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I have an extreme version of the birthday problem. I want to know:

  • The probability that $m$ individuals will share a birthday
  • The expected $m$ given the number of individuals

The slight complication is that:

  • An individual can have multiple birthdays
  • The number of "days" is almost infinite, meaning that each day has a very low probability
  • The probability space isn't uniform, e.g. Jan 1 is slightly more probable than Jan 2.

This is an abstracted explanation of the problem in this paper (equations 3 and 4). If someone can please help me understand equations 3 and 4 that would be super helpful.

In particular, what is a generating function in the context of the paper? It's explained as...

$M_m$s are the coefficients of the polynomial $G(x,{N i})$, and can be calculated just by expanding the polynomial in $x$ and summing over $s$.

Throughout the manuscript the term $x$ isn't really discussed, so I don't know where it comes from all of a sudden. Also, what is it supposed to represent? The Github repo from this paper doesn't say anything either. Thanks in advance. If there's any suggestions on code implementation that would be also helpful!

Anonymous Scientist
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  • Generating functions are a way to encode sequences of numbers as the coefficients of a polynomial. They are useful in combinatorics, for instance. The $x$ there has no meaning, it's just the polynomial variable. https://en.wikipedia.org/wiki/Generating_function – learner Sep 24 '20 at 11:03
  • Thanks @D... I'm still a bit confused though. – Anonymous Scientist Sep 24 '20 at 11:30
  • Does that mean $G(x, {N_i})$ as written in the paper (equation 4) is another way of rewriting equation 3? Also, what does that mean in terms of code? I feel like I can write equation 3 in code as something like ``` for s in S: for j in comb(J, m): prod(1-e^ps) * prod(e^ps) ``` But in the case of equation 4, I got up to ``` for s in S: prod(e^ps + (1-e^ps)) ``` And... I'm just really stumped. I'm so sorry. Where would the coefficient of the polynomial go here? – Anonymous Scientist Sep 24 '20 at 11:32

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