Suppose that is a random variable that follows Gamma where i

Suppose that is a random variable that follows Gamma(,), where is an integer and > 0, and suppose that, conditional on , X follows Poisson(). Show that the unconditional distribution of + X is NegativeBinomial(, 1/1+)

Hint: The MGF of X Poisson() is MX(t) = e(et1).

Please also use the MGF of X Gamma(, ), MX(t) = (1 t)^

and

X NegativeBinomial(r,p),   MX(t)= [(pe^t)/(1(1p)et)]^r

Solution

Answer:

We are given E[X 7] = 3, E[(X 7)2 ] = 11, and E[(X 7)3 ] = 15. Expanding the first equation gives E[X 7] = E[X] 7 = µ 7 = 3, and therefore µ = 10. Continuing the calculations, E[(X µ) 2 ] = E[(X 10)2 ] = E [(X 7) 3]2 = E[(X 7)2 6(X 7) + 9] = E[(X 7)2 ] 6E[X 7] + 9 = 11 18 + 9 = 2. E[(X µ) 3 ] = E[(X 10)3 ] = E [(X 7) 3]3 = E[(X 7)3 ] 9E[(X 7)2 ] + 27E[X 7] 27 = 15 99 + 81 27 = 30.