$X_{1}, ..., X_{10}$ ~ $Pois(\theta)$
Observations: $x_{1} = x_{3} = x_{6} = x_{8} = x_{9} = 0$; $x_{2} = x_{5} = x_{10} = 1$; $x_{4} = 2$; $x_{7} = 3$.
I want to determine an exact (numerical) and an approximate (numerical) confidence interval for $\theta$ of confidence level 0.9:
exact confidence interval: I don't know how to do this. On the internet I can only find things about an approximate confidence interval.
approximate confidence interval:
$X = (X_{1}, ..., X_{10})$
MLE: $\hat{\theta} = \bar{X} = 0.8$
And: $Var(X) = \theta$. So an estimate for the variance is also $\bar{X} = 0.8$.
Then according to the answer: An approximate confidence interval is $\bar{X}$ $\pm$ $\sqrt{\bar{X}/n}$ $\xi_{1 - \alpha/2}$.
But this means that: T = $\sqrt{n}$ $\frac{\theta - E[X]}{\sqrt{Var(X)}}$ ~ N(0,1). But this does not hold for X ~ Pois($\theta$), right? Because a Poisson distribution does not have to be symmetric.