31 Chapter 5 Exercises

31.1 Question 1

Exchangeability with known model parameters: For each of the following three examples, answer: (i) Are observations \(y_1\) and \(y_2\) exchangeable? (ii) Are observations \(y_1\) and \(y_2\) independent? (iii) Can we act as if the two observations are independent?

a) A box has one black ball and one white ball. We pick a ball \(y_1\) at random, put it back, and pick another ball \(y_2\) at random.

The observations are exchangeable, independent, and we can act as if they are independent.

b) A box has one black ball and one white ball. We pick a ball \(y_1\) at random, we do not put it back, then we pick ball \(y_2\).

The observations are exchangeable because we don’t have information about which ball is most likely to be picked first, but they are not independent because with \(y_1\), we know the result for \(y_2\). I don’t think we can treat the observations as independent

c) A box has a million black balls and a million white balls. We pick a ball \(y_1\) at random, we do not put it back, then we pick ball \(y_2\) at random.

The observations are exchangeable for the same reason as in the answer to (b). The observations are not independent because we will know that, after \(y_1\), the other color is slightly more likely to be picked. Since there are so many balls, we can likely treat the observations are independent.