What is that $t$ in $E[e^{itX}]$?

Question

I'm looking at the Characteristic Function Of A Random Variable. I don't understand why it's less than or equals to 1, since I see Mean Function on Random Variable ($E[e^{iX}]$), which might be bigger than 1. And t, which I don't understand in the first place is my 1st suspect. Could someone explain me the role of that $t$?

Referring to my latest comment after research on my own question, on How do the complex functions get built (here: How from Fourier Transform with imaginary numbers we get the one with real numbers? ).

Answer 1

This comes from two facts:

1. $\left|E[X]\right| \leq E[|X|]$ for any complex-valued random $X$
2. If $x$ is a real number, then $\left|e^{ix}\right| = 1$

In particular, this means $\left|E[e^{iX}]\right| \leq 1$ cannot be larger than $1$ when $X$ is real-valued.

As for your later question, the $t$ allows us to see the characteristic as a function of $t,$ so we may, among other things, integrate the characteristic function (e.g. the inverse transform) and differentiate the characteristic function (e.g. to find moments)