Suppose that X i ∼ Gamma(α i , β) independently for i = 1, . . . , N. The mgf(moment generating function) of X i is M Xi (t) = (1 − (t/β) ) −αi . (a)Use the mgf of X i to derive the mgf of ∑ i=1 X i ....

Suppose that X_i ∼ Gamma(α_i , β) independently for i = 1, . . . , N. The mgf(moment generating function) of X_iis M_Xi(t) = (1 − (t/β) )^−αi . (a)Use the mgf of X_i to derive the mgf of ∑_i=1 X_i .

Determine the distribution of ∑_i=1 X_i based on its mgf.