A risk manager rarely has a P&L's full distribution — usually only an estimated mean and variance. The Markov and Chebyshev inequalities are exactly the tools for that situation: distribution-free bounds on tail probabilities that require nothing beyond a mean (Markov) or a mean and variance (Chebyshev). They are the opening move of every limit theorem in this topic — the weak law of large numbers, proved in the next lesson, is a direct corollary of Chebyshev's inequality applied to the sample mean.
Learning objectives. By the end of this note you should be able to:
- state and derive the Markov inequality for a nonnegative random variable;
- state and derive the Chebyshev inequality, and its standard-deviation form;
- compute an upper bound on the variance of a random variable known only to lie in a bounded interval;
- recognise when these bounds are loose versus reasonably tight.
In this lesson
- The Markov inequality
- The Chebyshev inequality
- Bounding an unknown variance
Before you start
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