# The probability distribution of a maximum-likelihood-estimated parameter

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!

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.
• Thank you Zach. But since this data point is all I have, $x$ is MLE estimator of $\lambda$. I wonder if it is possible to determine this estimator's confidence interval. – Vincent Aug 18 at 20:16
• Your n is not really high enough to use the central limit theorem to construct a confidence interval. Typically, with a high enough number of observations n, the distribution of $\epsilon$ = Poisson MLE - true $\lambda$ is aymptotically (as n->$\inf$) normally distributed about zero with variance = MLE/n. So if you really wanted to say, "here is a confidence interval," you would have to have variance = your MLE. – Zach Favakeh Aug 19 at 16:41