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On Facebook, I can see which friends have birthdays today. Sometimes there are 1, sometimes more than 1, and sometimes zero friends. What's the average number of birthdays today? To formalize:

Problem statement

I have $n$ friends. Each of their birthdays are equally likely to occur on any day. Let $X$ be the number of friends with birthdays today. Ignore leap years.

Clearly, $0 \leq X \leq n$, with the mean being $n/365$.

What does the probability mass function of $X$ look like? Is this is a Poisson distribution? If yes, what is $\lambda$? If no, how do I describe the distribution?

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Under these specifics, i.e. that there are $n$ independent variables $Y_j$ uniformly distributed on $\{1,\cdots,365\}$ and, for some predetermined $m$, you are considering $X=\#\{j\,:\, Y_j=m\}$, then the pdf of $X$ is $p_X(k)=\binom nk 365^{-k}(1-365^{-1})^{n-k}$, i.e. $X\sim\operatorname{Binom}(n,365^{-1})$. If we decide to make the dependence of $X$ on $m$ explicit, we obtain that all the $X_m$-s are identically distributed, but clearly not independent.

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  • $\begingroup$ Welcome to math SE! My question did not have m - what is m in the context of the question? $\endgroup$
    – hongsy
    Commented Jan 6, 2022 at 4:18
  • $\begingroup$ @hongsy I suspect $m$ is today's date $\endgroup$
    – Henry
    Commented Jan 7, 2022 at 11:51

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