Actuarial Notes

CAS Exam MAS-II Spring 2019, Question 18

Study guide: Exam MAS-II

You are building the following model:

\begin{align*}
y_i &\sim \text{Exponential}(\theta) \\
\theta &\sim \text{Uniform}(0, \omega) \\
\omega &\sim \text{Uniform}(7, 10)
\end{align*}

Your buddy, Sarah Jane, tells you that the posterior distribution of \theta \mid \omega is:

p(\theta \mid \omega) =
\begin{cases}
\dfrac{\frac{1}{\theta}e^{-\sum y_i/\theta}}{\int_0^{\omega} \frac{1}{\theta}e^{-\sum y_i/\theta}\,d\theta} & \text{if } 0 < \theta < \omega \\[2ex]
0 & \text{otherwise}
\end{cases}

Determine which of the following statements is correct.

  • A) Sarah Jane is correct.
  • B) Sarah Jane is incorrect because the bounds of the integral are incorrect.
  • C) Sarah Jane is incorrect because she didn't include the prior density.
  • D) Sarah Jane is incorrect because the variable of integration (d\theta) is incorrect.
  • E) Sarah Jane is incorrect for another reason not listed in (B), (C), or (D).

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