Probability distribution of the expectation value of a poisson variable given an observed value.

what is probability density $P(\lambda|Poisson(\lambda) = N)$?

In other words, if I have a poisson variable $X(\lambda)$, where $\lambda$ is unknown, and I observe $X=N$, what is the probability density function of $\lambda$? (I don't know $P(N)$ or $P(\lambda)$.)

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Got something from an answer below? – Did Jul 19 '13 at 0:56

One assumes that the distribution of $X$ conditionally on some parameter $\Lambda$ is Poisson $\Lambda$ and that the distribution of $\Lambda$ has density $g$.
Then the density $h_n$ of the conditional distribution of $\Lambda$ conditionally on $X=n$ is such that $$h_n(\lambda)=\frac{g(\lambda)p_\lambda(n)}{G(n)},\qquad G(n)=\int_0^\infty g(\lambda)p_\lambda(n)\mathrm d\lambda,$$ where $p_\lambda$ is the Poisson distribution of parameter $\lambda$, that is, $p_\lambda(n)=\mathrm e^{-\lambda}\lambda^n/n!$ for every integer $n\geqslant0$. Alternatively, $$h_n(\lambda)=\frac{g(\lambda)\mathrm e^{-\lambda}\lambda^n}{\int_0^\infty g(t)\mathrm e^{-t}t^n\mathrm dt}.$$ Of course, the result depends very much on the density $g$ of the prior distribution of $\Lambda$.