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Free probability is a mathematical theory that studies non-commutative random variables. The "freeness" or free independence property is the analogue of the classical notion of independence , and it is connected with free products .
When the image (or range) of is finitely or infinitely countable, the random variable is called a discrete random variable [5]: 399 and its distribution is a discrete probability distribution, i.e. can be described by a probability mass function that assigns a probability to each value in the image of .
Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric.It was introduced by Charles Stein, who first published it in 1972, [1] to obtain a bound between the distribution of a sum of -dependent sequence of random variables and a standard normal distribution in the Kolmogorov (uniform ...
This is a list of probability topics.It overlaps with the (alphabetical) list of statistical topics.There are also the outline of probability and catalog of articles in probability theory.
Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is ...
In more formal probability theory, a random variable is a function X defined from a sample space Ω to a measurable space called the state space. [ 2 ] [ a ] If an element in Ω is mapped to an element in state space by X , then that element in state space is a realization.
The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively.
the product of two random variables is a random variable; addition and multiplication of random variables are both commutative; and; there is a notion of conjugation of random variables, satisfying (XY) * = Y * X * and X ** = X for all random variables X,Y and coinciding with complex conjugation if X is a constant.