LLMS estimation with random sums

Let N be a random variable with mean E[N]=m, and Var(N)=v; let A1, A2, ...be a sequence of i.i.d random variables, all independent of N, with mean1

and variance1; letB1, B2,...be another sequence of i.i.d. random variables, all independent ofN and ofA1,A2,..., also with mean1 and variance1.

LetA=i=1NAi and B=i=1NBi.

• Find the following expectations using the law of iterated expectations. Express each answer in terms ofm andv, using standard notation.

E[AB]=

E[NA]=

• Let N^ =c1A + c2 be the LLMS estimator ofN givenA. Findc1 andc2 in terms ofm andv.

c1=

c2=