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A random variate is a variable chosen from a uniform distribution of pseudorandom numbers. Random variates are often referred to when simulating stochastic models. Pseudorandom numbers generated on a PC are random variates. Random variates may have non-uniform or uniform distributions. Non-uniform random variates are also known as random deviates.

In probability theory, a random variable is a measurable function from a probability space to a measurable space of values the variable can take on. Those values are known as a random variates (occasionally: random deviates), particularly in the context of random variate generation.

Devroye^[1] defines a random variate generation algorithm (for real numbers) as follows:

Assume that

1. Computers can manipulate real numbers.
2. 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. Most computers lack a source of true randomness (like certain hardware random number generators), and instead use pseudorandom number sequences.)

The distinction between random variable and random variate is subtle and is not always made in the literature. It is useful when one wants to distinguish between a random variable itself with an associated probability distribution on the one hand, and random draws from that probability distribution on the other, in particular when those draws are ultimately derived by floating-point arithmetic from a pseudorandom sequence.

• inversion method
• rejection method
• alias method

• random number generation (this is a bit of a misnomer, but a popular alternative to "random variate generation")