Average number of birthdays today
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.