in a normal distribution, one standard deviation above and below the mean encompasses 68% of data. Can this also apply to Poisson distribution? in not, how can you tell how much of data falls between standard deviations for Poisson distribution? a reference to the answer would be great too.
Thanks
Best Answer
It depends strongly on the parameter $\lambda$. For large $\lambda$, it's close to the normal version by the Central Limit Theorem. For small $\lambda$, it's quite irregular, with jumps every time $\lambda+\sqrt{\lambda}$ or $\lambda-\sqrt{\lambda}$ crosses an integer.
Here's a graph of that behavior for $\lambda < 4$. The vertical axis is how much of the Poisson distribution is within one standard deviation of the mean, and the horizontal axis is the parameter $\lambda$.