Suppose ξ1,ξ2, . . . are independent and identically distributed random variables having mean μ and variance σ2. Form the random sum SN = ξ1+· · ·+ξN.
(a) Derive the mean and variance of SN when N has a Poisson distribution with parameter λ.
(b) Determine the mean and variance of SN when N has a geometric distribution with mean λ = (1−p)/p.
(c) Compare the behaviors in (a) and (b) as λ→[infinity].