Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2024 / Paper 1 / A1 / b / Solution

For a moment of order $q$ with $\theta +q>-1$, integration of the probability density function givesThus the expected value and variance areThe cumulative distribution function on $(0,1)$ is $F(x)=x^{\theta +1}$. Its unique median $m$ satisfies $F(m)=1/2$, giving