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These numbers have been proved prime by computer with a primality test for their form, for example the Lucas–Lehmer primality test for Mersenne numbers. “!” is the factorial, “#” is the primorial, and () is the third cyclotomic polynomial, defined as + +.
Nontrivial examples are distributions that are subject to multiple constraints that are different from the assignment of the entropy. These are often found by starting with the same procedure ln p ( x ) → f ( x ) {\displaystyle \ln {p(x)}\rightarrow f(x)} and finding that f ( x ) {\displaystyle f(x)} can be separated into parts.
The zeta distribution has uses in applied statistics and statistical mechanics, and perhaps may be of interest to number theorists. It is the Zipf distribution for an infinite number of elements. The Hardy distribution, which describes the probabilities of the hole scores for a given golf player.
The principle of maximum entropy states that the probability distribution which best represents the current state of knowledge about a system is the one with largest entropy, in the context of precisely stated prior data (such as a proposition that expresses testable information).
Two bits of entropy: In the case of two fair coin tosses, the information entropy in bits is the base-2 logarithm of the number of possible outcomes — with two coins there are four possible outcomes, and two bits of entropy. Generally, information entropy is the average amount of information conveyed by an event, when considering all ...
Examples: Throwing dice, experiments with decks of cards, random walk, and tossing coins. Classical definition: Initially the probability of an event to occur was defined as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space: see Classical definition of probability.
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In mathematics, Kummer's theorem is a formula for the exponent of the highest power of a prime number p that divides a given binomial coefficient. In other words, it gives the p-adic valuation of a binomial coefficient. The theorem is named after Ernst Kummer, who proved it in 1852 (Kummer 1852).