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1. From RosenBook

Do Exercises 5.3.26, 5.3.30, and Supplementary Exercise 5.20 (on pages 396-397) from RosenBook.

Hint: For Supplementary Exercise 5.20, it may be possible to eliminate a summation by expressing one of your answers in terms of the harmonic numbers Hn, where

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Further clarification: for parts (c) and (d) of Supplementary Exercise 5.20, do not assume any limit on how many times you put a ball into a bin; in each case you only stop when the condition (either filling a particular bin or filling all bins) is satisfied.

2. Not from RosenBook

Suppose that n basketball players, no two of whom have the same height, are tossed into an urn and then sampled uniformly at random without replacement until none are left. Let Xi be the height of the i-th basketball player removed from the urn.

  1. As a function of i, compute the probability that Xi > Xj for all j < i; i.e., the probability that the i-th basketball player removed from the urn sets a new height record.

  2. Let Y be the number of times that a new height record is set. Compute E[Y].
  3. Compute the exact probability that Y=n and compare this to the upper bound you get applying Markov's inequality to the value of E[Y] you just computed.

2014-06-17 11:57