Let Y1, Y2,...,Yn be independent Poisson random variables with means ƛ1, ƛ2, ...,ƛn, respectively.
Find the: Probability function of Yi using the method of moment generating functions.
I am unsure of how to do this question so any help would be great.
Well I know that the mgf of Poisson is: exp[ƛ(e^t - 1)]
I also realize that since they are indepedent you can multiply them together. ie. My1(t) = exp[ƛ1(e^t - 1)], My2(t) = exp[ƛ2(e^t - 1)],....,Myn(t) = exp[ƛn(e^t - 1)] ------> My1(t)*My2(t)*...*Myn(t)
I just don't know how that when multiplying these together you obtain the Poisson probability function: p(n) = (ƛ^n*e^-ƛ)/n!