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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!

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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.

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  • $\begingroup$ 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. $\endgroup$ – Vincent Aug 18 at 20:16
  • $\begingroup$ 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. $\endgroup$ – Zach Favakeh Aug 19 at 16:41
  • $\begingroup$ Yes, I know that. The problem is exactly that I only have one measurement. However, there's nothing I can do to measure more times, because this is a real experiment and my experimental set-up doesn't allow me to measure more than once. But thank you Zach anyway! $\endgroup$ – Vincent Aug 21 at 13:21
  • $\begingroup$ May I ask what it is you are measuring? Maybe there's a creative way to obtain more measurements. $\endgroup$ – Zach Favakeh Aug 21 at 15:40
  • $\begingroup$ I'm measuring the atom number of an atomic cloud by imaging, which is suffered from photon shot noise, which means the photon I can receive may deviate from the mean value. Photon shot noise obeys Poisson distribution. The thing is the atom number is also fluctuating and also obeys Poisson distribution. So, if I measure twice, I don't know whether the difference comes from photon shot noise or from the atom number fluctuation. Actually, what I care about is the atom number fluctuation rather than the atom number itself. So I want to know how large the influence of shot noise is. Thank you! $\endgroup$ – Vincent Aug 21 at 22:39

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