Joint pdf question and mgf

silkdigital

Hi guys,

I'm really stuck on the following questions, not sure as to how to approach it:

Let X and Y be random variables for which the joint pdf is as follows:

f(x,y) = 2(x+y) for 0 <= x <= y <= 1
and 0 otherwise.

Find the pdf of Z = X + Y

And also:

Suppose that X is a random variable for which the mgf is as follows:

/u(t) = e^(t^2 + 3t) for minus infinity < t < infinity

Find the mean and variance for X.
I know that the answers are 3 and 2 respectively, but was unsure how they got to the answer, do I need to integrate by parts?

Any help would be appreciated! Thanks guys :)

mathman

I'll address the second question only. The moments are obtained from the moment generating function by simply taking derivatives and setting t = 0. As you must be aware, the variance is the second moment minus square of first moment.

silkdigital

Figured out second question now, pretty straightforward in hindsight. Any help on the first one? ;)

chiro

Have you tried a transformation? Let u = x + y. Now use that transformation to get a integral in terms of u, take into account limits and then use transformation theorem to relate g(u) = 2(x+y) = 2u to another PDF f(u) which represents the distribution of Z.