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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. Markov's inequality can also be used to upper bound the expectation of a non-negative random variable in terms of its distribution function.
In mathematics, the Markov brothers' inequality is an inequality, proved in the 1890s by brothers Andrey Markov and Vladimir Markov, two Russian mathematicians.This inequality bounds the maximum of the derivatives of a polynomial on an interval in terms of the maximum of the polynomial. [1]
Andrey Andreyevich Markov [a] (14 June 1856 – 20 July 1922) was a Russian mathematician best known for his work on stochastic processes. A primary subject of his research later became known as the Markov chain .
The first moment method is a simple application of Markov's inequality for integer-valued variables. For a non-negative, integer-valued random variable X, we may want to prove that X = 0 with high probability.
In mathematical analysis, the Chebyshev–Markov–Stieltjes inequalities are inequalities related to the problem of moments that were formulated in the 1880s by Pafnuty Chebyshev and proved independently by Andrey Markov and (somewhat later) by Thomas Jan Stieltjes. [1]
Jensen's inequality provides a simple lower bound on the moment-generating function: (), where is the mean of X. The moment-generating function can be used in conjunction with Markov's inequality to bound the upper tail of a real random variable X.
They are closely related, and some authors refer to Markov's inequality as "Chebyshev's First Inequality," and the similar one referred to on this page as "Chebyshev's Second Inequality." Chebyshev'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 ...
Any function belongs to L 1,w and in addition one has the inequality ‖ ‖, ‖ ‖. This is nothing but Markov's inequality (aka Chebyshev's Inequality). The converse is not true. For example, the function 1/x belongs to L 1,w but not to L 1.