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=∑Ni=1Ai and B=∑Ni=1Bi.
1) Find the E[AB] and E[NA] using the law of iterated expectations.
2) Let N^=c1A+c2 be the LLMS estimator of N given A. Find c1 and c2 in terms of p.