We flip a fair coin If it is heads then we roll a die n time

We flip a fair coin. If it is heads then we roll a die n times, if it is tails we sample a number n times from the set {1, 2, 3, 4} with replacement. We denote the resulting n numbers by X1, . . . , Xn. Show that these random variables are exchangeable, but not independent.

Solution

The above experiment involves two events. First is flipping a coin (Head or Tail) and the second event is rolling a die n times, given the first outcome is Head, or sampling a number from a set n times, given the first outcome is Tail. Rolling a die has a outcome from the set {1,2,3,4,5,6} and sampling a number has a outcome from {1,2,3,4}. Since, sampling a number is a subset of rolling a die, these random variables are exchangable. The occurence of the first event affects the outcome of the second event so that the probability is changed. Hence the outcomes are dependent.