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In probability theory and statistics, the Zipf–Mandelbrot law is a discrete probability distribution.Also known as the Pareto–Zipf law, it is a power-law distribution on ranked data, named after the linguist George Kingsley Zipf, who suggested a simpler distribution called Zipf's law, and the mathematician Benoit Mandelbrot, who subsequently generalized it.
The 'discrete case' given above is the special case arising when X takes on only countably many values and μ is a probability measure. In fact, the discrete case (although without the restriction to probability measures) is the first step in proving the general measure-theoretic formulation, as the general version follows therefrom by an ...
In probability theory and statistics, the Poisson binomial distribution is the discrete probability distribution of a sum of independent Bernoulli trials that are not necessarily identically distributed. The concept is named after Siméon Denis Poisson.
In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of successes (random draws for which the object drawn has a specified feature) in draws, without replacement, from a finite population of size that contains exactly objects with that feature, wherein each draw is either a success or a failure.
These are also the three discrete distributions among the six members of the natural exponential family with quadratic variance functions (NEF–QVF). More general distributions can be defined by fixing some initial values of p j and applying the recursion to define subsequent values. This can be of use in fitting distributions to empirical data.
The efficiency of accessing a key depends on the length of its list. If we use a single hash function which selects locations with uniform probability, with high probability the longest chain has ( ) keys. A possible improvement is to use two hash functions, and put each new key in the shorter of the two lists.
In the discrete time case, if the planning horizon is finite, the problem can also be easily solved by dynamic programming. When the underlying process is determined by a family of (conditional) transition functions leading to a Markov family of transition probabilities, powerful analytical tools provided by the theory of Markov processes can ...
The problem of estimating the maximum of a discrete uniform distribution on the integer interval [,] from a sample of k observations is commonly known as the German tank problem, following the practical application of this maximum estimation problem, during World War II, by Allied forces seeking to estimate German tank production.