= Poisson limit for occupancy fractions
{c}
{title2=$\mathbb E[N_j]/m\to e^{-\lambda}\lambda^j/j!$}
In a <uniform balls-in-bins allocation> with $m\to\infty$ and $n/m\to\lambda\in(0,\infty)$, a fixed bin count has <convergence in distribution> to a <Poisson distribution> variable of mean $\lambda$ by the <Poisson limit theorem>. If $N_j$ counts bins containing exactly $j$ balls, symmetry and the <linearity of expectation> give $\mathbb E[N_j]/m=\mathbb P(B_1=j)\to e^{-\lambda}\lambda^j/j!$ for each fixed $j$. In particular the expected empty fraction tends to $e^{-\lambda}$, and the expected fraction with at least two balls tends to $1-(1+\lambda)e^{-\lambda}$. These are statements about expectations; convergence of the random fractions requires an additional argument.
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