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We know the PMF of poisson distribution is $P_K(k) = e^{-\lambda} \frac{\lambda ^k}{k!}$, now, given $k$ arrivals in a unit time, what is the PDF of the arriving rate being $\lambda$?

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    It means $P(K=k)$, $K$ stands for the random variable, $k$ stands for the value of $K$.2012-12-04

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There's no such thing unless you specify a prior. Wikipedia has a section that gives a conjugate prior for this problem.

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    @CravingSpirit: Yes, that just about summarizes it :-)2012-12-04