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Expectation and variance of a random number of random variables

  1. Jan 26, 2010 #1
    1. The problem statement, all variables and given/known data

    Let X1...XN be independent and identically distributed random variables, N is a non-negative integer valued random variable. Let Z = X1 + ... + XN (assume when N=0 Z=0).
    1. Find E(Z)
    2. Show var(Z) = var(N)E(X1)2 + E(N)var(X1)

    2. Relevant equations

    E(Z) = EX (E(X|Z))
    Law of total variance: var(Z) = EX (var(Z|X)) + VarX (E(Z|X))

    3. The attempt at a solution

    1. I think I have managed this, I got E(N)E(X)
    2. I'm unsure how to tackle this one, I know var(Z) = E(Z2) - E(Z)2, and I know E(Z)2 but I don't know how to calculate the other, or if I should be using the equation above, and if so, how.
  2. jcsd
  3. Jan 26, 2010 #2
    2. follows immediately from the total variance formula.

    var Z=E ( var (Z|N)) + var (E (Z|N))

    E ( var (Z|N))=E(N var (X1))=var(X1) E(N) -- by fixing N, Z is a sum of fixed number of Xi-s

    var (E (Z|N))=var (N E(X1))=[E(X1)]^2*var(N)
  4. Jan 27, 2010 #3
    Thanks :)
  5. Jan 27, 2010 #4
    The question has a second part which I've just attempted but am also struggling with:

    The number of calls received each day at an emergency centre, N, has a poisson distribution, with mean [tex]\mu[/tex]. Each call has probability p of requiring immediate police response. Let Z be the random bariable representing the number of calls involving police response.

    a) What is the probability mass function of Z given N=n?
    b) What is the probability generating function of Z, given that we know N=n?
    c) Find E(sZ) (use the partition theorem for expectation)
    d) Deduce the unconditional distribution of Z and write down var(Z).
    e) How is this related to the formula we already worked out?

    a) If we know N=n, can we model Z on a binomial distribution with parameters (n,p) so pZ(n) = ([tex]^{N}_{n}[/tex])pn(1-p)N-n =pn
    b) The binomial p.g.f. is (q+ps)n
    c) E(sZ) = [tex]\sum[/tex][tex]^{N}_{n=1}[/tex] E(sZ | N=n)P(N=n) = [tex]\sum[/tex][tex]^{N}_{n=1}[/tex] (q+ps)npn = [tex]\sum[/tex][tex]^{N}_{n=1}[/tex] (pq + p2s)n

    I don't understand where I go from here if any of that is correct. I'm not sure that I should have modelled Z on a binomial r.v.
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