Generating random numbers of prescribed distribution using physical sources

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 . 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 (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 /2.We compare this deviation with the error resulting from the use of biased bits B_j with P{B_j = 1{ = (where ] – [) 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 = (|Q()|^p)^1/p (p 1) we have P_Y – P⁰_Y^TV_p (c_n · )^1/p with c_n p for n . For the distribution function F_Y – F⁰_Y 2(1 – 2^–n)|| holds.