= Poisson cumulative probability at its growing mean
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{title2=$\mathbb P(X_\lambda\leq\lfloor\lambda\rfloor)\longrightarrow1/2$}
If $X_\lambda$ has a <Poisson distribution>, then $\mathbb P(X_\lambda\leq\lfloor\lambda\rfloor)\to1/2$ as $\lambda\to\infty$. The <Poisson central limit theorem> standardizes by mean $\lambda$ and <variance> $\lambda$; the cutoff $(\lfloor\lambda\rfloor-\lambda)/\sqrt\lambda$ tends to zero. Squeezing between fixed nearby cutoffs and using continuity of the standard <normal distribution> function gives the limit.
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