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. Probability theory is the mathematical study of phenomena characterized by randomness or uncertainty. ... In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. ... This article does not cite its references or sources. ... A negative number is a number that is less than zero, such as −3. ... Partial plot of a function f. ... A random variable is a mathematical function that maps outcomes of random experiments to numbers. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... Andrey (Andrei) Andreyevich Markov (Russian: ) (June 14, 1856 N.S. – July 20, 1922) was a Russian mathematician. ...

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. In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than...

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 In mathematics, a measure is a function that assigns a number, e. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics, measurable functions are well-behaved functions between measurable spaces. ... The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ...

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

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.

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, given a>0,

Therefore

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

Thus we have

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

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 In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ...

Therefore

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

Thus we have

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

Q.E.D. Q.E.D. is an abbreviation of the Latin phrase (literally, which was to be demonstrated). In simple terms, the use of this Latin phrase is to indicate that something has been definitively proven. ...

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 If X is the cardinality of some set, then this proves that that set is not empty. Thus one gets existence proofs.