Discussion
Let X ~ Poisson (\(\lambda\)) and a single realization of X resulted in 3. If \(\lambda\) has a prior distribution
\[ P(\lambda) = q\, \lambda\, e^{-3\lambda},\ \lambda \gt 0, \]
then the posterior mean is given by :
Answer: C
The posterior mean is 5/4.
The prior \(q\lambda e^{-3\lambda}\) is a Gamma density with shape 2 and rate 3. With one Poisson observation x = 3 the likelihood is proportional to \(\lambda^{3}e^{-\lambda}\), so
\[ \begin{aligned} \pi(\lambda \mid x) &\propto \lambda^{3}e^{-\lambda} \cdot \lambda e^{-3\lambda} \\ &= \lambda^{4}e^{-4\lambda} \end{aligned} \]
which is Gamma with shape 5 and rate 4. Its mean is \(5/4\).
4/5 inverts the shape and rate, and 3/5 and 2/5 use the wrong shape or rate.
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