Random Function

Also found in: Wikipedia.

random function

[′ran·dəm ′fəŋk·shən]
A function whose domain is an interval of the extended real numbers and has range in the set of random variables on some probability space; more precisely, a mapping of the cartesian product of an interval in the extended reals with a probability space to the extended reals so that each section is a random variable.

Random Function


If a function of an arbitrary argument t is defined on the set T of the values of t and assumes numerical values or, more generally, values from some vector space, the function is said to be a random function if its values are determined by some trial and can differ depending on the outcome of the trial. It is also required that there exist a definite probability distribution function for the values. If the set T is finite, the random function is a finite set of random variables; this set can be regarded as a single random vector quantity.

The most thoroughly studied random function with an infinite T is the important special case where t assumes numerical values and is time. The random function X(t) in this case is called a stochastic process; when t assumes only integral values, the terms “random sequence” and “time series” are also sometimes applied to X(t). If the values of t are points in some region of a multidimensional space, the random function is called a random field. Typical examples of random functions that are not stochastic processes are the velocity, pressure, and temperature fields of a turbulent flow of a liquid or gas and the height z of the agitated surface of the sea or the surface of an artificial rough plate.

The mathematical theory of random functions coincides with the theory of probability distribution functions in the function space of the values of X(t). These distribution functions can be specified by the set of finite-dimensional probability distribution functions for the sets of random variables X(t1), X(t2), …, X(tn) corresponding to all possible finite subsets (t1, t2,…, tn) of points of T. Alternatively, the distribution functions can be specified by the characteristic functional of the random function X(t); this characteristic functional is the mathematical expectation of the random variable il[X(t)], where l[X(t)] is a linear functional of X(t) of general form. Much progress has been made in the theory of homogeneous random fields, which are a special class of random functions; this class is a generalization of the class of stationary stochastic processes.


Vybrosy sluchainykh polei: Sb. st. Moscow, 1972.
Yaglom (Iaglom), A. M. “Second-order Homogeneous Random Fields.” In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, vol. 2. Berkeley-Los Angeles, 1961.
Whittle, P. “Stochastic Processes in Several Dimensions.” Bulletin of the Institute of Statistics, 1963, vol. 40.


References in periodicals archive ?
As it is well known, a random field [PHI](x) is a 2-dimensional, real-valued random function.
Thus, the correlation function of the random function X(t) is called not random function of two arguments [K.
Let X be the set of possible sequences of random bits and random function values that take the adversary up to the first forking point where A asks for H([y.
Simulation algorithms differ in the underlying random function model (multiGaussian or nonparametric), the amount and type of information that can be accounted for, and the computer requirements (Goovaerts 1997).
In this method the assumption is made that the image is a random function of brightness with the ergodicity property.
Hence, we need a strong random function of n/2 bits for a wide variety of n.
Also features FM/AM analogue radio and a programmable CD player with random function, as well as a player for all your old cassette tapes.
As I played the CD using my random function, scores of silent two-second tracks clipped by, prompting a jolt of surprise when each burst of the eight-part album sequence came around.
We have generated the points using a stratified random function (Idrisi software) which assures a better selection of the samples regarding the spatial distribution.
Recall that a Gaussian process is a random function Z(t) such that the value for a fixed t has a normal distribution, and the values Z([t.