# Random numbers

## Random numbers Topics

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### Random variable

A random variable is a measurable function from a probability space into a measurable space known as the state space (Doob 1996). Papoulis (1984, p. 88) gives the slightly different definition of a random variable as a real function whose domain is the probability space and such that: 1. The set is an event for any real number . 2. The probability of the events and equals zero. The abbreviation "r.v." is sometimes used to denote a random variable.

### Local discrepancy

Given a point set in the -dimensional unit cube , the local discrepancy is defined as is the content of .

### Random number

A random number is a number chosen as if by chance from some specified distribution such that selection of a large set of these numbers reproduces the underlying distribution. Almost always, such numbers are also required to be independent, so that there are no correlations between successive numbers. Computer-generated random numbers are sometimes called pseudorandom numbers, while the term "random" is reserved for the output of unpredictable physical processes. When used without qualification, the word "random" usually means "random with a uniform distribution." Other distributions are of course possible. For example, the Box-Muller transformation allows pairs of uniform random numbers to be transformed to corresponding random numbers having a two-dimensional normal distribution.It is impossible to produce an arbitrarily long string of random digits and prove it is random. Strangely, it..

### Linear congruence method

A method for generating random (pseudorandom) numbers using the linear recurrence relationwhere and must assume certain fixed values, is some chosen modulus, and is an initial number known as the seed.

### Variate

A variate is a generalization of the concept of a random variable that is defined without reference to a particular type of probabilistic experiment. It is defined as the set of all random variables that obey a given probabilistic law.It is common practice to denote a variate with a capital letter (most commonly ). The set of all values that can take is then called the range, denoted (Evans et al. 2000, p. 5). Specific elements in the range of are called quantiles and denoted , and the probability that a variate assumes the value is denoted .

### Van der corput sequence

Van der Corput sequences are a means of generating sequences of points that are maximally self-avoiding (a.k.a. quasirandom sequences). In the one-dimensional case, the simplest approach to generate such a sequence is to simply divide the interval into a number of equal subintervals. Similarly, one can divide an -dimensional volume by uniformly partitioning each of its dimensions. However, these approaches, have a number of drawbacks for numerical integration, especially for high dimensions.Like quasirandom sequences, "permuted" van der Corput sequences are constrained by a low-discrepancy requirement, which has the net effect of generating points in a highly correlated manner (i.e., the next point "knows" where the previous points are).For example, the ordinary van der Corput sequence in base 3 is given by 1/3, 2/3, 1/9, 4/9, 7/9, 2/9, 5/9, 8/9, 1/27, ......

### Quasirandom sequence

A sequence of -tuples that fills n-space more uniformly than uncorrelated random points, sometimes also called a low-discrepancy sequence. Although the ordinary uniform random numbers and quasirandom sequences both produce uniformly distributed sequences, there is a big difference between the two. A uniform random generator on will produce outputs so that each trial has the same probability of generating a point on equal subintervals, for example and . Therefore, it is possible for trials to coincidentally all lie in the first half of the interval, while the st point still falls within the other of the two halves with probability 1/2. This is not the case with the quasirandom sequences, in which the outputs are constrained by a low-discrepancy requirement that has a net effect of points being generated in a highly correlated manner (i.e., the next point "knows" where the previous points are).Such a sequence is extremely useful..

### Stochastic process

Doob (1996) defines a stochastic process as a family of random variables from some probability space into a state space . Here, is the index set of the process.Papoulis (1984, p. 312) describes a stochastic process as a family of functions.

### Discrete discrepancy

Given a point set in the -dimensional unit cube , the star discrepancy is defined as(1)where the local discrepancy is defined as(2) is the content of , and is the class of all discrete subintervals of of the form(3)with .

### Noise sphere

A mapping of random number triples to points in spherical coordinates according to(1)(2)(3)in order to detect unexpected structure indicating correlations between triples. When such structure is present (note that this does not include the expected bunching of points along the -axis according to the factor in the spherical volume element), numbers may not be truly random.

### Star discrepancy

Given a point set in the -dimensional unit cube , the star discrepancy is defined as(1)where the local discrepancy is defined as(2) is the content of , and is the class of all -dimensional subintervals of of the form(3)with for . Here, the term "star" refers to the fact that the -dimensional subintervals have a vertex at the origin.

### Cliff random number generator

A random number generator produced by iteratingfor a seed . This simple generator passes the noise sphere test for randomness by showing no structure.

### Martingale

A sequence of random variates , , ... with finite means such that the conditional expectation of given , , , ..., is equal to , i.e.,(Feller 1971, p. 210). The term was first used to describe a type of wagering in which the bet is doubled or halved after a loss or win, respectively. The concept of martingales is due to Lévy, and it was developed extensively by Doob.A one-dimensional random walk with steps equally likely in either direction () is an example of a martingale.