The relationship between bits and int/float random number generator

Question

I want to figure out the relationship between bits and RNG for int or float.

(By random I mean uniformly distributed)

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I am given a perfect boolean random generator, and I am asked to implement a random 32 bits integer generator (including negative, zero and positive). What I want to do is generate a random boolean for each of the 32 bits, and concat them together to be a random int.

Am I doing the right thing?

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Also from the other way around, if I am given a perfect random 32 bits integer generator, can I say each bit can be considered as uniformly distributed over 0 and 1?

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how about float (not only between 0 and 1, but the full range of float)?

Can I use the same way to generate random IEEE 745 float?

Answer 1

Yes, you are indeed doing it right, using 32 draws of the different numbers, will give you a uniform distributed random variable.

Explanation: Each number can be generated by a unique combination of 32 0/1 draws. No 2 numbers are generated from the same combination, and no number is generated from 2 combinations - meaning, the probability for each number is 1/2^32 - as expected.

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Yes, Same principle applies here. There are 2^32 'ways' to choose 32 bits number, and similarly to the previous question - you can see that the numbers are independently uniformly distributed on {0,1} per bit.

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A random uniformly distributed float in range [0,1] can be generated by randUnsignedInt()/(2^32-1). An alternative is drawing an int and just re-interpret it ad float - assuming both are using the same number of bits (basically - both are 32 bits number, they only vary in the way you interpret them...) Note that the alternative is NOT in range [0,1].

Answer 2

Yes. This is exactly what random.org does. In this case, the domains are fairly easy to map across -- things like a 6 sided die are harder.