1. * Suppose n balls are distributed into N urns according to the Maxwell-Boltzmann scheme and that the probabilities that a particular ball drops into the N urns are p1; p2;::: ; pN , respectively. Prove that the expected value of the number of empty urns is minimized when each pi D 1.
2. * Suppose n balls are distributed into N urns according to the Bose- Einstein scheme. Find a formula for the mean and the variance of M0, the number of empty urns. Hint: For the variance, try EŒM0.M0 - 1/] first.
3. Fifty balls are distributed into ten urns according to the M-B scheme. Find the expected number of urns with k balls for k D 2; 5; 10. Repeat if the balls are distributed according to the B-E scheme.