How Do You Find the PDF of Z and Calculate Mean and Variance Using 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]

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 :)

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]

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

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.