I am having trouble answering the following question. Any help would be much appreciated.
Let X1, X2... be i.i.d. random variables according to a distribution with a moment generation function mx(t). Also let N be a random variable which is independent of the Xi's and has a negative binomial distribution with parameters k and q, so that its mgf is:
mn(t)={q/[1-(1-q)e^t]}^k.
Similarly let X1(hat), X2(hat) be i.i.d. random variables according to a distribution with mgf mx(hat)(t) and let N(hat) be independent of the Xi(hat) and have a Poisson distribution such that E(N(hat))=E(N). Define the random sums S= sum of Xi's from i=1 to N, and S(hat) = sum of Xi(hat) from i=1 to N(hat)
Find a choice of mx(hat)(t) such that the distributions of S and S(hat) are the same, and what is the structure and implication of the result.
Let X1, X2... be i.i.d. random variables according to a distribution with a moment generation function mx(t). Also let N be a random variable which is independent of the Xi's and has a negative binomial distribution with parameters k and q, so that its mgf is:
mn(t)={q/[1-(1-q)e^t]}^k.
Similarly let X1(hat), X2(hat) be i.i.d. random variables according to a distribution with mgf mx(hat)(t) and let N(hat) be independent of the Xi(hat) and have a Poisson distribution such that E(N(hat))=E(N). Define the random sums S= sum of Xi's from i=1 to N, and S(hat) = sum of Xi(hat) from i=1 to N(hat)
Find a choice of mx(hat)(t) such that the distributions of S and S(hat) are the same, and what is the structure and implication of the result.