CAS Exam MAS-II Fall 2019, Question 29
Study guide: Exam MAS-II
You are told that \theta represents the probability of a positive outcome for the random variable X. Accordingly, you specify the following model:
\begin{align*}
X &\sim \text{Bernoulli}(\theta) \\
p(\theta) &\sim 1/\theta^2; \quad \theta > 0
\end{align*}You observe X for one trial and note that the outcome is positive. You select the following proposal distribution for the Metropolis algorithm:
\theta_{prop} =
\begin{cases}
\theta_{curr} + 0.1 & \text{; probability} = 0.5 \\
\theta_{curr} - 0.1 & \text{; probability} = 0.5
\end{cases}\theta_{prop} and \theta_{curr} represent the proposed and current parameter values.
The algorithm is seeded with an initial value for \theta of 0.45.
Calculate the probability that after two iterations of the algorithm, \theta is at 0.45.
- A) Less than 0.15
- B) Greater than or equal to 0.15 but less than 0.25
- C) Greater than or equal to 0.25 but less than 0.35
- D) Greater than or equal to 0.35 but less than 0.45
- E) Greater than or equal to 0.45
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