Why is it that if for independent random variables $X_i \ni i $ is finite, then the expectation of:
$$E\bigg[Z^{\sum_i X_i}\bigg] = \Pi_i \ E\bigg[Z^{X_i}\bigg]$$
Would this be true for the expectation of any function of $X_i$? And what conditions must be satisfied for this to be true? i.e. continuity etc...
So My problem is that $X_i$ are independent. Not the functions of them. Is there a theorem or property that says that joint functions of independent variables equal to the product of the partials in case of independent r.v. If so, what conditions do we put on the functions? Thanks
Maybe I'm missing something subtle here, but $Z^{x_1+x_2}=Z^{x_1}Z^{x_2}$. So it's just a rearrangement, writing the Z factor as a multiplication instead of a sum in the exponent. If they are independent, then $$E[Z^{x_1}Z^{x_2}...]=E[Z^{x_1}]\cdot E[Z^{x_2}]...$$which is sort of the definition of independence (see: https://en.wikipedia.org/wiki/Independence_(probability_theory).