Double-precision Floats have more values between 0.0 and 1.0, than between 1.0 and 2.0. In fact, roughly half of ALL double-precision floats exist between -1.0 and 1.0, a very small minority of them exist between 1.0 and 2.0.
To generate unbiased random numbers between 0.0 and 2.0, it therefore requires you to either reject a significant amount of numbers in the 0.0 to 1.0 range, or perform some kind of many-to-few mapping in the 1.0 to 2.0 range.
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With regards to arbitrary INTEGER ranges, the proof is even easier. A random bitstream has 2^number-of-bits possible random values. Which does NOT divide evenly into an arbitrary integer range.
For example, 5-random bits will represent 32-different values. There's no way to map 32-values and divide them evenly into 0-9 (10 numbers).
Comments
Trivial proof. Pigeon hole principle.
Double-precision Floats have more values between 0.0 and 1.0, than between 1.0 and 2.0. In fact, roughly half of ALL double-precision floats exist between -1.0 and 1.0, a very small minority of them exist between 1.0 and 2.0.
To generate unbiased random numbers between 0.0 and 2.0, it therefore requires you to either reject a significant amount of numbers in the 0.0 to 1.0 range, or perform some kind of many-to-few mapping in the 1.0 to 2.0 range.
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With regards to arbitrary INTEGER ranges, the proof is even easier. A random bitstream has 2^number-of-bits possible random values. Which does NOT divide evenly into an arbitrary integer range.
For example, 5-random bits will represent 32-different values. There's no way to map 32-values and divide them evenly into 0-9 (10 numbers).