Moment Generating Functions
So, the moment generating function of a random variable is a quick way to derive moments of the random variable through taking multiple derivatives and evaluating at t = 0.
Is there ever an instance where you would evaluate it at some t other than 0? What is the significance of doing so?
EDIT: I guess the I have the same question for joint distributions as well... I'm guessing they have similar significance but clearly the joint probabilities are a litttttle bit more complicated.
Re: Moment Generating Functions
Think of the mgf as a sum, like here : Moment-generating function - Wikipedia, the free encyclopedia (the formula with 1+tm1+tēm2/2...). If you differentiate with respect to t, you'll get m1+tm2+... and so on. So if you don't set t=0, you'll have a residue that won't make you get the moments.