K
kylie jane
Member
So we have
S=X_1+X_2+...+X_n
E(S|N=n)=nE(X)
var(S|N=n) = nvar(X)
E(S)=E(E(S|N)) = E(NE(X))=E(N)E(E(X))=E(N)E(X)
then for var(S)
var(S) = E(var(S|N))+var(E(S|N))
= E(Nvar(X))+var(NE(X))
= E(N)var(X) +var(N)[E(X)]^2
How do we get that
var(NE(X))=var(N)[E(X)]^2
S=X_1+X_2+...+X_n
E(S|N=n)=nE(X)
var(S|N=n) = nvar(X)
E(S)=E(E(S|N)) = E(NE(X))=E(N)E(E(X))=E(N)E(X)
then for var(S)
var(S) = E(var(S|N))+var(E(S|N))
= E(Nvar(X))+var(NE(X))
= E(N)var(X) +var(N)[E(X)]^2
How do we get that
var(NE(X))=var(N)[E(X)]^2