Let X1,…,Xn be independent, and put Yn := X1 +···+Xn.
(a) If Xi ∼ N(mi,σ2 i ), show that Yn ∼ N(m,σ2), and identify m and σ2. In other words, “The sum of independent Gaussian random variables is Gaussian.”
(b) If Xi ∼ Cauchy(λi), show that Yn ∼ Cauchy(λ), and identify λ. In other words, “The sum of independent Cauchy random variables is Cauchy.”
(c) If Xi is a gamma random variable with parameters pi and λ (same λ for all i), show that Yn is gamma with parameters p and λ, and identify p. In other words, “The sum of independent gamma random variables (with the same scale factor) is gamma (with the same scale factor).”
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