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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]
Gauss–Markov theorem (brief pointer to proof) ... Markov's inequality (proof of a generalization) Mean value theorem; Multivariate normal distribution (to do)
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.
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 ...
The second-to-last inequality is Markov's inequality. The last inequality holds since ... Proof: It is sufficient to let ...
Chernoff bounds may also be applied to general sums of independent, bounded random variables, regardless of their distribution; this is known as Hoeffding's inequality. The proof follows a similar approach to the other Chernoff bounds, but applying Hoeffding's lemma to bound the moment generating functions (see Hoeffding's inequality).
Proof. Let X be the number cycles of length less than g. The number of cycles of length i in the complete graph on n vertices is ! ()! and each of them is present in G with probability p i. Hence by Markov's inequality we have