Find posterior distribution of Poisson process knowing that the prior is Exponential$(1)$.

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This is the problem: Bus arrival times form a Poisson process with intensity measured in buses per hour. Your prior distribution on is that is an exponential random variable, Exponential$(1)$. Suppose that you observe buses in hours. Find the density function of the posterior distribution for . Identify this distribution.

What I thought is to write: $$f_{ \lambda | X}\left( \lambda | x\right) \propto f_{\lambda }\left( \lambda \right) \cdot f_{ X| \lambda }\left( x| \lambda \right)$$ where $f_{\lambda }\left( \lambda \right) = e^{-\lambda}$ . I have a problem calculating $f_{ X| \lambda }\left( x| \lambda \right)$, since I don't know how to use the fact $k$ buses are seen in $n$ hours. I would appreciate any advice on how to proceed or a possible solution. Thank you.

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Using your notation $X$ is the random variable denoting the number of buses observed in $n$ hours. Since the arrivals follow a Poisson process, $(X \mid \lambda) \sim \mathrm{Poisson}(n \lambda)$. Now following your idea to use Bayes' rule, what can you say about $f_{X \mid \lambda}(x \mid \lambda)$?