An urn contains an unknown number N of identical unlabeled balls. In order to get an idea of the size N, we randomly mark one of the balls from the urn. Next, we select at random a ball from the urn. If this is the marked ball we stop, otherwise we return the ball to the urn, and we randomly select a second one and so on. We stop when the selected ball is the marked one. Let X be the number of times a ball was selected.
1. Show that X has a Geo(1/N) distribution.
2. We will use X to find a maximum likelihood estimate of N. Suppose that x1, ..., xn is a dataset, which is a realization of a random sample from a Geo(1/N) distribution. Determine the maximum likelihood estimate for N.