Fractional Moments of the Geometric Distribution

In one of my papers, I needed an upper bound on the fractional moments of the geometric distribution. The integer moments have a nice-looking upper bound involving factorials. In addition, Mathematica showed that the same upper bound holds for fractional moments once we use Gamma function instead of factorials. The problem was, I could not prove it. Worse, I could not find a proof after much looking on the Internet.

This post contains a proof that for any real \lambda > 1, the \lambdath moment of a geometric random variable with success probability \displaystyle p \geq 1/2 is at most \displaystyle \Gamma(\lambda + 1)/p^\lambda.

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