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In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability . [ 1 ] [ 2 ] : 10 It is also called a probability matrix , transition matrix , substitution matrix , or Markov matrix .
The word stochastic is used to describe other terms and objects in mathematics. Examples include a stochastic matrix, which describes a stochastic process known as a Markov process, and stochastic calculus, which involves differential equations and integrals based on stochastic processes such as the Wiener process, also called the Brownian ...
Doubly stochastic matrix — a non-negative matrix such that each row and each column sums to 1 (thus the matrix is both left stochastic and right stochastic) Fisher information matrix — a matrix representing the variance of the partial derivative, with respect to a parameter, of the log of the likelihood function of a random variable.
In mathematics, especially in probability and combinatorics, a doubly stochastic matrix (also called bistochastic matrix) is a square matrix = of nonnegative real numbers, each of whose rows and columns sums to 1, i.e.,
The term stochastic process first appeared in English in a 1934 paper by Joseph Doob. [60] For the term and a specific mathematical definition, Doob cited another 1934 paper, where the term stochastischer Prozeß was used in German by Aleksandr Khinchin, [63] [64] though the German term had been used earlier, for example, by Andrei Kolmogorov ...
In mathematics and statistics, a probability vector or stochastic vector is a vector with non-negative entries that add up to one.. The positions (indices) of a probability vector represent the possible outcomes of a discrete random variable, and the vector gives us the probability mass function of that random variable, which is the standard way of characterizing a discrete probability ...
In probability theory, a Markov kernel (also known as a stochastic kernel or probability kernel) is a map that in the general theory of Markov processes plays the role that the transition matrix does in the theory of Markov processes with a finite state space.
In 1926, Van der Waerden conjectured that the minimum permanent among all n × n doubly stochastic matrices is n!/n n, achieved by the matrix for which all entries are equal to 1/n. [17] Proofs of this conjecture were published in 1980 by B. Gyires [ 18 ] and in 1981 by G. P. Egorychev [ 19 ] and D. I. Falikman; [ 20 ] Egorychev's proof is an ...