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In probability theory, Popoviciu's inequality, named after Tiberiu Popoviciu, is an upper bound on the variance σ 2 of any bounded probability distribution.Let M and m be upper and lower bounds on the values of any random variable with a particular probability distribution.
The Vysochanskij–Petunin inequality generalizes Gauss's inequality, which only holds for deviation from the mode of a unimodal distribution, to deviation from the mean, or more generally, any center. [42] If X is a unimodal distribution with mean μ and variance σ 2, then the inequality states that
This theorem makes rigorous the intuitive notion of probability as the expected long-run relative frequency of an event's occurrence. It is a special case of any of several more general laws of large numbers in probability theory. Chebyshev's inequality. Let X be a random variable with finite expected value μ and finite non-zero variance σ 2.
Hoeffding's inequality was proven by Wassily Hoeffding in 1963. [1] Hoeffding's inequality is a special case of the Azuma–Hoeffding inequality and McDiarmid's inequality. It is similar to the Chernoff bound, but tends to be less sharp, in particular when the variance of the random variables is small. [2]
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. ... (by Jensen's inequality), ...
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.
Consider the sum, Z, of two independent binomial random variables, X ~ B(m 0, p 0) and Y ~ B(m 1, p 1), where Z = X + Y.Then, the variance of Z is less than or equal to its variance under the assumption that p 0 = p 1 = ¯, that is, if Z had a binomial distribution with the success probability equal to the average of X and Y 's probabilities. [8]
In mathematics, the Bhatia–Davis inequality, named after Rajendra Bhatia and Chandler Davis, is an upper bound on the variance σ 2 of any bounded probability distribution on the real line. Statement