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On the convolution of gamma distributions. (English) Zbl 1069.60012
Summary: We give a formula for the distribution of the sum of $$n$$ independent random variables with gamma distributions. A formula for such a sum was provided by A. M. Mathai [Ann. Inst. Stat. Math. 34, 591–597 (1982; Zbl 0505.60099)]. But it was complicated. H. Jasiulewicz and W. Kordecki [Demonstr. Math. 36, No. 1, 231–238 (2003; Zbl 1017.62105)] derived a formula for the particular case of independent random variables having Erlang distributions by using Laplace transform. Our method is based on elementary computations and our result (Theorem 2.1) is expressed by means of the generalized beta function.

##### MSC:
 60E05 Probability distributions: general theory 60G50 Sums of independent random variables; random walks