# Math Help - Proof of MGF Theorm

1. ## Proof of MGF Theorm

I'm stuck on trying to prove three theorms in the moment generating function below. The image is a screen shot of what I've done (I believe the first part I've proven, but the second and third theorms I don't really understand how to prove after it out to the points in the screenshot. Can someone show me how to prove them? Thanks!

2. Part 2: Define Y = Xb. $M_{Xb} (t) = M_Y (t) = E[e^{Yt}] = E[e^{Xbt}] = E[e^{X(bt)}] = M_X (bt)$. Remember that when we write down $M_X (t)$, t is just some argument. If we wanted to evaluate $M_X (bt)$, we would just use the definition of the MGF to end up with $E[e^{X(bt)}]$.

Part 3: Do the same thing basically, remembering that, with respect to the expectation operator, everything other than X is a constant, and can be moved in an out with the usual rules. You should also use this method to redo part 1, since your method of just replacing X with whatever function of X you want hasn't exactly been justified.