Generating random numbers of prescribed distribution using physical sources

Abstract

When constructing uniform random numbers in [0, 1] from the output of a physical device, usually n independent and unbiased bits B _j are extracted and combined into the machine number $Y{\text{ :}} = \sum {_{j = 1}^n {\text{ }}2^{ - j} B_j }$. In order to reduce the number of data used to build one real number, we observe that for independent and exponentially distributed random variables X _n (which arise for example as waiting times between two consecutive impulses of a Geiger counter) the variable U _n : = X _{2n − 1}/(X _{2n − 1} + X _2n ) is uniform in [0, 1]. In the practical application X _n can only be measured up to a given precision ϑ (in terms of the expectation of the X _n ); it is shown that the distribution function obtained by calculating U _n from these measurements differs from the uniform by less than ϑ/2.

We compare this deviation with the error resulting from the use of biased bits B _j with P ^ε{B _j = 1{ = $\frac{1}{2} + {\varepsilon }$ (where ε ∈] − $\frac{1}{2},\frac{1}{2}$[) in the construction of Y above. The influence of a bias is given by the estimate that in the p-total variation norm ‖Q‖^TV _p = ($\sum {_\omega ^{} }$|Q(ω)|^p)^1/p (p ≥ 1) we have ‖P ^ε _Y − P ⁰ _Y ‖^TV _p ≤ (c _n $\sqrt n$ · ε)^1/p with c _n → p $\sqrt {8/\pi }$ for n → ∞. For the distribution function ‖F ^ε _Y − F ⁰ _Y ‖ ≤ 2(1 − 2⁻ⁿ)|ε| holds.