Another neat property of Poisson processes is that when raced against one another, they win in proportion to their underlying rates. This property is the basis of a clever random sampling algorithm that works well in SQL:
SELECT *
FROM Population
WHERE weight > 0
ORDER BY -LN(1.0 - RANDOM()) / weight
LIMIT 100 -- Sample size.
(author of OP) That post of yours was actually what got me tooling around with this stuff again :) it's a really excellent one
Thanks! That was very kind of you to say. Whenever I write stuff like that, I wonder, "Does anyone find this useful?" It helps to hear every once in a while that the answer is sometimes yes.
Splitting a Poisson process into two lower rate processes is a neat trick. Even better is that you can do the same to convert a Poisson process into one with a variable rate, provided that rate is lower than the original (original may be variable as well).
And the fact that the partial sums of a bunch of exponential distributions results in the same distribution of values as picking Poisson(lambda * time) values uniformly at random is pure magic.