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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

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