it's indeed a product of normal distributions
Maybe this will work.
I'll use X and Z as standard normal rvs.
Let U=XZ. You want ...
What I'm doing is fixing the X which makes the argument now Xt.
So plug in Xt into the MGF of Z and see what you get.
Using the MGF of Z and plugging in Xt for t, I get
and now using the density of X...
which you can compute without integrating by observing the st deviation here as
Wow, thank you very much.
I've been flirting around with the "given X" part of your answer, I just hadn't pinpointed that yet haha.
I'm just a tad confused at the end - how did you solve that final integral? The integral is equal to the MGF of U, right?
Thanks a million for your persisted help in this problem