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I've been looking everywhere for a method of finding a given percentile from a Poisson distribution but can't seem to find much, so I figured I'd ask here if anyone knows.

How can I find a given percentile (say the 20th percentile) of a poisson distribution given the value of its parameter $\lambda$.

For example, if $X\sim \text{Pois}(4.5)$, how can I find the first 20th percentile of the distribution?

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For a discrete distributions like the Poisson, it depends on how you define "percentiles".

Most statistics software programs will compute probabilities for the Poisson distribution. For example, using R, we can compute $P = P(X \le x)$ like this:

> x <- 0:4
> P <- ppois(x, lambda=4.5)
> names(P) <- x
> P
     0      1      2      3      4 
0.0111 0.0611 0.1736 0.3423 0.5321 

So you can see that the 20th percentile is between 2 and 3.

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