Question

Let X₁, X₂, ... be i.i.d. r.v. and N an independent nonnegative integer valued r.v. Let S_N=X₁ +...+ X_N.
Assume that the m.g.f. of the X_i, denoted M_X(t), and the m.g.f. of N, denoted M_N(t) are finite in some interval (-δ, δ) around the origin.
1. Express the m.g.f. M_{S_N}(t) of S_N in terms ofM_X(t) and M_N(t).
2. Give an alternate proof of Wald's identity by computing the expectation E[S_N] as M'_{S_N}(0).
3. Express the second moment E[S_N²] in terms of the moments of X and N.
4. Express the variance of S_N in terms of the means and variances of X and N.

Answer 1

a)

since X1, X2, ... be i.i.d. r.v. and N an independent nonnegative integer valued r.v. thus

b)  

thus satisfying the wald's identity  

c)

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d)

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