Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability / 7 / a

Multiplying the beta distribution density by $x$ gives a constant multiple of the normalized density with parameters $(p+1,q)$:Integrating and using the fact that the latter probability density function integrates to one givesThis is a normalization-ratio argument, requiring no explicit integration of the polynomial density.

For $q>1$, decreasing the second parameter by one gives another valid beta distribution density andNormalization therefore givesThe restriction $q>1$ is essential for this finite expected value; when $q=1$, the singularity near $1$ instead makes the expectation infinite.