Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 12 / 1 / i / Solution

Let $Z$ be a real random variable with finite expectation $m$ and finite variance ${\sigma }^2$. The Chebyshev inequality isThe proof is the pointwise comparisonTaking expectations and using the definition of variance givesDivision by the positive number $t^2$ proves the Chebyshev inequality, including the case ${\sigma }^2=0$. Equivalently, this is the Markov inequality applied to the nonnegative random variable $(Z-m{)}^2$.