Mixed power moments of a beta distribution ($\mathbb{E}[X^s(1-X{)}^t]=B(\alpha +s,\beta +t)/B(\alpha ,\beta )$)

For $X\sim \mathrm{Beta}(\alpha ,\beta )$ with positive parameters, and real $s>-\alpha$, $t>-\beta$, multiply its density by $x^s(1-x{)}^t$. The resulting integral is a beta function with shifted parameters, henceThe restrictions on the powers are exactly the endpoint-integrability conditions. This includes ordinary moments, reciprocal moments when they exist, and quantities involving both the occupied fraction $X$ and remaining fraction $1-X$. In particular, $s=1,t=0$ gives $\mathbb{E}[X]=\alpha /(\alpha +\beta )$.^[ref]