The probability distribution of a maximum-likelihood-estimated parameter

Question

I have a discrete random variable $X$, which obeys the Poission distribution $X \sim \mathcal{P}(\lambda)$ with $\lambda$ being its mean value. $\lambda$ is unknown and to be estimated. Now I carry out only one measurement of $X$ and get the result $x$. The maximum-likelihood-estimation (MLE) of $\lambda$ is $x$. But how do I know how good this estimation is? Or, I want to obtain the probability distribution of $\lambda$.

I know that if I have many measurements I can estimate this with a $\chi^2$ distribution. But this is extracted from a real problem, which only allows me to do one measurement.

Thanks!

Answer 1

If n=1, the only probability distribution you will be able to come up with will be the Poisson distribution with $\lambda=x$. You cannot characterize a distribution describing the rate of an event with only one data point.