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.
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.
Computers necessarily lack the ability to manipulate real numbers, typically using floating point representations instead.
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 pseudo-random sequence.
For the generation of non-uniform random variates, see Pseudo-random number sampling.