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 mean 1 and variance 1; let B1,B2,… be another sequence of i.i.d. random variables, all independent of N and of A1,A2,…, also with mean 1 and variance 1. Let A= ∑_(i=1)^NAi and B= ∑_(i=1)^NBi

Question

1. Find the following expectations using the law of iterated expectations. Express each answer in terms of m and v, using standard notation.
E[AB]=
E[NA]=
2. Let N^=c1A+c2 be the LLMS estimator of N given A. Find c1 and c2 in terms of m and v.
c1=
c2=

Answer 1

part 1 both answers are v+m^2
part 2
c1 = v/(m+v)
c2 = m^2/(m+v)