# Thread: Moment generating function problem

1. ## 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 $M_x (t) = e^{3t + 8t^2 }$ , find the moment generating function of the random variable $Z = \frac{1}
{4}(X - 3)$
, 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 $\mu$ and $\sigma ^2$ 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

2. ## moment generating function

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