I am stuck on this problem:
I am given the CDF, and unless there is a shortcut I am not remembering, I need to find the PDF before I can get the moment generating function and solve the problem.
In this case, it looks like the only part that "matters" is the second part, because if f(x) = (d/dx)F(x), the x<1 and x>=2 pieces are just zero. Thus, I think I am left with this PDF: f(x) = (x-1) for 1 <= x < 2. To get the moment generating function, then, I would do this, right? :
If I calculated right, this integral gives me:
I think the second derivative of this would have t stuff in the bottom, so how could I evaluate it at zero (if that is what is being asked for)? Otherwise, where did I go wrong in approaching this?