1. (Simple Application of Ewens' Formula). In a population with n D 10 animals, compute the probability that there are two species, each with five animals, and the probability that there are five species, each with two animals.
2. * (Ewens' Formula). In a population with n animals, derive an expression for the probability that there are no species with strictly larger than two animals.
3. (Use Your Computer). Plot the Po´lya-Eggenberger distribution for each of the following cases and then superimpose the Beta density function f .p/ as defined in the text for each corresponding case. Comment on the accuracy.
(a) a D b D 5; c D 1; n D 10I
(b) a D b D 5; c D 1; n D 25I
(c) a D 5; b D 20; c D 5; n D 100: