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A characterization of the gamma distribution. (English) Zbl 0579.60013

This note first establishes that if \(\alpha >0\), there exists a positive random variable (rv) Y such that (*) \(E(Y^ s)=(1+(s/\alpha))^{(\alpha +s)}\), for all \(s>0\). The main result shows that if X and Y are independent rvs and condition (*) holds, then XY exp(-X/\(\alpha)\) and X are identically distributed if, and only if, X is a gamma rv with parameter \(\alpha\).
Reviewer: H.N.Nagaraja

MSC:

60E05 Probability distributions: general theory
62E10 Characterization and structure theory of statistical distributions
60E10 Characteristic functions; other transforms
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