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The term Chebyshev's inequality may also refer to Markov's inequality, especially in the context of analysis. 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."
In probability theory, the coupon collector's problem refers to mathematical analysis of "collect all coupons and win" contests. It asks the following question: if each box of a given product (e.g., breakfast cereals) contains a coupon, and there are n different types of coupons, what is the probability that more than t boxes need to be bought ...
In fact, Chebyshev's proof works so long as the variance of the average of the first n values goes to zero as n goes to infinity. [15] As an example, assume that each random variable in the series follows a Gaussian distribution (normal distribution) with mean zero, but with variance equal to 2 n / log ( n + 1 ) {\displaystyle 2n/\log(n+1 ...
Consider the sum = = = (). The two sequences are non-increasing, therefore a j − a k and b j − b k have the same sign for any j, k.Hence S ≥ 0.. Opening the brackets, we deduce:
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]
Chebyshev's sum inequality, about sums and products of decreasing sequences Chebyshev's equioscillation theorem , on the approximation of continuous functions with polynomials The statement that if the function π ( x ) ln x / x {\textstyle \pi (x)\ln x/x} has a limit at infinity, then the limit is 1 (where π is the prime-counting function).
In probability theory, the multidimensional Chebyshev's inequality [1] is a generalization of Chebyshev's inequality, which puts a bound on the probability of the event that a random variable differs from its expected value by more than a specified amount.
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.