Genuinely discrete discrete probability

Summary

Genuinely discrete discrete probability

What “genuinely discrete” methods can be used with discrete random variables?

When we first learn probability, we start with discrete random variables – experiments whose outcomes take values are non-negative whole numbers. But we then move on to continuous random variables, where we have richer techniques and results – like moments, scaling, shifting, the law of large numbers, the vital importance of the normal distribution, and the central limit theorem.

But what if we instead try to stay in those “early days” of learning probability, and stick firmly to the integers, looking for “genuinely discrete” ideas? We might then look at factorial moments, the probability generating function, “thinning”, the vital importance of the Poisson distribution, and the “law of thin numbers”.

There are plenty of easy-to-state problems regarding “genuinely discrete” discrete probability that have not received the attention they deserve. What is the discrete equivalent of the normal distribution? Which distributions are changed the least by adding independent copies? What symmetries can we observe between high-variance and low-variance distributions? Which discrete random variables are the “most random”?

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