Markov's inequality

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In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant. It is named after the Russian mathematician Andrey Markov.

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.

1 Statement
2 Proofs
- 2.1 Special case: probability theory
- 2.2 General case: measure theory
3 Examples

[edit] Statement

In the language of measure theory, Markov's inequality states that if (X,Σ,μ) is a measure space, f is a measurable extended real-valued function, and t > 0, then

$\mu(\{x\in X|\,|f(x)|\geq t\}) \leq {1\over t}\int_X f\,d\mu.$

For the special case where the space has measure 1 (i.e., it is a probability space), it can be restated as follows: if X is any random variable and a > 0, then

$\textrm{Pr}(|X| \geq a) \leq \frac{\textrm{E}(|X|)}{a}.$

[edit] 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.

[edit] Special case: probability theory

For any event E, let I_E be the indicator random variable of E, that is, I_E = 1 if E occurs and = 0 otherwise. Thus I_(X ≥ a) = 1 if the event X ≥ a occurs, and I_(X ≥ a) = 0 if X < a. Then

$aI_{(X \geq a)} \leq X.\,$

Therefore

$\operatorname{E}(aI_{(|X| \geq a)}) \leq \operatorname{E}(|X|).\,$

Now observe that the left side of this inequality is the same as

$a\operatorname{E}(I_{(|X| \geq a)})=a\Pr(|X| \geq a).\,$

Thus we have

$a\Pr(|X| \geq a) \leq \operatorname{E}(|X|)\,$

and since a > 0, we can divide both sides by a.

[edit] General case: measure theory

For any measurable set A, let 1_A be its indicator function, that is, 1_A(x) = 1 if x ∈ A, and 0 otherwise. If A_t is defined as A_t = {x ∈ X| |f(x)| ≥ t}, then

$0\leq t\,1_{A_t}\leq |f|1_{A_t}\leq |f|.$

Therefore

$\int_X t\,1_{A_t}\,d\mu\leq\int_{A_t}|f|\,d\mu\leq\int_X |f|\,d\mu.$

Now, note that the left side of this inequality is the same as

$t\int_X 1_{A_t}\,d\mu=t\mu(A_t).$

Thus we have

$t\mu(\{x\in X|\,|f(x)|\geq t\}) \leq \int_X|f|\,d\mu,$

and since t > 0, both sides can be divided by t, obtaining

$\mu(\{x\in X|\,|f(x)|\geq t\}) \leq {1\over t}\int_X|f|\,d\mu.$

Q.E.D.

[edit] Examples

Markov's inequality is used to prove Chebyshev's inequality.
If X is a non-negative integer valued random variable, as is often the case in combinatorics, then taking a = 1 in Markov's inequality gives that $\textrm{Pr}(X \neq 0) \leq \textrm{E}(X).$ If X is the cardinality of some set, then this proves that that set is not empty. Thus one gets existence proofs.