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The main objective of interval arithmetic is to provide a simple way of calculating upper and lower bounds of a function's range in one or more variables. These endpoints are not necessarily the true supremum or infimum of a range since the precise calculation of those values can be difficult or impossible; the bounds only need to contain the function's range as a subset.
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 ...
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.
In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events. This inequality provides an upper bound on the probability of occurrence of at least one ...
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 C ~ {\displaystyle {\tilde {C}}} such that C ~ ( u ) = W ( u ) {\displaystyle {\tilde {C}}(u)=W(u)} .
The rational number line Q does not have the least upper bound property. An example is the subset of rational numbers = {<}. This set has an upper bound. However, this set has no least upper bound in Q: the least upper bound as a subset of the reals would be √2, but it does not exist in Q.
= [0.2, 0.25] × [0.1, 0.3] = [0.2 × 0.1, 0.25 × 0.3] = [0.02, 0.075] so long as A and B can be assumed to be independent events. If they are not independent, we can still bound the conjunction using the classical Fréchet inequality. In this case, we can infer at least that the probability of the joint event A & B is surely within the interval
In probability theory, a Chernoff bound is an exponentially decreasing upper bound on the tail of a random variable based on its moment generating function. The minimum of all such exponential bounds forms the Chernoff or Chernoff-Cramér bound , which may decay faster than exponential (e.g. sub-Gaussian ).
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