I need some help in understanding moment generating functions and transformations of continuous random variables. I've found various webpages that provide pieces of the puzzle, but I can't find anything that explains the big picture in terms that are easy to understand for someone who isn't very mathematically inclined. If anyone can provide links to material that can help explain what MGFs are and how to do transformations I would be very happy.
And although I do need to gain an overall understanding of these concepts, there are two specific problems I am wrestling with right now. The first is finding the variance of the geometric distribution by taking the natural log of the moment generating function. According to The Geometric Distribution, the variance for a geometric series should be q/p^2. I have calculated the MGF to be ((p*e^t)/(1-q*e^t)), however, when I calculate the derivative of ln[((p*e^t)/(1-q*e^t))] I get 1/(1-q*e^t), so even when t=0 the value I get for variance is 1/(1-q), when it should be q/p^2. I would appreciate any insight into what I am doing wrong here.
The second problem I keep butting my head up against is how to calculate the density function U of two different density functions X and Y. I dont' even have an idea of how to approach this problem, so I would love any guidance here.