exercise:F06ecc053b: 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> A census in the United States is an attempt to count everyone in the country. It is inevitable that many people are not counted. The U. S. Census Bureau proposed a way to estimate the number of people who were not counted by the latest census....") |
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A census in the United States is an attempt to count everyone in the country. It is inevitable that many people are not counted. The U. | |||
everyone in the country. It is inevitable that many people are not counted. The U. | |||
S. Census Bureau proposed a way to estimate the number of people who were not | S. Census Bureau proposed a way to estimate the number of people who were not | ||
counted by the latest census. Their proposal was as follows: In a given locality, | counted by the latest census. Their proposal was as follows: In a given locality, | ||
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locality, and <math>n_2</math> people were counted. In addition, <math>n_{12}</math> people were counted | locality, and <math>n_2</math> people were counted. In addition, <math>n_{12}</math> people were counted | ||
both times. | both times. | ||
<ul><li> Given <math>N</math>, <math>n_1</math>, and <math>n_2</math>, let <math>X</math> denote the number of people counted both | <ul style="list-style-type:lower-alpha"><li> Given <math>N</math>, <math>n_1</math>, and <math>n_2</math>, let <math>X</math> denote the number of people counted both | ||
times. Find the probability that <math>X = k</math>, where <math>k</math> is a fixed positive integer | times. Find the probability that <math>X = k</math>, where <math>k</math> is a fixed positive integer | ||
between 0 and <math>n_2</math>. | between 0 and <math>n_2</math>. | ||
Latest revision as of 00:02, 14 June 2024
A census in the United States is an attempt to count everyone in the country. It is inevitable that many people are not counted. The U. S. Census Bureau proposed a way to estimate the number of people who were not counted by the latest census. Their proposal was as follows: In a given locality, let [math]N[/math] denote the actual number of people who live there. Assume that the census counted [math]n_1[/math] people living in this area. Now, another census was taken in the locality, and [math]n_2[/math] people were counted. In addition, [math]n_{12}[/math] people were counted both times.
- Given [math]N[/math], [math]n_1[/math], and [math]n_2[/math], let [math]X[/math] denote the number of people counted both times. Find the probability that [math]X = k[/math], where [math]k[/math] is a fixed positive integer between 0 and [math]n_2[/math].
- Now assume that [math]X = n_{12}[/math]. Find the value of [math]N[/math] which maximizes the expression in part (a). Hint: Consider the ratio of the expressions for successive values of [math]N[/math].