Chebyshev's Inequality
Chebyshev's inequality, named after Pafnuty Chebyshev, states that if
and
then the following inequality holds:
.
On the other hand, if and
then:
.
Proof
Chebyshev's inequality is a consequence of the Rearrangement inequality, which gives us that the sum is maximal when
.
Now, by adding the inequalities:
...
we get the initial inequality.