1.5 Exercises

  1. The Court Case: The Blue or Green Cab

    A cab was involved in a hit-and-run accident at night. There are two cab companies in the town: Blue and Green. The Blue company has 150 cabs, while the Green company has 850 cabs. A witness stated that a blue cab was involved in the accident. The court tested the reliability of the witness under similar circumstances and found that the witness correctly identified the color of the cab 80% of the time, but made an incorrect identification 25% of the time. What is the probability that the cab involved in the accident was actually blue, given that the witness said it was blue?

  2. The Monty Hall Problem

    What is the probability of winning a car in the Monty Hall problem if you switch your decision, when there are four doors, three goats, and one car? Solve this problem both analytically and computationally. What if there are \(n\) doors, \(n-1\) goats, and one car?

  3. Solve the health insurance example using a Gamma prior in the rate parametrization, that is, \(\pi(\lambda) = \frac{\beta_0^{\alpha_0}}{\Gamma(\alpha_0)} \lambda^{\alpha_0 - 1} \exp\left\{-\lambda \beta_0\right\}\).

  4. Suppose you are analyzing the decision to buy car insurance for the next year. To make a better decision, you want to know: What is the probability that you will have a car claim next year? You have the records of your car claims over the last 15 years, \(\mathbf{y} = \left\{ 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0 \right\}\).

    Assume that this is a random sample from a data-generating process (statistical model) that is Bernoulli, \(Y_i \sim \text{Ber}(p)\). Your prior beliefs about \(p\) are well described by a Beta distribution with parameters \(\alpha_0\) and \(\beta_0\), i.e., \(p \sim B(\alpha_0, \beta_0)\). You are interested in calculating the probability of a claim the next year, \(P(Y_0 = 1 \mid \mathbf{y})\).

    Solve this using both an empirical Bayes approach and a non-informative approach where \(\alpha_0 = \beta_0 = 1\) (uniform distribution).

  5. Show that, given the loss function \(L(\theta, a) = |\theta - a|\), the optimal decision rule minimizing the risk function, \(a^*(\mathbf{y})\), is the median.