Thread: Moment generating function problem

Hello this is my first post in this forum. I hope I can give back some help in my spare time. I'll go on with my question.

Given the moment generating function , find the moment generating function of the random variable , and use it to determine the mean and the variance of Z.

I am a quite confused since I don't know exactly what are they asking from me. The only thing that closely resembles an answer that I have come up with is to calculate and using the given generating function and use the standard deviation obtained as the integration limits for finding the mgf of Z. If my approach ok? I'm i totally wrong?

Edit: I should probably add that I calculated the mean and sd and arrived at a mean of 3 and sd of 4. This just doesn't seem right to me.

edit 2: Please disregard. The answer is here

Set ϕ(t)=M_{X}(t)=Ee^{tX} and ψ(t)=M_{Z}(t)=Ee^{tZ}=Ee^{t((X/4)-(3/4))}=Ee^{(t/4)X}e^{-((3t)/4)}=e^{-((3t)/4)}Ee^{(t/4)X}=e^{-((3t)/4)}ϕ((t/4))=e^{-((3t)/4)}e^{3(t/4)+8((t²)/(16))}=e^{((t²)/2)}.
We have ψ′(t)=te^{((t²)/2)} and ψ′′(t)=e^{((t²)/2)}+t²e^{((t²)/2)}, and so EZ=ψ′(0)=0, EZ²=ψ′′(0)=1 and VarZ=EZ²-(EZ)²=1


Best regards, Aurel Spataru