For an observation of N Poisson events, what is the PMF of a Poisson parameter? If $N$ events are observed resulting from a Poisson distribution with an unknown Poisson parameter, what is the distribution of the Poisson parameter? 
So if 
$X \sim \text{Poisson}(\mu)$ 
and a sample from the distribution $X$ gives $N$ events, what is the probability distribution function for $\mu$ (given a flat prior for $\mu$)?
How would that differ from the sampling distribution of the maximum likelihood estimator for $\mu$ in a frequentist analysis?
 A: (I presume the $N$ events are for observing the process for 1 unit of time or at least that we want the rate per the observation interval, otherwise we have an exposure-time coming in as well.)
$f(\mu|X=N) \propto f(X=N|\mu)f(\mu)$
$\,\,\:\quad\qquad\qquad \propto \exp(-\mu)\mu^N\,,\quad \mu>0$
which is (up to a normalizing constant) the density of a $\text{Gamma}(N+1,1)$ distribution; it has mean $N+1$ and variance $N+1$.
By contrast, the sampling distribution of $\hat{\mu}$ is $\text{Poisson}(\mu)$.
A confidence interval for $\mu$ would also be based on quantiles of a gamma distribution (specifically a $\text{Gamma}(N,1)$ as shown at the link).
Note that if you use a prior proportional to $\mu^{-1}$ -- a fairly common low-information conjugate (improper) prior for $\mu$ in the Bayesian setup -- you will get a $\text{Gamma}(N,1)$ posterior. (There's a connection between the confidence distributions in a frequentist analysis and the posteriors in a Bayesian analysis, at least when a suitably non-informative conjugate prior is applied.)
