Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / M / 3 / c / Solution

Let ${\overline{X}}_n=n^{-1}{\sum }_i{X}_i$ and write ${\sigma }^2=\mathrm{Var}(X_1)<\infty$. Independence gives $\mathbb{E}{\overline{X}}_n=\mu$ and $\mathrm{Var}({\overline{X}}_n)={\sigma }^2/n$. Apply part (b) to the nonnegative variable $({\overline{X}}_n-\mu {)}^2$, which is the Chebyshev inequality:Thus ${\overline{X}}_n\to \mu$ in probability, proving the finite-variance weak law of large numbers.