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In probability theory and information theory, adjusted mutual information, a variation of mutual information may be used for comparing clusterings. [1] It corrects the effect of agreement solely due to chance between clusterings, similar to the way the adjusted rand index corrects the Rand index.
A unique representation of e can be found within the structure of Pascal's Triangle, as discovered by Harlan Brothers. Pascal's Triangle is composed of binomial coefficients, which are traditionally summed to derive polynomial expansions. However, Brothers identified a product-based relationship between these coefficients that links to e.
where is the Kullback–Leibler divergence, and is the outer product distribution which assigns probability () to each (,).. Notice, as per property of the Kullback–Leibler divergence, that (;) is equal to zero precisely when the joint distribution coincides with the product of the marginals, i.e. when and are independent (and hence observing tells you nothing about ).
The sum-product conjecture informally says that one of the sum set or the product set of any set must be nearly as large as possible. It was originally conjectured by Erdős in 1974 to hold whether A is a set of integers, reals, or complex numbers. [3] More precisely, it proposes that, for any set A ⊂ ℂ, one has
In other words, if X n converges in probability to X sufficiently quickly (i.e. the above sequence of tail probabilities is summable for all ε > 0), then X n also converges almost surely to X. This is a direct implication from the Borel–Cantelli lemma. If S n is a sum of n real independent random variables:
The sum of squared deviations needed to calculate sample variance (before deciding whether to divide by n or n − 1) is most easily calculated as = From the two derived expectations above the expected value of this sum is
The purpose of this page is to catalog new, interesting, and useful identities related to number-theoretic divisor sums, i.e., sums of an arithmetic function over the divisors of a natural number , or equivalently the Dirichlet convolution of an arithmetic function () with one:
and E α n as above. In particular, E α n are the binomial coefficients of power −1 − α. Then the (C, α) sum of Σa n is defined as above. If Σa n has a (C, α) sum, then it also has a (C, β) sum for every β > α, and the sums agree; furthermore we have a n = o(n α) if α > −1 (see little-o notation).