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Markov's inequality (and other similar inequalities) relate probabilities to expectations, and provide (frequently loose but still useful) bounds for the cumulative distribution function of a random variable.
An example of an application of Markov's inequality is the fact that (assuming incomes are non-negative) no more than 1/5 of the population can have more than 5 times the average income.
Statement
If X is a nonnegative random variable and a > 0, then
Extended version for monotonically increasing functions
If φ is a monotonically increasing function for the nonnegative reals, X is a random variable, a ≥ 0, and φ(a) > 0, then
Proofs
We separate the case in which the measure space is a probability space from the more general case because the probability case is more accessible for the general reader.
Proof In the language of probability theory
For any event , let be the indicator random variable of , that is, if occurs and otherwise.
Using this notation, we have if the event occurs, and if . Then, given ,
which is clear if we consider the two possible values of . If , then , and so . Otherwise, we have , for which and so .
Since is a monotonically increasing function, taking expectation of both sides of an inequality cannot reverse it. Therefore,
Now, using linearity of expectations, the left side of this inequality is the same as
Thus we have
and since a > 0, we can divide both sides by a.
In the language of measure theory
We may assume that the function is non-negative, since only its absolute value enters in the equation. Now, consider the real-valued function s on X given by