Let be two i.i.d. random variables and is a variable. How to show that the distribution functions of and are identical?
Let's call Z the rv .
We'll use mgf's and conditional expectations :
(expectations with the rv's in subscript mean that it's the expectation with respect to the distributions of the rv's in the subscript - to avoid any confusion)
And then comparing to the other one...
Same mgf implies same distribution.
P.S. : being of the lazy type, some steps have been put in silent mode