I'm really struggling with this question:
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)
E(Z) = EX (E(X|Z))
Law of total variance: var(Z) = EX (var(Z|X)) + VarX (E(Z|X))
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.
Expectation and variance of a random number of random variables Watch
- Thread Starter
- 26-01-2010 17:35
- 26-01-2010 19:51
When i =/= j, X_i and X_j are independent, and so E[X_i X_j] = E[X_i]E[X_j] = E[X]^2
When i = j, E[X_i X_j] = E[X^2] = Var[X]+E[X]^2
Use this to find a simple expression for E[(X_1+X_2+...+X_n)^2] (for fixed n) and then use this to find E[Z^2].
- 26-01-2010 19:53
Let X1,...,XN be independent identically distributed random variables, where N is a
non-negative integer valued random variable. Let Z = X1 + ... + XN, assuming that Z = 0 if N = 0.
Find E(Z) and show that
var(Z) = var(N)E(X1)2 + E(N)var(X1)
(Original post by The Muon)
- 26-01-2010 20:20
Let X1,...,XN be independent identically distributed random variables...
(Original post by DFranklin)
- 26-01-2010 21:46