Why do characteristic functions use $e^{ix}$ and not $e^{-ix}$? Does it matter?

Question

I've heard the characteristic function be described as the Fourier-Stieltjes Transform of the distribution measure of a r.v., but I was curious as to why it's written as $E[e^{ix}]$ and not the typical $E[e^{-ix}]$ that I'm used to seeing.

Is there a purpose or is it just difference in notation?

Answer 1

I think it reflects the fact that even for the Fourier transform there is no clear convention.. Some use $e^{ix}$, other $e^{-ix}$ or $e^{2i\pi x}$

you can read more about it on wikipedia

The main properties are of course the same, there is going to be a difference however in the actual calculations