/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q16E It is said that a random variabl... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Q16E (page 326)

It is said that a random variable X has the Pareto distribution with parameters\({{\bf{x}}_{\bf{0}}}\,{\bf{and}}\,{\bf{\alpha }}\) if X has a continuous distribution for which the pdf\({\bf{f}}\left( {{\bf{x|}}\,{{\bf{x}}_{\bf{0}}}{\bf{,\alpha }}} \right)\) is as follows

\(\begin{array}{l}{\bf{f}}\left( {{\bf{x|}}\,{{\bf{x}}_{\bf{0}}}{\bf{,\alpha }}} \right){\bf{ = }}\frac{{{\bf{\alpha }}{{\bf{x}}_{\bf{0}}}^{\bf{\alpha }}}}{{{{\bf{x}}^{{\bf{\alpha + 1}}}}}}\,{\bf{,x}} \ge {{\bf{x}}_{\bf{0}}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\bf{ = }}\,{\bf{0}}\,\,{\bf{,x < }}{{\bf{x}}_{\bf{0}}}\end{array}\)

Show that if X has this Pareto distribution, then the random variable\({\bf{log}}\left( {{\bf{X|}}\,{{\bf{x}}_{\bf{0}}}} \right)\)has the exponential distribution with parameter 伪.

Short Answer

Expert verified

The random variable\({\rm{log}}\left( {X|\,{x_0}} \right)\) has the exponential distribution with parameter 伪.

Step by step solution

01

Given information

X has the Pareto distribution with parameters \({x_0}\,{\rm{and}}\,\alpha \). We need to prove that the random variable \({\rm{log}}\left( {X|\,{x_0}} \right)\) has the exponential distribution with parameter 伪.

02

Proof of random variable log(X|xo) has the exponential distribution with parameter α.

X follows has the Pareto distribution with parameters \({x_0}\,{\rm{and}}\,\alpha \) then the CDF of X is given by

\(P\left( {X \le x} \right) = 1 - {\left( {\frac{{{x_0}}}{x}} \right)^\alpha }\)

Let \(Y = {\rm{log}}\left( {\frac{X}{{{x_0}}}} \right)\)then in order to show that the random variable \(\log \left( {X|\,{x_0}} \right)\) has the exponential distribution with parameter 伪 we consider the following

\(\begin{array}{l}P\left( {Y \le y} \right) = P\left( {{\rm{log}}\left( {\frac{X}{{{x_0}}}} \right) \le y} \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = P\left( {\frac{X}{{{x_0}}} \le {e^y}} \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = P\left( {X \le {x_0}{e^y}} \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 1 - {\left( {\frac{{{x_0}}}{{{x_0}{e^y}}}} \right)^\alpha }\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 1 - {e^{ - \alpha y}}\end{array}\)

This indicates that the random variable \({\rm{log}}\left( {X|\,{x_0}} \right)\) has the exponential distribution with parameter 伪.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose that n items are being tested simultaneously, the items are independent, and the length of life of each item has the exponential distribution with parameter\(\beta \).Determine the expected length of time until three items have failed. Hint: The required value is\(E\left( {{Y_1} + {Y_2} + {Y_3}} \right)\)in the notation of Theorem 5.7.11.

Suppose that 15,000 people in a city with a population of 500,000 are watching a certain television program. If 200 people in the city are contacted at random, what is the approximate probability that fewer than four of themare watching the program?

Let X have the normal distribution whose p.d.f. is given by (5.6.6). Instead of using the m.g.f., derive the variance of X using integration by parts.

Suppose that the random variables\({X_1},...,{X_k}\)are independent and\({X_i}\)has the exponential distribution with parameter\({\beta _i}\left( {i = 1,...,n} \right)\). Let\(Y = \min \left\{ {{X_{1,...,}}{X_k}} \right\}\)Show that Y has the exponential distribution with parameter\({\beta _1} + .... + {\beta _k}\).

Suppose that F is a continuous c.d.f. on the real line, and let \({{\bf{\alpha }}_{\bf{1}}}\,{\bf{and}}\,{{\bf{\alpha }}_{\bf{2}}}\)be numbers such that \({\bf{F}}\left( {{{\bf{\alpha }}_{\bf{1}}}} \right)\,{\bf{ = 0}}{\bf{.3}}\,{\bf{and}}\,{\bf{F}}\left( {\,{{\bf{\alpha }}_{\bf{2}}}} \right){\bf{ = 0}}{\bf{.8}}{\bf{.}}\). Suppose 25 observations are selected at random from the distribution for which the c.d.f. is F. What is the probability that six of the observed values will be less than \({{\bf{\alpha }}_{\bf{1}}}\), 10 of the observed values will be between \({{\bf{\alpha }}_{\bf{1}}}\) and \({{\bf{\alpha }}_{\bf{2}}}\), and nine of the observed values will be greater than \({{\bf{\alpha }}_{\bf{2}}}\)?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.