Past exam of the year 1 of the computer science course of the University of Oxford / 2014 / Discrete mathematics, continuous mathematics and probability / 8 / b / i / Solution

Given $X=k$, the score $Y$ has a binomial distribution with success probability $1/2$. Its conditional expected value and variance are $k/2$ and $k/4$. ThusThis is binomial thinning of the first-round count; the second formula uses the identity $\mathbb{E}[Y^2\mid X]=\mathrm{Var}(Y\mid X)+\mathbb{E}[Y\mid X{]}^2$.