# Distribution of binary digits in moduli

Considering the (infinite) set of all positive integers that are a product of $2$ primes only, represented in binary $100...01$.

Question: is the distribution of the proportion of $0,1$ digits "uniform" (meaning $Pr(0) = Pr(1)$) ? Or should we expect an asymmetric distribution (eg. the relative frequency of "$1$"s to exceed that of the "$0$"s) ?

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Which probability distribution are you using for picking numbers to sample bits from? It can't be uniform, because there's no such thing possible for a countably infinite set. –  Henning Makholm Dec 30 '12 at 14:06
@ Henning Makholm Well i am working in the space of "all set of all positive integers that are a product of 2 primes". There is no sampling scheme here. I am looking for a statement regarding the entire population of moduli. –  Pam Dec 30 '12 at 14:58
My personal hunch is that there should be an asymmetry slighly favoring the "$1$"s. But I am having hard time to find a way to justify that in a rigorous way. –  Pam Dec 30 '12 at 15:04
Yes I see you're working on that set. But you cannot define averages over such a set without first selecting a distribution to average over. –  Henning Makholm Dec 30 '12 at 15:11
@ Henning Makholm The "distribution" is here the "popolation distribution". There is no sampling scheme here. (Even the original population has a distribution with its own mean, variance, etc.) [The set where you "average over" is the entire set of the moduli.] –  Pam Dec 30 '12 at 15:13