Inégalité de Markov

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, 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. 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: Let (where ); then we can rewrite the previous inequality as In the language of measure theory, Markov's inequality states that if (X, Σ, μ) is a measure space, is a measurable extended real-valued function, and ε > 0, then This measure-theoretic definition is sometimes referred to as Chebyshev's inequality. If φ is a nondecreasing nonnegative function, X is a (not necessarily nonnegative) random variable, and φ(a) > 0, then An immediate corollary, using higher moments of X supported on values larger than 0, is 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. where is larger than or equal to 0 as the random variable is non-negative and is larger than or equal to because the conditional expectation only takes into account of values larger than or equal to which r.v. can take. Hence intuitively , which directly leads to . Method 1: From the definition of expectation: However, X is a non-negative random variable thus, From this we can derive, From here, dividing through by allows us to see that Method 2: For any event , let be the indicator random variable of , that is, if occurs and otherwise.