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ProbabilityHard

Failed settlements per day are Poisson with rate $λ$ , and the posterior for $λ$ is a Gamma distribution with shape $3$ and rate $4$ . What is the posterior predictive chance of no failed settlement tomorrow? Give a fraction.

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Hint

Average $exp (- λ)$ over the posterior. Integrating out $λ$ gives a power of $\frac{b}{b + 1}$ .

Solution

The chance is $E [exp (- λ)]$ , which is $(\frac{b}{b + 1})^{a}$ for shape $a$ and rate $b$ .
With shape $3$ and rate $4$ that is $(\frac{4}{5})^{3} = \frac{64}{125}$ .

Answer: $\frac{64}{125}$

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