# Can I use a confidence interval of poisson mean for its variance

In a Poisson distribution the mean equals the variance. I would like to find a confidence interval of the variance. Is my reasoning below correct?
Using the central limit theorem I construct a 95% confidence interval for the mean $\mu$
$L \leq \mu \leq U$
$\mu=\sigma^2$
Therefore
$L \leq \sigma^2 \leq U$
It would seem to me that the inequality should work like any other inequality in mathematics but statistics can throw a curve ball sometimes so I'm not certain about it. I can't find any papers discussing if this approach is valid.
Another good example of this is a confidence interval for the mean and median of a normal distribution. The mean confidence interval is smaller but the median confidence interval is more robust so either might be preferred as an estimate of the other.

• You're forming a confidence interval for the parameter $\mu$. That confidence interval applies equally whether you're looking at the variance, the mean or any other population quantity which is equal to $\mu$. Your last paragraph suggests you're confusing the parameter and a sample statistic. A confidence interval for $\mu$ in the normal (which is the population median) is just a confidence interval for $\mu$, however you construct it. When you say 'median confidence interval' do you mean 'a confidence interval for that median, $\mu$ based on sample quantiles'? Commented Oct 1, 2013 at 22:56