1.5 Exercises: Chapter 1

1. The court case: the blue or green cap

A cab was involved in a hit and run accident at night. There are two cab companies in the town: blue and green. The former has 150 cabs, and the latter 850 cabs. A witness said that a blue cab was involved in the accident; the court tested his/her reliability under the same circumstances, and got that 80% of the times the witness correctly identified the color of the cab. what is the probability that the color of the cab involved in the accident was 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 switching the decision if there are four doors, where there are three goats and one car? Solve this problem 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 that you are analyzing to buy a car insurance next year. To make a better decision you want to know what is the probability that you have a car claim next year? You have the records of your car claims in 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 binomial, \(Y_i\sim Beroulli(p)\), and your probabilistic prior beliefs about \(p\) are well described by a beta distribution with parameters \(\alpha_0\) and \(\beta_0\), \(p\sim B(\alpha_0, \beta_0)\), then, you are interested in calculating the probability of a claim the next year \(P(Y_0 = 1|\mathbf{y})\).

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

5. Show that given the loss function, \(L({\theta},a)=|{\theta}-a|\), then \({\delta}(\mathbf{y})\) is the median.