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Chapter 4: Q10E (page 147)

A family of pdf鈥檚 that has been used to approximate the distribution of income, city population size, and size of firms is the Pareto family. The family has two parameters, \({\rm{k}}\) and \({\rm{\theta }}\), both\({\rm{ > 0}}\), and the pdf is

\({\rm{f(x;\theta ) = \{ }}\begin{array}{*{20}{c}}{\frac{{{\rm{k}} \cdot {{\rm{\theta }}^{\rm{k}}}}}{{{{\rm{x}}^{{\rm{k + 1}}}}}}}&{{\rm{x}} \ge {\rm{\theta }}}\\{\rm{0}}&{{\rm{x < \theta }}}\end{array}\)

a. Sketch the graph of \({\rm{f(x;\theta )}}\).

b. Verify that the total area under the graph equals \({\rm{1}}\).

c. If the rv \({\rm{X}}\) has pdf \({\rm{f(x;\theta )}}\), for any fixed \({\rm{b > \theta }}\), obtain an expression for \({\rm{P(X}} \le {\rm{b)}}\).

d. For \({\rm{\theta < a < b}}\) obtain an expression for the probability \({\rm{P(a}} \le {\rm{X}} \le {\rm{b)}}\).

Short Answer

Expert verified

(a) The graph for\({\rm{f(x;\theta )}}\)is -

(b) It is verified that the total area under the graph equals \({\rm{1}}\).

(c) The expression for\({\rm{P(x}} \le {\rm{b)}}\)is obtained as\(\frac{{{{\rm{b}}^{\rm{k}}}{\rm{ - }}{{\rm{\theta }}^{\rm{k}}}}}{{{{\rm{b}}^{\rm{k}}}}}\).

(d) The expression for \({\rm{P(a}} \le {\rm{x}} \le {\rm{b)}}\) is obtained as \(\frac{{{{\rm{\theta }}^{\rm{k}}} \cdot \left( {{{\rm{b}}^{\rm{k}}}{\rm{ - }}{{\rm{a}}^{\rm{k}}}} \right)}}{{{{\rm{b}}^{\rm{k}}} \cdot {{\rm{a}}^{\rm{k}}}}}\).

Step by step solution

01

Concept Introduction

Probability refers to the likelihood of a random event's outcome. This word refers to determining the likelihood of a given occurrence occurring.

02

The graph for \({\rm{f(x;\theta )}}\)

(a)

The graph of \({\rm{f(x;k,\theta )}}\) is shown below.

Therefore, the graph is obtained.

03

Total Area under the graph

(b)

The pdf of\({\rm{f(x)}}\)is given as 鈥�

\({\rm{f(x) = }}\left\{ {\begin{array}{*{20}{l}}{\frac{{{\rm{k}} \cdot {{\rm{\theta }}^{\rm{k}}}}}{{{{\rm{x}}^{{\rm{k + 1}}}}}}}&{{\rm{X}} \ge {\rm{\theta }}}\\{\rm{0}}&{{\rm{X < \theta }}}\end{array}} \right.\)

It is known that for any pdf \({\rm{f(x)}}\)鈥�

\({\rm{Area under graph of f(x) = }}\int_\infty ^\infty {\rm{f}} {\rm{(x)}} \cdot {\rm{dx}}\)

From the given pdf, it can be written 鈥�

\(\begin{array}{c}\int_\infty ^\infty {\rm{f}} {\rm{(x)}} \cdot {\rm{dx = }}\int_{\rm{\theta }}^\infty {\frac{{{\rm{k}} \cdot {{\rm{\theta }}^{\rm{k}}}}}{{{{\rm{x}}^{{\rm{k + 1}}}}}}} \cdot {\rm{dx}}\\{\rm{ = k}} \cdot {{\rm{\theta }}^{\rm{k}}}\int_{\rm{\theta }}^\infty {{{\rm{x}}^{{\rm{ - (k + 1)}}}}} \cdot {\rm{dx}}\\{\rm{ = k}} \cdot {{\rm{\theta }}^{\rm{k}}}\left( {\frac{{{{\rm{x}}^{{\rm{ - k}}}}}}{{{\rm{ - k}}}}} \right)_{\rm{\theta }}^\infty \\{\rm{ = }}\frac{{{\rm{k}} \cdot {{\rm{\theta }}^{\rm{k}}}}}{{{\rm{ - k}}}}\left( {\frac{{\rm{1}}}{{{{\rm{x}}^{\rm{k}}}}}} \right)_{\rm{\theta }}^\infty \\{\rm{ = - }}{{\rm{\theta }}^{\rm{k}}}\left( {\frac{{\rm{1}}}{{{\infty ^{\rm{k}}}}}{\rm{ - }}\frac{{\rm{1}}}{{{{\rm{\theta }}^{\rm{k}}}}}} \right)\\{\rm{ = - }}{{\rm{\theta }}^{\rm{k}}}\left( {{\rm{0 - }}\frac{{\rm{1}}}{{{{\rm{\theta }}^{\rm{k}}}}}} \right)\\\int_\infty ^\infty {\rm{f}} {\rm{(x)}} \cdot {\rm{dx = 1}}\end{array}\)

Here \(\frac{{\rm{1}}}{{{\infty ^{\rm{k}}}}}{\rm{ = 0}}\) is used since it is given that \({\rm{k > 0}}\).

