RANDOM VARIABLES WITH INDEPENDENT BINARY DIGITS
BOEING SCIENTIFIC RESEARCH LABS SEATTLE WA MATHEMATICS RESEARCH LAB
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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 logpnqn squared diverges, and that X has a density that is positive on some interval if and only if logpnqn is a geometric sequence with ratio 12 for n greater than some k, and in that case the fractional part of 2 to the power kX 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,qnconverges, but unless the tail of the sequence logpnqn is geometric, ratio 12, the density is a weird one that vanishes at least once in every interval.
- Statistics and Probability