A box contains an unknown number N of identical bolts. In order to get an idea of the size N, we randomly mark one of the bolts from the box. Next we select at random a bolt from the box. If this is...

A box contains an unknown number N of identical bolts. In order to get an idea of the size N, we randomly mark one of the bolts from the box. Next we select at random a bolt from the box. If this is the marked bolt we stop, otherwise we return the bolt to the box, and we randomly select a second one, etc. We stop when the selected bolt is the marked one. Let X be the number of times a bolt was selected. Later (in Exercise 21.11) we will try to find an estimate of N. Here we look at the probability distribution of X.

a. What is the probability distribution of X? Specify its parameter(s)!

b. The drawback of this approach is that X can attain any of the values 1, 2, 3,... , so that if N is large we might be sampling from the box for quite a long time. We decide to sample from the box in a slightly different way: after we have randomly marked one of the bolts in the box, we select at random a bolt from the box. If this is the marked one, we stop, otherwise we randomly select a second bolt (we do not return the selected bolt). We stop when we select the marked bolt. Let Y be the number of times a bolt was selected