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# Expectation and variance of a random number of random variables watch

1. 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.
2. 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].
3. 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)
4. (Original post by The Muon)
Let X1,...,XN be independent identically distributed random variables...
See above...
5. (Original post by DFranklin)
See above...
My bad for the posting the same topic, should learn to use the search

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