Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability / 9 / c / i / Solution

Write the card values as $a_1,\dots ,a_{1000}$. Uniform selection givesThus $\mathbb{E}[X]=0$ and $\mathrm{Var}(X)=1$, by the variance identity $\mathrm{Var}(X)=\mathbb{E}[X^2]-\mathbb{E}[X{]}^2$.