# Random number generation: divisibility

I'm interested in generating random numbers.

I have a distribution [for simplicity let it be uniform distribution] of some event occurrence, i.e. the event (visits to a doctor) occurs between $1$ to $3$ times per year.

I want to change resolution of events to be on per month basis. We know how to do it for one tracked person.

In case there are more than one person (e.g. I want to simulate a small town of $200,000$ inhabitants visiting my favorite clinic), this imposes a formidable difficulty, as from programming point of view, I need now to track each visitor with a dedicated PRNG.

My interest is to obtain a single PRNG, which in streaming mode, aka the one used for a per year event generation, will produce events in the needed or extremely close distribution.

E.g., for per year generation, I can generate a number of visits in a loop, to asses a total number of visits per year. Each call for PRNG produces a number of visits for the next person.

Is it possible from mathematical point of view to create a single per month PRNG?

• That's an average of 400,000 visits per year, so 33,333 per month. A normal distribution with mean = variance = 33,333 should be good enough – Hagen von Eitzen Sep 17 '18 at 9:17
• Do you need to know which inhabitants visit the clinic that month, or do you just need the total number of visits? – David K Sep 17 '18 at 12:27
• I need to track specific ones – dEmigOd Sep 17 '18 at 14:14
• I can't figure out what the problem is. The existing machinery doesn't know what a year vs. a month is. So why would switching to monthly require to track every individual ? – Yves Daoust Sep 18 '18 at 9:02
• Turned out, this is the requirement of at least $1$ visit per person per year – dEmigOd Sep 18 '18 at 10:25

After more thoughts, it would be better to model arrivals, as a Poisson process, then due to infinite divisibility property we can model arrivals per month as independent $\frac{1}{12}$th of per year process. And they are all independent (we can assume this).