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In probability theory, Cantelli's inequality (also called the Chebyshev-Cantelli inequality and the one-sided Chebyshev inequality) is an improved version of Chebyshev's inequality for one-sided tail bounds. [1] [2] [3] The inequality states that, for >,
The bounds these inequalities give on a finite sample are less tight than those the Chebyshev inequality gives for a distribution. To illustrate this let the sample size N = 100 and let k = 3. Chebyshev's inequality states that at most approximately 11.11% of the distribution will lie at least three standard deviations away from the mean.
The theorem refines Chebyshev's inequality by including the factor of 4/9, made possible by the condition that the distribution be unimodal. It is common, in the construction of control charts and other statistical heuristics, to set λ = 3 , corresponding to an upper probability bound of 4/81= 0.04938..., and to construct 3-sigma limits to ...
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 ...
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.
One-sided relationships can trigger or exacerbate stress, anxiety and depression. "Stress can cause people to feel exhausted and lead to significant physical and emotional symptoms," Dr. Trotter says.
Such inequalities are of importance in several fields, including communication complexity (e.g., in proofs of the gap Hamming problem [13]) and graph theory. [14] An interesting anti-concentration inequality for weighted sums of independent Rademacher random variables can be obtained using the Paley–Zygmund and the Khintchine inequalities. [15]
Harry Styles is seen after the funeral service for One Direction singer Liam Payne, at St. Mary's Church in Amersham, Buckinghamshire, England, Nov. 20, 2024. / Credit: Andrew Matthews/PA Images/Getty