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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.
Bennett's inequality, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount; Bhatia–Davis inequality, an upper bound on the variance of any bounded probability distribution; Bernstein inequalities (probability theory) Boole's inequality; Borell–TIS ...
Bennett's inequality; Van den Berg–Kesten inequality; Bernstein inequalities (probability theory) Berry–Esseen theorem; Bobkov's inequality; Boole's inequality; Borell–TIS inequality; Bretagnolle–Huber inequality; BRS-inequality
For instance, to solve the inequality 4x < 2x + 1 ≤ 3x + 2, it is not possible to isolate x in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yielding x < 1 / 2 and x ≥ −1 respectively, which can be combined into the final solution −1 ≤ x < 1 / 2 .
In many cases where the principle could give an exact formula (in particular, counting prime numbers using the sieve of Eratosthenes), the formula arising does not offer useful content because the number of terms in it is excessive. If each term individually can be estimated accurately, the accumulation of errors may imply that the inclusion ...
In probability theory, Bernstein inequalities give bounds on the probability that the sum of random variables deviates from its mean. In the simplest case, let X 1, ..., X n be independent Bernoulli random variables taking values +1 and −1 with probability 1/2 (this distribution is also known as the Rademacher distribution), then for every positive ,
In probability theory, Kolmogorov's inequality is a so-called "maximal inequality" that gives a bound on the probability that the partial sums of a finite collection of independent random variables exceed some specified bound.
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] It is similar to, but incomparable with, one of Bernstein's inequalities.