[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Assume that, every time you buy a box of Wheaties, you

receive a picture of one of the [math]n[/math] players for the New York Yankees (see Exercise \ref{sec 3.2}.). Let [math]X_k[/math] be the number of additional boxes you have to buy, after you have obtained [math]k - 1[/math] different pictures, in order to obtain the next new picture. Thus [math]X_1 = 1[/math], [math]X_2[/math] is the number of boxes bought after this to obtain a picture different from the first pictured obtained, and so forth.

• Show that [math]X_k[/math] has a geometric distribution with [math]p = (n - k + 1)/n[/math].
• Simulate the experiment for a team with 26 players (25 would be more accurate but we want an even number). Carry out a number of simulations and estimate the expected time required to get the first 13 players and the expected time to get the second 13. How do these expectations compare?
• Show that, if there are [math]2n[/math] players, the expected time to get the first half of the players is
  [[math]] 2n \left( \frac 1{2n} + \frac 1{2n - 1} +\cdots+ \frac 1{n + 1} \right)\ , [[/math]]
  and the expected time to get the second half is
  [[math]] 2n \left( \frac 1n + \frac 1{n - 1} +\cdots+ 1 \right)\ . [[/math]]
• In Example we stated that
  [[math]] 1 + \frac 12 + \frac 13 +\cdots+ \frac 1n \sim \log n + .5772 + \frac 1{2n}\ . [[/math]]
  Use this to estimate the expression in (c). Compare these estimates with the exact values and also with your estimates obtained by simulation for the case [math]n = 26[/math].