Confidence interval for a small number of iid Poisson I want to calculate the confidence interval for $\lambda$ from a small ($n=10$) set of repeated observations from a Poisson distribution. That is, I have $X_1, \dots, X_{10}$ which I believe are i.i.d. and I want a 95% confidence interval given these observations for $\lambda$.
I know I can use $\bar x \pm 1.96\frac{s}{\sqrt{n}}$ when $n$ is large but that doesn't seem to be a good assumption here. I also know for a single instance $y$ you can use 
qchisq(0.025,2*y)/2, qchisq(0.975,2*(y+1))/2

so my problem appears to be somewhere between the two.
 A: Since it is small sample you can calculate exact distribution to arrive at confidence interval.
$Z= X_1+ \cdots + X_{10}  \sim $  Poisson($10 \lambda$). 
The mle of $\lambda = \hat{\lambda} = \bar{X}= Z/10$. From here you can find the confidence interval.
EDIT: Confidence interval
$Z= 10* \bar{X} \sim$ Poisson($10 \lambda$). 
Let $\lambda_L$ and $\lambda_U$ be the confidence limits of $\lambda$. 
And $\lambda_{L^{*}}$ and $\lambda_{U^{*}}$ be the confidence limits of $10\lambda$. 
$ \lambda_{U^{*}}=10\lambda_U, \quad $ and $ \lambda_{L^{*}}=10\lambda_L \quad $
Then
$\exp(-\lambda_{L^{*}}) \sum\limits_{j=z}^{\infty}\frac{\lambda_{L^{*}}^j}{j!}= \frac{\alpha}{2}, \quad $ and  $ \quad \exp(-\lambda_{U^{*}}) \sum\limits_{j=0}^{z}\frac{\lambda_{U^{*}}^j}{j!}= \frac{\alpha}{2}$
From the relationship between poisson and gamma (chisqure) it can write as
$10*\lambda_L =0.5 \chi^2_{2z,\alpha/2}, \quad$
$10*\lambda_L =0.5 \chi^2_{2(z+1),1-\alpha/2}$
which leads the CI of $\lambda$ as
(qchisq(0.025,2*z)/(2*10), qchisq(0.975,2*(z+1))/(2*10)) 
