Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability / 7 / a / iii / Solution

By linearity of expectation, $\mathbb{E}(S_n)={\sum }_{i=1}^{n}p=np$. Part (ii) gives $\mathrm{Var}(X_i)=\mathbb{E}(X_i^2)-[\mathbb{E}(X_i){]}^2=p-p^2$. Independent random variables have zero pairwise covariance, so the variance of their sum is the sum of their variances. Therefore