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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} .
is called a sample function, a realization, or, particularly when is interpreted as time, a sample path of the stochastic process {(,):}. [50] This means that for a fixed ω ∈ Ω {\displaystyle \omega \in \Omega } , there exists a sample function that maps the index set T {\displaystyle T} to the state space S {\displaystyle S} . [ 28 ]
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 ...
If a stochastic process is strict-sense stationary and has finite second moments, it is wide-sense stationary. [2]: p. 299 If two stochastic processes are jointly (M + N)-th-order stationary, this does not guarantee that the individual processes are M-th- respectively N-th-order stationary. [1]: p. 159
Stochastic dominance is a partial order between random variables. [1] [2] It is a form of stochastic ordering.The concept arises in decision theory and decision analysis in situations where one gamble (a probability distribution over possible outcomes, also known as prospects) can be ranked as superior to another gamble for a broad class of decision-makers.
The order in probability notation is used in probability theory and statistical theory in direct parallel to the big O notation that is standard in mathematics.Where the big O notation deals with the convergence of sequences or sets of ordinary numbers, the order in probability notation deals with convergence of sets of random variables, where convergence is in the sense of convergence in ...
A Markov chain with two states, A and E. In probability, a discrete-time Markov chain (DTMC) is a sequence of random variables, known as a stochastic process, in which the value of the next variable depends only on the value of the current variable, and not any variables in the past.
Consider the real line with its usual Borel topology. Let denote the Dirac measure, a unit mass at the point in .The collection := {|} is not tight, since the compact subsets of are precisely the closed and bounded subsets, and any such set, since it is bounded, has -measure zero for large enough .