In probability theory, Markov's inequality gives an upper bound on the probability that a non-negative random variable is greater than or equal to some positive constant. Markov's inequality is tight in the sense that for each chosen positive constant, there exists a random variable such that the inequality is in fact an equality. It is named after the Russian mathematician Andrey Markov, although it appeared earlier in the work of Pafnuty Chebyshev (Markov's teacher), and many sources, especially in analysis, refer to it as Chebyshev's inequality (sometimes, calling it the first Chebyshev inequality, while referring to Chebyshev's inequality as the second Chebyshev inequality) or Bienaymé's inequality. 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. Mitzenmacher and Upfal note that Markov's inequality "is often too weak to yield useful results, but it is still a fundamental tool in developing more sophisticated bounds". Markov's inequality can also be used to upper bound the expectation of a non-negative random variable in terms of its distribution function.
Statement If X is a nonnegative random variable and a > 0, then the probability that X is at least a is at most the expectation of X divided by a:
P ( X ≥ a ) ≤ E ( X ) a . {\displaystyle \operatorname {P} (X\geq a)\leq {\frac {\operatorname {E} (X)}{a}}.}
When E ( X ) > 0 {\displaystyle \operatorname {E} (X)>0} , we can take a = a ~ ⋅ E ( X ) {\displaystyle a={\tilde {a}}\cdot \operatorname {E} (X)} for a ~ > 0 {\displaystyle {\tilde {a}}>0} to rewrite the previous inequality as
P ( X ≥ a ~ ⋅ E ( X ) ) ≤ 1 a ~ . {\displaystyle \operatorname {P} (X\geq {\tilde {a}}\cdot \operatorname {E} (X))\leq {\frac {1}{\tilde {a}}}.}
In the language of measure theory, Markov's inequality states that if (X, Σ, μ) is a measure space, f {\displaystyle f} is a measurable extended real-valued function, and ε > 0, then
μ ( { x ∈ X : | f ( x ) | ≥ ε } ) ≤ 1 ε ∫ X | f | d μ . {\displaystyle \mu (\{x\in X:|f(x)|\geq \varepsilon \})\leq {\frac {1}{\varepsilon }}\int _{X}|f|\,d\mu .}
This measure-theoretic definition is sometimes referred to as Chebyshev's inequality.
Extended version for nondecreasing functions If φ is a nondecreasing nonnegative function, X is a (not necessarily nonnegative) random variable, and φ(a) > 0, then
P ( X ≥ a ) ≤ E ( φ ( X ) ) φ ( a ) . {\displaystyle \operatorname {P} (X\geq a)\leq {\frac {\operatorname {E} (\varphi (X))}{\varphi (a)}}.}
An immediate corollary, using higher moments of X supported on values larger than 0, is
P ( | X | ≥ a ) ≤ E ( | X | n ) a n . {\displaystyle \operatorname {P} (|X|\geq a)\leq {\frac {\operatorname {E} (|X|^{n})}{a^{n}}}.}
The respective measure-theoretic versions are
μ ( { x ∈ X : | f ( x ) | ≥ ε } ) ≤ 1 φ ( ε ) ∫ X φ ( | f | ) . {\displaystyle \mu (\{x\in X:|f(x)|\geq \varepsilon \})\leq {\frac {1}{\varphi (\varepsilon )}}\int _{X}\varphi (|f|).}
and
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