exercise:Ff5e075972: Difference between revisions
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(Created page with "<div class="d-none"><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></div> 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 addition...") |
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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:5ffc34493a |Exercise]]). 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. | |||
<ul style="list-style-type:lower-alpha"><li> Show that <math>X_k</math> has a geometric distribution with <math>p = (n - k + 1)/n</math>. | |||
receive a picture of one of the <math>n</math> players for the New York Yankees (see | |||
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. | |||
<ul><li> Show that <math>X_k</math> has a geometric distribution with <math>p = (n - k + 1)/n</math>. | |||
</li> | </li> | ||
<li> Simulate the experiment for a team with 26 players (25 would be more accurate | <li> Simulate the experiment for a team with 26 players (25 would be more accurate | ||
Latest revision as of 17:13, 14 June 2024
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). 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].