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13934 and other numbers x such that x ≥ 13934 would be an upper bound for S. The set S = {42} has 42 as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that S. Every subset of the natural numbers has a lower bound since the natural numbers have a least element (0 or 1, depending on ...
The bounds often also enclose distributions that are not themselves possible. For instance, the set of probability distributions that could result from adding random values without the independence assumption from two (precise) distributions is generally a proper subset of all the distributions enclosed by the p-box computed for the sum. That ...
The upper and lower bounds on the elastic modulus of a composite material, as predicted by the rule of mixtures. The actual elastic modulus lies between the curves. In materials science , a general rule of mixtures is a weighted mean used to predict various properties of a composite material .
The intuition behind the CDF-based approach is that bounds on the CDF of a distribution can be translated into bounds on statistical functionals of that distribution. Given an upper and lower bound on the CDF, the approach involves finding the CDFs within the bounds that maximize and minimize the statistical functional of interest.
Then has an upper bound (, for example, or ) but no least upper bound in : If we suppose is the least upper bound, a contradiction is immediately deduced because between any two reals and (including and ) there exists some rational , which itself would have to be the least upper bound (if >) or a member of greater than (if <).
The upper bound is sharp: M is always a copula, it corresponds to comonotone random variables. The lower bound is point-wise sharp, in the sense that for fixed u, there is a copula ~ such that ~ = ().
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 previous bounds are upper bounds for each root separately. Landau's inequality provides an upper bound for the absolute values of the product of the roots that have an absolute value greater than one.
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