Average number of birthdays today

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?

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.

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