# How to calculate a confidence level for a Poisson distribution?

Would like to know how confident I can be in my $\lambda$. Anyone know of a way to set upper and lower confidence levels for a Poisson distribution?

• Observations ($n$) = 88
• Sample mean ($\lambda$) = 47.18182

what would the 95% confidence look like for this?

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 This should be posted to the statistics SE site. – Iterator Sep 9 '11 at 12:32

## migrated from stackoverflow.comSep 9 '11 at 14:57

For poisson, the mean and the variance are both $\lambda$. If you want the confidence interval around lambda, you can calculate the standard error as $\sqrt{\lambda / n}$.

The 95-percent confidence interval is $\lambda \pm 1.96\sqrt{\lambda / n}$.

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That will do it. Should have gotten that way earlier...was looking right at the formula for the last 20 minutes. Thanks again :) – Travis Sep 9 '11 at 12:47
This is fine when $n \lambda$ is large, for then the Poisson is adequately approximated by a Normal distribution. For smallish values or higher confidence, better intervals are available. See math.mcmaster.ca/peter/s743/poissonalpha.html for two of them along with an analysis of their actual coverage. (Here, the "exact" interval is (45.7575, 48.6392), the "Pearson" interval is (45.7683, 48.639), and the Normal approximation gives (45.7467, 48.617): it's a little too low, but close enough, because $n \lambda = 4152$.) – whuber Sep 9 '11 at 15:34
For others confused like I was: here is a description of where the 1.96 comes from. – mjibson Apr 17 '12 at 19:26
How did you calculate the exact interval for this problem given the information on that website given by whuber? I couldn't follow because that site seems to only indicate how to proceed when you have one sample. Maybe I'm just not understanding something simple but my distribution has a much smaller value of lambda(n) so I can't use the normal approximation and I don't know how to compute the exact value. Any help would be greatly appreciated. Thanks! – user12849 Jul 25 '12 at 17:59

This paper discusses 19 different ways to calculate a confidence interval for the mean of a Poisson distribution.

http://www.ine.pt/revstat/pdf/rs120203.pdf

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You might also consider bootstrapping your estimates -- here's a short tutorial on bootstrapping:

http://www.ats.ucla.edu/stat/r/library/bootstrap.htm

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