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.

Thanks,

James