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Concept# Random variate

Summary

In probability and statistics, a random variate or simply variate is a particular outcome of a random variable; the random variates which are other outcomes of the same random variable might have different values (random numbers).
A random deviate or simply deviate is the difference of a random variate with respect to the distribution central location (e.g., mean), often divided by the standard deviation of the distribution (i.e., as a standard score).
Random variates are used when simulating processes driven by random influences (stochastic processes). In modern applications, such simulations would derive random variates corresponding to any given probability distribution from computer procedures designed to create random variates corresponding to a uniform distribution, where these procedures would actually provide values chosen from a uniform distribution of pseudorandom numbers.
Procedures to generate random variates corresponding to a given distribution are known as procedures for (uniform) random number generation or non-uniform pseudo-random variate generation.
In probability theory, a random variable is a measurable function from a probability space to a measurable space of values that the variable can take on. In that context, those values are also known as random variates or random deviates, and this represents a wider meaning than just that associated with pseudorandom numbers.
Devroye defines a random variate generation algorithm (for real numbers) as follows:
Assume that
Computers can manipulate real numbers.
Computers have access to a source of random variates that are uniformly distributed on the closed interval [0,1].
Then a random variate generation algorithm is any program that halts almost surely and exits with a real number x. This x is called a random variate.
(Both assumptions are violated in most real computers. Computers necessarily lack the ability to manipulate real numbers, typically using floating point representations instead.

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