The other day I was solving some problems relating to distribution functions and expected values.
There was a question asking about the "meaning" of the moment generating function depending on a random variable.
There was a random variable A, which I solved and the $E(X^\alpha) = $undefined and $E(X^\beta) = ∞$.
Does this fact tell us anything about the moment generating function?
I have come across the fact that when $E(X^\alpha)$ is undefined, then the "$\alpha$"th moment of X does not exist. However I do not know what is means for the moment generating function of X when $E(X^\beta)$ = ∞
It would be nice if I can get an explanation about this.
Best Answer
Note that the MGF is defined as $\int_{-\infty}^{\infty} e^{tw}f_W(w)dw$ where $f_W$ is the PDF of the function and is valid for all values of $t$ for which the expected value exists. To get an i'th moment, you take i derivatives of this integral wrt t, and plug in t = 0.
If you're not getting any valid number, then that i'th derivative does not exist, and so the i'th moment does not exist.