Let (Y1, Y2, ...,Yc) denote independentPoissonrandomvariables,withparameters (1,2, ...,c). (a) Explainwhythejointprobabilitymassfunctionfor {Yi} is c i=1 [exp(i)yi i ~yi!]

Let (Y1, Y2, ...,Yc) denote independentPoissonrandomvariables,withparameters

(μ1,μ2, ...,μc).

(a) Explainwhythejointprobabilitymassfunctionfor {Yi} is cΠ

i=1

[exp(−μi)μyi i ~yi!]

for allnonnegativeintegervalues (y1, y2, ...,yc).

(b) Section 3.2.6 explains thatthesum n = Σi Yi also hasaPoissondistribution,withpa-

rameter Σi μi. ForindependentPoissonrandomvariables,ifweconditionon n, explain why {Yi} are nolongerindependentandnolongerhavePoissondistributions.

(c) Showthattheconditionalprobabilityof {yi}, conditionalon Σi yi = n, isthe multinomial

(2.14), characterizedbythesamplesize n and probabilities ™πi = μi~‰Σj μjŽž.