Accession Number : AD0705642


Title :   RANDOM VARIABLES WITH INDEPENDENT BINARY DIGITS


Corporate Author : BOEING SCIENTIFIC RESEARCH LABS SEATTLE WA MATHEMATICS RESEARCH LAB


Personal Author(s) : Marsaglia, George


Full Text : http://www.dtic.mil/get-tr-doc/pdf?AD=AD0705642


Report Date : Jan 1970


Pagination or Media Count : 15


Abstract : Let X = .b1b2b3... be a random variable with independent binary digits bn taking values 0 or 1 with probabilities pn and qn. When does X have a density function. A continuous density function. A singular distribution. This note proves that the distribution X is singular is and only if the tail of the series Summation (log(pn/qn)) squared diverges, and that X has a density that is positive on some interval if and only if log(pn/qn) is a geometric sequence with ratio 1/2 for n greater than some k, and in that case the fractional part of (2 to the power k)X has an exponential density (increasing or decreasing with the uniform density a special case). It gives a sufficient condition for X to have a density, (Summation log (2 max (pn,qn))converges), but unless the tail of the sequence log(pn/qn) is geometric, ratio 1/2, the density is a weird one that vanishes at least once in every interval.


Descriptors :   *RANDOM VARIABLES, CONVERGENCE, EXPONENTIAL FUNCTIONS, PROBABILITY, PROBABILITY DENSITY FUNCTIONS, SERIES(MATHEMATICS), STATISTICAL DISTRIBUTIONS, THEOREMS


Subject Categories : Statistics and Probability


Distribution Statement : APPROVED FOR PUBLIC RELEASE