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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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I don't know this notation: What does $K$ stands for? – math Dec 4 '12 at 13:48
It means $P(K=k)$, $K$ stands for the random variable, $k$ stands for the value of $K$. – CravingSpirit Dec 4 '12 at 14:07

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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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Thanks. This is kind of deep for me at the moment. – CravingSpirit Dec 4 '12 at 14:08
@CravingSpirit: OK; to put it in more mundane terms: What you think about $\lambda$ after you learn that there were $k$ arrivals in a unit time depends on what you that about $\lambda$ before you learned that. For instance, if you think a priori that very low values of $\lambda$ are very unlikely, then even after you observe $0$ arrivals the probability you assign a posteriori to low values of $\lambda$ will be less than the probability you would have assigned a posteriori after observing $0$ arrivals if you had a priori believed these low values to be quite likely. – joriki Dec 4 '12 at 14:30
Ok, you mean the probability of $\lambda$ depends both on our a priori judgment and the value of $k$. – CravingSpirit Dec 4 '12 at 14:35
@CravingSpirit: Yes, that just about summarizes it :-) – joriki Dec 4 '12 at 14:35

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