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If $B \sim B(1, \beta) $ is a beta random variable. We know that its $k$ raw moment is

$$ E(B^k) = {\binom {\beta + k} k}^{-1} $$

But how can we calculate $E(B^k)$ when $k > 0$ is not an Integer but a real number? Can we still get a closed format?

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up vote 2 down vote accepted

The formula is essentially the same:

$$E(B^k) = \int_0^1 \beta t^k (1-t)^{\beta-1}\ dt = \beta B(\beta,k+1) = \frac{\Gamma(\beta+1) \Gamma(k+1)}{\Gamma(\beta+k+1)}$$

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