In this exercise, the r.v.s X1,..., Xk are independent with distributions as indicated, and X = X1 +....+ Xk. Then use Exercise 13 in Chapter 10, the inversion formula (Theorem 2), and the appropriate ch.f.s in order to show that:
(i) if Xj ~ B(nj,p),j = 1....... k, then X ~ B(n, p), where n = n1 +....+ nk.
(ii) If Xj ~ P(λj), j = 1.....k, then X ~ P(λ), where λ = λ1 +....+ λk.
(iii) If Xj ~ N(μj,σ2j) j = 1....... k, then X ~ N(μ,σ2), where μ = μ1 + ... + μk and σ2 = σ12 + .... + σk2. Also, c1X1 + ... + ckXk ~ N(c1μ1 +...... + ckμk c12σ12+....+ck2σk2, where c1,..., ck are constants.
(iv) If Xj ~ Gamma with parameters αj and β, j = 1....k then X ~ Gamma with parameters α = α1 + ... + αk and β. In particular, if has the Negative Exponential distribution with parameter λ, j = 1,..., k, then X ~ Gamma with parameters α = l: and β = 1/λ whereas, if Xj ~ Xr2 ,j = 1, ...,k then X ~ Xr2 where r = r1 + .... + rk.