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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, the theory of stochastic processes is an important contribution to probability theory, [29] and continues to be an active topic of research for both theory and applications. [30] [31] [32] The word stochastic is used to describe other terms and objects in mathematics.
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.Two events are independent, statistically independent, or stochastically independent [1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds.
In probability theory and statistics, a stochastic order quantifies the concept of one random variable being "bigger" than another. These are usually partial orders , so that one random variable A {\displaystyle A} may be neither stochastically greater than, less than, nor equal to another random variable B {\displaystyle B} .
In mathematics and statistics, a stationary process (also called a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose statistical properties, such as mean and variance, do not change over time.
Statistical models are often used even when the data-generating process being modeled is deterministic. For instance, coin tossing is, in principle, a deterministic process; yet it is commonly modeled as stochastic (via a Bernoulli process). Choosing an appropriate statistical model to represent a given data-generating process is sometimes ...
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 ...
A stochastic simulation is a simulation of a system that has variables that can change stochastically (randomly) with individual probabilities. [ 1 ] Realizations of these random variables are generated and inserted into a model of the system.