Therefore, the area is verified.

04

Expression for \({\rm{P(X}} \le {\rm{b)}}\) 

(c)

From the given pdf, it can be written 鈥�

\(\begin{array}{c}{\rm{P(x}} \le {\rm{b) = }}\int_{\rm{\theta }}^{\rm{b}} {\frac{{{\rm{k}} \cdot {{\rm{\theta }}^{\rm{k}}}}}{{{{\rm{x}}^{{\rm{k + 1}}}}}}} \cdot {\rm{dx}}\\{\rm{ = k}} \cdot {{\rm{\theta }}^{\rm{k}}}\int_{\rm{\theta }}^{\rm{b}} {{{\rm{x}}^{{\rm{ - (k + 1)}}}}} \cdot {\rm{dx}}\\{\rm{ = k}} \cdot {{\rm{\theta }}^{\rm{k}}}\left( {\frac{{{{\rm{x}}^{{\rm{ - k}}}}}}{{{\rm{ - k}}}}} \right)_{\rm{\theta }}^{\rm{b}}\\{\rm{ = }}\frac{{{\rm{k}} \cdot {{\rm{\theta }}^{\rm{k}}}}}{{{\rm{ - k}}}}\left( {\frac{{\rm{1}}}{{{{\rm{x}}^{\rm{k}}}}}} \right)_{\rm{\theta }}^{\rm{b}}\\{\rm{ = - }}{{\rm{\theta }}^{\rm{k}}}\left( {\frac{{\rm{1}}}{{{{\rm{b}}^{\rm{k}}}}}{\rm{ - }}\frac{{\rm{1}}}{{{{\rm{\theta }}^{\rm{k}}}}}} \right)\\{\rm{ = - }}{{\rm{\theta }}^{\rm{k}}}\left( {\frac{{{{\rm{\theta }}^{\rm{k}}}{\rm{ - }}{{\rm{b}}^{\rm{k}}}}}{{{{\rm{\theta }}^{\rm{k}}} \cdot {{\rm{b}}^{\rm{k}}}}}} \right)\\{\rm{ = }}\frac{{{{\rm{b}}^{\rm{k}}}{\rm{ - }}{{\rm{\theta }}^{\rm{k}}}}}{{{{\rm{b}}^{\rm{k}}}}}\end{array}\)

Therefore, the expression is obtained as \(\frac{{{{\rm{b}}^{\rm{k}}}{\rm{ - }}{{\rm{\theta }}^{\rm{k}}}}}{{{{\rm{b}}^{\rm{k}}}}}\).

05

Expression for \({\rm{P(a}} \le {\rm{X}} \le {\rm{b)}}\) 

(d)

From the given pdf, it can be written 鈥�

\(\begin{array}{c}{\rm{P(a}} \le {\rm{x}} \le {\rm{b) = }}\int_{\rm{a}}^{\rm{b}} {\frac{{{\rm{k}} \cdot {{\rm{\theta }}^{\rm{k}}}}}{{{{\rm{x}}^{{\rm{k + 1}}}}}}} \cdot {\rm{dx}}\\{\rm{ = k}} \cdot {{\rm{\theta }}^{\rm{k}}}\int_{\rm{a}}^{\rm{b}} {{{\rm{x}}^{{\rm{ - (k + 1)}}}}} \cdot {\rm{dx}}\\{\rm{ = k}} \cdot {{\rm{\theta }}^{\rm{k}}}\left( {\frac{{{{\rm{x}}^{{\rm{ - k}}}}}}{{{\rm{ - k}}}}} \right)_{\rm{a}}^{\rm{b}}\\{\rm{ = }}\frac{{{\rm{k}} \cdot {{\rm{\theta }}^{\rm{k}}}}}{{{\rm{ - k}}}}\left( {\frac{{\rm{1}}}{{{{\rm{x}}^{\rm{k}}}}}} \right)_{\rm{a}}^{\rm{b}}\\{\rm{ = - }}{{\rm{\theta }}^{\rm{k}}}\left( {\frac{{\rm{1}}}{{{{\rm{b}}^{\rm{k}}}}}{\rm{ - }}\frac{{\rm{1}}}{{{{\rm{a}}^{\rm{k}}}}}} \right)\\{\rm{ = - }}{{\rm{\theta }}^{\rm{k}}}\left( {\frac{{{{\rm{a}}^{\rm{k}}}{\rm{ - }}{{\rm{b}}^{\rm{k}}}}}{{{{\rm{a}}^{\rm{k}}} \cdot {{\rm{b}}^{\rm{k}}}}}} \right)\\{\rm{P(a}} \le {\rm{x}} \le {\rm{b) = }}\frac{{{{\rm{\theta }}^{\rm{k}}} \cdot \left( {{{\rm{b}}^{\rm{k}}}{\rm{ - }}{{\rm{a}}^{\rm{k}}}} \right)}}{{{{\rm{b}}^{\rm{k}}} \cdot {{\rm{a}}^{\rm{k}}}}}\end{array}\)

