The sum of a finite number of independent Poisson random variables is a Poisson random variable. .Prove for two Poi r.v.'s.That is, assuming X ? Poi(?) and Y ? Poi(?),

independently, show that S = X + Y ? Poi(? + ?).

Hint: Use the law of total probability with Ex = {X1 = k} to find ps(j) = Ek-o .... Starting with the law of total probability for ps(), and then moving all factors that do not involve ? outside the sum we get ps (j) = > px, (k) px (j - k) = k k! k K = 0 K =0 (j - k)! J! K = 0 Let RHS denote the final right hand side above. Completing the proof from here, which of the following argument proves the claim. (a) RHS - e (1tx2) (with2) ho (1 ) p*(1 - p)j k with p = p1/(/1 + 12) and ER... = 1. (b) RHS = e (#1 +x2) J! " since Ek o () Ming * = 1 (c) Use np = > = /1 + /2 for p = #1/ (M1 + /2) (d) lim; RHS = e (12) (Hitun) (e) none of these