Suppose I have a Poisson distribution with parameter $\lambda$. On average, the fraction of observations with $X=0$ events will tend to $e^{-\lambda}$, i.e. $P(X=0; \lambda) = e^{-\lambda}$ as the number of observations increases. Likewise, for single events, $X=1$, the overall fraction across all events for this subset should tend to $\lambda e^{-\lambda}$ as the number of observations increases.
However, this is only true for a large number of samples, $n$. How do I express the error or distribution of this subset of observations as the number of $n$ observations increases? Is there some relationship like $e^{-\lambda} + 1/n$ for the 0 event case, which tends to the average expectation for this event count as $n$ increases?