A statistical characterization of the gamma function (2022-12-04)

The gamma function \(\Gamma(z)\) is a weird thing for the mortal other than number theorists; \[\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} dt.\]Its most usual introduction is that it is the factorial, extended to the reals; for a positive integer \(n\), \[\Gamma(n) = (n-1)!.\]

This fact is nice, but provides no intuition for any practical application. The classic written by Emil Artin is highly recommended for those who want deeper understanding.

Here we show an alternative, unpopular characterization of the gamma function of the real argument from a statistical point of view. The statement is that, for a positive real \(z\), \(\Gamma(z)\) is the expected value of an exponential random variable raised to the power \(z-1\), i.e., \[\Gamma(z) = \mathbb{E}[X^{z-1}]\] where \(X\) is an exponential random variable of unit rate \(\lambda = 1\). It is proved by the law of the unconscious statistician.

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