Realization (stochastics)

A (coincidental) realization , sometimes incorrectly called realization , is a term from stochastics , a branch of mathematics . Realization is a concrete value that a random variable assumes, comparable to a function value of a function for a given argument. If the random variable describes a fair dice, a realization of this random variable would correspond to a rolled number. Random variables are i. d. Typically noted with capital letters and their realizations with lower case letters.

definition

Given a random variable on a probability space . Then means for ${\ displaystyle X}$ ${\ displaystyle (\ Omega, {\ mathcal {A}}, P)}$ ${\ displaystyle \ omega \ in \ Omega}$

{\ displaystyle x = X (\ omega)}

a realization of . ${\ displaystyle X}$

Examples and usage

Two example paths of a standard Wiener process

If there is a binomially distributed random variable for the parameters and , each natural number would be less than or equal to a possible implementation. If it is normally distributed , then every real number is a possible realization. ${\ displaystyle X}$ ${\ displaystyle n}$ ${\ displaystyle p}$ ${\ displaystyle n}$ ${\ displaystyle X}$

In mathematical statistics realizations play an important role of random variables. There samples are understood as the realization of a random variable with an unknown distribution. On the basis of this realization, an attempt is then made to make statements about the distribution of the random variable.

In the theory of stochastic processes , paths similar to the realizations appear, which are used, among other things, to illustrate processes. The resulting picture spaces are then very large. Accordingly, the realizations are not a number, but a continuous function or something similar.

literature

David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , pp. 456-457 , doi : 10.1007 / b137972 .