# Poisson Distribution Confidence Interval

Suppose I have 6 pieces of data from a poisson experiment:

4, 6, 8, 10, 12, and 14.

$\hatλ$ = 9. I want to establish a 95% confidence interval for this poisson experiment, and z* = 1.960. So I do the following:

Upper boundary:

$λ+1.960\sqrt{λ}$

$9+1.960\sqrt{9}$

$14.88$

Lower boundary:

$λ-1.960\sqrt{λ}$

$9-1.960\sqrt{9}$

$3.12$

Is this work mathematically sound?

===

As for interpreting the confidence interval, I'm lost. What does the confidence interval mean?

• I learned a ton about the Garwood Confidence Interval and have ended up using that. Very interesting how it approximates it with the chi square distribution. – Matthew Anderson Mar 23 '17 at 3:53

What you have is not a confidence interval. A confidence interval should be based on data. If you know that $X_i \sim \mathsf{Pois}(\lambda = 9),$ what is the point of getting an interval estimate of $\lambda?$

From the corrected data ($4,6,8,10,12,14$), you have $\bar X = \hat \lambda = 9.0.$

One simple style of approximate confidence interval for $\lambda$ is the so-called Wald 95% CI. It is of the form

$$\bar X \pm 1.96\sqrt{\bar X/n}.$$

This would be an approximate confidence interval based on data, and would give a somewhat similar result: $(6.60, 11.40).$

A somewhat better style of 95% CI for the Poisson mean $\lambda$ uses $T = n\bar X$ to get a CI for $n\lambda$ as $$T + 2 \pm 1.96\sqrt{T + 1},$$ and then divide the endpoints by $n$. The resulting 95% CI is $(7.1, 11.9).$

There are many styles of Poisson confidence intervals, you should use the one given in your text. (But I don't understand what the t distribution has to do with a Poisson CI.)

Note: Per Comments, I'm showing a graph of the PDF of $\mathsf{Pois}(9)$ compared with the density curve of $\mathsf{Norm}(\mu=9,\,\sigma=3).$

• My data was supposed to be 2, 4, 6, 8, 10, 12, 14. Making the mean equal to 9. So (lambda hat) would be equal to 9, correct? I'm still slightly confused. Like with the 'better style' what is T? And what does the confidence interval actually tell me? Why do you say that one of the styles is better than another? – Matthew Anderson Mar 12 '17 at 22:19
• And this other CI that you used. I was doing research on all of the types of CI (Wald CI, Score CI, etc.) and was confused as to what your CI is. Does it have a name, and what should I look up to learn more about it? – Matthew Anderson Mar 12 '17 at 22:31
• Changed my answer to match data 4,6,8,10,12,14 for an avg of $\bar X =9.$ There is much debate about what style of CI to use for Poisson mean $\lambda.$ For large $\bar X$ (say above 40), it doesn't make much difference which one uses. See the Wilipedia article on Poisson under estimation for some of them. The one I give is in the same spirit as the Agresti (or plus-four) binomial CI. Maybe the best is based on $\mathsf{Gamma}(T, n).$ For $n=6,T=54,$ the R code qgamma(c(.025,.975), 54, 6) gives $(6.7,11.6).$ "Good" means that the true coverage probability of the CI is the 'advertised' 95%. – BruceET Mar 12 '17 at 22:52
• Is there any inherent problem with the Wald CI? Would it be wrong to use it? – Matthew Anderson Mar 12 '17 at 22:58
• Wouldn't use Wald for means as small as 9. Wald has two approximations. One is to approx Pois by Norm. The other is to estimate std error by $\sqrt{\bar X/n}$ instead of $\sqrt{\lambda/n}$. Both approximations get better with larger $\lambda$ or larger $n$. Appending figure to my answ showing norm aprx to Pois(9). – BruceET Mar 12 '17 at 23:09