This Chebychev's inequality (there are several) is a simple extension of Markov (by setting phi(x) = |x|^2 - see the Wikipedia article on Markov's inequality).<p>There is another simple extension[0], much less known, of setting phi(x) = exp(-s*x), and taking the infimum over all s; it is often tractable and yields much, much sharper bounds.<p>[0] <a href="https://en.wikipedia.org/wiki/Chernoff_bound" rel="nofollow">https://en.wikipedia.org/wiki/Chernoff_bound</a>
If you have a hard time remembering exactly how Markov inequality goes (like I do), there's a great mnemonic from which you can construct the general version:<p>- if the average person is 6' tall, than at most 10% of the people are taller than 60'.