Historically, did pseudo RNGs have range [0,1] because of the ease of drawing from $f$ using $F^{-1}(p)$, or for some other reason? A simple, though not necessarily efficient, way to simulate random draws from a probability density function $f$, is to apply the inverse cumulative distribution function to a random variate distributed uniformly from 0 to 1. The pseudo random number generators (RNGs) I am (barely) familiar with produce pseudo-random values having this kind of distribution.
I do not really know much about the history of how RNGs were developed, but I wonder if RNG default behavior was expressly designed because of its suitability for simulating from arbitrary distributions (or at least from arbitrary distributions with computable inverse cumulative distribution functions)? Or are there other reasons why RNG algorithm output should range [0,1]—as opposed to [-1, 1], [0, floating point limit], or [0, long limit], etc.—and specifically with a uniform, as opposed to some other distribution?
I know that, statistical properties aside, computational algorithms have other properties, e.g., security, memory use, computational complexity, etc., which weigh on development and adoption of a RNG algorithm; any insights welcome.
 A: In principle, any continuous distribution can serve as a starting point for a random number generator.  Nevertheless, the standard continuous uniform distribution is a natural starting place for a pseudo-random number generator (PRNG) for a few main reasons:


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*The method of inverse transformation sampling allows us to generate any random variable from a standard uniform random variable, and the latter is a natural starting place for this computation.  In particular, the mathematics of transforming a standard uniform random variable to a random variable with another distribution is particularly simple and intuitive.

*The standard uniform distribution has particularly simple properties for the purposes of testing the accuracy of the PRNG method.  These methods are subject to a battery of tests to ensure that they have desirable properties for a random number generator.  These tests are particularly easy to frame for a PRNG that generates a standard continuous uniform random variable.  For example, standard occupancy tests are particularly easy to deploy for uniform random variables.

*Computational methods that generate real numbers are subject to rounding error.  In most platforms the numbers are stored in double-precision floating point format, and this format has a fixed level of accuracy in the fractional part.  When generating a standard continuous uniform random variable the interval between values of the fractional part have fixed probability, so no intervals are larger or smaller than others.  (Contra this reasoning, note that this is also arguably a reason to prefer an exponential random variable as the starting point for analysis, since the floating point format uses an exponent.)
