Is it a correct way to write a moment generating function?

Question

When I write the moment generating functions, can I write $M_{aX}(t)$ instead of $M_{X}(at)$ ?

Thank you really much!

Answer 1

I mean, it's true that $$M_{aX}(t) = \mathbb E[e^{(aX)t}] = \mathbb E[e^{atX}] = M_X(at).$$ This is a special case of the more general rule for linear combinations: if $S = \sum_{i=1}^n a_i X_i$, where the $X_i$ are independent random variables, then $$M_S(t) = \prod_{i=1}^n M_{X_i}(a_i t).$$ I guess there's connotations that, writing $M_{aX}(t)$ as opposed to $M_X(at)$, you care about the distribution of $aX$ as opposed to the distribution of $X$. But either is equally mathematically valid.

...If you wanted to be really weird, you could write $M_X(t) = M_{tX}(1)$, but nobody does that.