/*! 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} Q. 5.20 Verify thatVar(X)=伪位2聽whenX聽... [FREE SOLUTION] | 91影视

91影视

Find study content
Learning Materials

Discover learning materials by subject, university or textbook.

Explanations

Textbooks
All Subjects

Anthropology

Archaeology

Architecture

Art and Design

Bengali

Biology

Business Studies

Greek

Chemistry

Chinese

Combined Science

Computer Science

Economics

Engineering

English

English Literature

Environmental Science

French

Geography

German

History

Hospitality and Tourism

Human Geography

Italian

Japanese

Law

Macroeconomics

Marketing

Math

Media Studies

Medicine

Microeconomics

Music

Nursing

Nutrition and Food Science

Physics

Polish

Politics

Psychology

Religious Studies

Sociology

Spanish

Sports Sciences

Translation

All Subjects

Biology

Business Studies

Chemistry

Combined Science

Computer Science

Economics

English

Environmental Science

Geography

History

Math

Physics

Psychology

Sociology
Features
Features

Discover all of these amazing features with a free account.

Flashcards

91影视 AI

Notes

Study Plans

Study Sets
What鈥檚 new?

Flashcards
Study your flashcards with three learning modes.

Study Sets
All of your learning materials stored in one place.

Notes
Create and edit notes or documents.

Study Plans
Organise your studies and prepare for exams.
91影视
Discover

All the hacks around your studies and career - in one place.

Find a degree

Magazine
Featured

Magazine
Trusted advice for anyone who wants to ace their studies & career.

The largest student job board with the most exciting opportunities.

Verified student deals from top brands.

Discover our mobile app to take your studies anywhere.

Learning Materials

Features

Discover

A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 5: Q. 5.20 (page 216)

Verify that $Var (X) = \frac{伪}{位^{2}}$ when $X$ is a gamma random variable with parameters $伪$ and $位$

Short Answer

Expert verified

To get the Variance, find the generic expression for $E (X^{k})$ and use it.

Step by step solution

01

Determine the positive integral.

Let's find the first and second moments of $X$ if $X$ $~$ Gamma $(伪, 位)$

For any positive integer $k$ .

we have that

localid="1649619459546" $E (X^{k}) = {鈭�}_{0}^{鈭�} x^{k} 路 \frac{位^{伪}}{螕 (伪)} x^{伪 - 1} e^{- 位 x} d x = \frac{位^{伪}}{螕 (伪)} {鈭�}_{0}^{鈭�} x^{k + 伪 - 1} e^{- 位 x} d x$

02

Expression the integral.

in order to evaluate the integral, make the substitution $s = 位 x$

Hence, the expression above is equal to localid="1649619493912" $\frac{位^{伪}}{螕 (伪)} {鈭�}_{0}^{鈭�} x^{k + 伪 - 1} e^{- 位 x} d x = \frac{位^{伪}}{螕 (伪)} {鈭�}_{0}^{鈭�} {(\frac{s}{位})}^{k + 伪 - 1} e^{- s} \frac{d s}{位}$

localid="1649619476293"

= \frac{1}{位^{k} 路 螕 (伪)} {鈭�}_{0}^{鈭�} s^{k + 伪 - 1} e^{- s} d s = \frac{螕 (k + 伪)}{位^{k} 路 螕 (伪)}

So, we have obtained that,

$E (X^{k}) = \frac{螕 (k + 伪)}{位^{k} 路螕 (伪)}$

03

Implies the value.

The implies that

$E (X) = \frac{螕 (伪 + 1)}{位脳螕 (伪)} = \frac{伪脳螕 (伪)}{位脳螕 (伪)} = \frac{伪}{位}$

and

E (X^{2}) = \frac{螕 (伪 + 2)}{位^{2} 脳 螕 (伪)} = \frac{(伪 + 1) 伪 螕 (伪)}{位^{2} 脳 螕 (伪)} = \frac{(伪 + 1) 伪}{位^{2}}

Finally, we have that

$Var (X) = E (X^{2}) - E (X)^{2} = \frac{(伪 + 1) 伪}{位^{2}} - \frac{伪^{2}}{位^{2}} = \frac{伪}{位^{2}}$

Which has to be proved.

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

Compute the hazard rate function of $X$ when $X$ is uniformly distributed over. $(0, a)$

5.6. Compute $E [X]$ if $X$ has a density function given by

$(a) f (x) = \{\begin{cases} \frac{1}{4} x e^{\frac{- x}{2}} & x > 0 \\ 0 & o t h e r w i s e \end{cases}$ ;

$(b) f (x) = \{\begin{cases} c (1 鈭� x^{2}) & - 1 < x < 1 \\ 0 & o t h e r w i s e \end{cases}$ ;

$(c) f (x) = \{\begin{cases} \frac{5}{x^{2}} & x > 5 \\ 0 & x 鈮� 5 \end{cases}$ .

Let $X$ be a random variable that takes on values between $0$ and $c$ . That is $P \{0 鈮� X 鈮� c\} = 1$ .Show that

$V a r (X) 鈮� \frac{c^{2}}{4}$

Hint: One approach is to first argue that

localid="1646883602992" $E [X^{2}] 鈮� c E [X]$

and then use this inequality to show that

$V a r (X) 鈮� c^{2} [伪 (1 - 伪)] w h e r e 伪 = \frac{E [X]}{c}$

Compute the hazard rate function of a Weibull random variable and show it is increasing when $饾浗鈮� 1$ and decreasing when $饾浗鈮� 1$

The lung cancer hazard rate $位 (t)$ of a $t$ -year-old male smoker is such that

$位 (t) = . 027 + . 00025 {(t 鈭� 40)}^{2}$ $t 鈮� 40$

Assuming that a $40$ -year-old male smoker survives all other hazards, what is the probability that he survives to

$(a)$ age $50$ and $(b)$ age 60 without contracting lung cancer?

See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

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.