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Chebyshev's inequality then follows by dividing by k 2 σ 2. This proof also shows why the bounds are quite loose in typical cases: the conditional expectation on the event where | X − μ | < kσ is thrown away, and the lower bound of k 2 σ 2 on the event | X − μ | ≥ kσ can be quite poor.
In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if ...
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 mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. [1] It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions.
the most common choice for function h being either the absolute value (in which case it is known as Markov inequality), or the quadratic function (respectively Chebyshev's inequality). Another useful result is the continuous mapping theorem : if T n is consistent for θ and g (·) is a real-valued function continuous at point θ , then g ( T n ...
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.
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 ...
It is a sharper bound than the first- or second-moment-based tail bounds such as Markov's inequality or Chebyshev's inequality, which only yield power-law bounds on tail decay. However, when applied to sums the Chernoff bound requires the random variables to be independent, a condition that is not required by either Markov's inequality or ...