Therefore, the expression is obtained as \(\frac{{{{\rm{\theta }}^{\rm{k}}} \cdot \left( {{{\rm{b}}^{\rm{k}}}{\rm{ - }}{{\rm{a}}^{\rm{k}}}} \right)}}{{{{\rm{b}}^{\rm{k}}} \cdot {{\rm{a}}^{\rm{k}}}}}\).

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Most popular questions from this chapter

A sample of \({\rm{15}}\) female collegiate golfers was selected and the clubhead velocity (km/hr) while swinging a driver was determined for each one, resulting in the following data

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Let \({\rm{X}}\) be a continuous \({\rm{rv}}\) with cdf

\({\rm{F(x) = }}\left\{ {\begin{array}{*{20}{c}}{\rm{0}}&{{\rm{x}} \le {\rm{0}}}\\{\frac{{\rm{x}}}{{\rm{4}}}\left( {{\rm{1 + ln}}\left( {\frac{{\rm{4}}}{{\rm{x}}}} \right)} \right)}&{{\rm{0 < x}} \le {\rm{4}}}\\{\rm{1}}&{{\rm{x > 4}}}\end{array}} \right.\)

(This type of cdf is suggested in the article 鈥淰ariability in Measured Bedload Transport Rates鈥� (Water 91影视 Bull., \({\rm{1985:39 - 48}}\)) as a model for a certain hydrologic variable.) What is a. \({\rm{P(X}} \le {\rm{1)}}\)? b. \({\rm{P(1}} \le {\rm{X}} \le {\rm{3)}}\)? c. The pdf of \({\rm{X}}\)?

Rockwell hardness of a metal is determined by impressing a hardened point into the surface of the metal and then measuring the depth of penetration of the point. Suppose the Rockwell hardness of a particular alloy is normally distributed with a mean of\({\bf{70}}\)and a standard deviation of\({\bf{3}}\). a. If a specimen is acceptable only if its hardness is between 67 and 75, what is the probability that a randomly chosen specimen has an acceptable hardness? b. If the acceptable range of hardness is\(\left( {{\bf{70}} - {\bf{c}},{\rm{ }}{\bf{70}} + {\bf{c}}} \right)\), for what value of c would\({\bf{95}}\% \)of all specimens have acceptable hardness? c. If the acceptable range is as in part (a) and the hardness of each of ten randomly selected specimens is independently determined, what is the expected number of acceptable specimens among the ten? d. What is the probability that at most eight of ten independently selected specimens have a hardness of less than\({\bf{73}}.{\bf{84}}\)?

In commuting to work, a professor must first get on a bus near her house and then transfer to a second bus. If the waiting time (in minutes) at each stop has a uniform distribution with \({\rm{A = 0}}\) and \({\rm{B = 5}}\), then it can be shown that the total waiting time \({\rm{Y}}\) has the pdf

\({\rm{f(x) = \{ }}\begin{array}{*{20}{c}}{\frac{{\rm{1}}}{{{\rm{25}}}}{\rm{y}}}\\{\frac{{\rm{2}}}{{\rm{5}}}{\rm{ - }}\frac{{\rm{1}}}{{{\rm{25}}}}{\rm{y}}}\\{\rm{0}}\end{array}\begin{array}{*{20}{c}}{{\rm{0}} \le {\rm{y < 5}}}\\{{\rm{5}} \le {\rm{y}} \le {\rm{10}}}\\{{\rm{y < 0ory > 10}}}\end{array}\)

a. Sketch a graph of the pdf of \({\rm{Y}}\).

b. Verify that \(\int_{{\rm{ - }}\infty }^\infty {{\rm{f(y)dy = 1}}} \).

c. What is the probability that total waiting time is at most \(3\) min?

d. What is the probability that total waiting time is at most \(8\) min?

e. What is the probability that total waiting time is between \(3\) and \(8\) min?

f. What is the probability that total waiting time is either less than \(2\) min or more than \(6\) min?

If the temperature at which a certain compound melts is a random variable with mean value \({\rm{12}}{{\rm{0}}^{\rm{^\circ }}}{\rm{C}}\) and standard deviation \({{\rm{2}}^{\rm{^\circ }}}{\rm{C}}\), what are the mean temperature and standard deviation measured in \(^{\rm{^\circ }}{\rm{F}}\)? (Hint: \(^{\rm{^\circ }}{\rm{F = 1}}{\rm{.}}{{\rm{8}}^{\rm{^\circ }}}{\rm{C + 32}}{\rm{.}}\))
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