/*! 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.7.64 Typei light bulbs function for a... [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 7: Q.7.64 (page 357)

Type $i$ light bulbs function for a random amount of time having mean $渭 i$ and standard deviation $蟽 i, i = 1, 2$ . A light bulb randomly chosen from a bin of bulbs is a type $1$ bulb with probability $p$ and a type $2$ bulb with probability $1 鈭� p$ . Let $X$ denote the lifetime of this bulb. Find
(a) $E [X]$ ;
(b) $V a r (X)$ .

Short Answer

Expert verified

The values of $E [X]$ and $V a r (X)$ is

a) $E [X] = p 渭_{1} + (1 鈭� p) 渭_{2}$

b) $Var (X) = [p 蟽_{1}^{2} + (1 鈭� p) 蟽_{2}^{2}] + p (1 鈭� p) {(渭_{1} 鈭� 渭_{2})}^{2}$

Step by step solution

01

Step 1:Given Information(part a)

Given that $X$ denotes the lifetime of the bulb.

$Y = \{\begin{cases} 1 鈥呪赌呪赌呪赌� type 1 bulb with probability p \\ 2 鈥呪赌呪赌呪赌� type 2 bulb with probability (1 鈭� p) \end{cases}$

role="math" localid="1647349813765" $P (Y = 1) = p$ , $P (Y = 2) = (1 鈭� p)$

02

Step 2:Explanation(part a)

$E [X] = E (X 鈭� Y = 1) 鈰� P (Y = 1) + E (X 鈭� Y = 2) 鈰� P (Y = 2)$

$= p 渭_{1} + (1 鈭� p) 渭_{2}$

03

Step 3:Final Answer(part a)

The $E [X]$ is

\begin{aligned} E [X] = p 渭_{1} + (1 鈭� p) 渭_{2} \end{aligned}

04

Step 4:Given Information(part b)

Given that $X$ as the lifetime of this bulb.

$Var (X) = E [X^{2}] 鈭� (E [X])^{2}$

05

Step 5:Explanation(Step 5)

$E [X^{2}] = E (X^{2} 鈭� Y = 1) 鈰� P (Y = 1) + E (X^{2} 鈭� Y = 2) 鈰� P (Y = 2)$

$= p (渭_{1}^{2} + 蟽_{1}^{2}) + (1 鈭� p) (渭_{2}^{2} + 蟽_{2}^{2})$

$= [p 蟽_{1}^{2} + (1 鈭� p) 蟽_{2}^{2}] + [p 渭_{1}^{2} + (1 鈭� p) 渭_{2}^{2}]$

$Var (X) = [p 蟽_{1}^{2} + (1 鈭� p) 蟽_{2}^{2}] + [p 渭_{1}^{2} + (1 鈭� p) 渭_{2}^{2}] 鈭� p 渭_{1} + {[(1 鈭� p) 渭_{2}]}^{2}$

$= [p 蟽_{1}^{2} + (1 鈭� p) 蟽_{2}^{2}] + p (1 鈭� p) [渭_{1}^{2} + 渭_{2}^{2} + 2 渭_{1} 渭_{2}]$

$= [p 蟽_{1}^{2} + (1 鈭� p) 蟽_{2}^{2}] + p (1 鈭� p) {(渭_{1} 鈭� 渭_{2})}^{2}$

06

Step 6:Final Answer(part b)

The $V a r (X)$ is

Var (X) = [p 蟽_{1}^{2} + (1 鈭� p) 蟽_{2}^{2}] + p (1 鈭� p) {(渭_{1} 鈭� 渭_{2})}^{2}

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

Let $X_{1}, X_{2}, 鈥�, X_{n}$ be independent random variables having an unknown continuous distribution function $F$ and let $Y_{1}, Y_{2}, 鈥�, Y_{m}$ be independent random variables having an unknown continuous distribution function $G$ . Now order those $n + m$ variables, and let

$I_{i} = \{\begin{cases} 1 鈥呪赌呪赌呪赌� if the i th smallest of the n + m \\ 鈥呪赌呪赌呪赌� variables is from the X sample \\ 0 鈥呪赌呪赌呪赌� otherwise \end{cases}$

The random variable $R = {鈭�}_{i = 1}^{n + m} i I_{i}$ is the sum of the ranks of the $X$ sample and is the basis of a standard statistical procedure (called the Wilcoxon sum-of-ranks test) for testing whether $F$ and $G$ are identical distributions. This test accepts the hypothesis that $F = G$ when $R$ is neither too large nor too small. Assuming that the hypothesis of equality is in fact correct, compute the mean and variance of $R$ .

Hint: Use the results of Example 3e.

A group of 20 people consisting of 10 men and 10 women is randomly arranged into 10 pairs of 2 each. Compute the expectation and variance of the number of pairs that consist of a man and a woman. Now suppose the 20 people consist of 10 married couples. Compute the mean and variance of the number of married couples that are paired together.

Urn $1$ contains $5$ white and $6$ black balls, while urn $2$ contains $8$ white and $10$ black balls. Two balls are randomly selected from urn $1$ and are put into urn $2$ . If $3$ balls are then randomly selected from urn $2$ , compute the expected number of white balls in the trio.

Hint: LetXi = $1$ if the i th white ball initially in urn $1$ is one of the three selected, and let Xi = $0$ otherwise. Similarly, let Yi = $1$ if the i the white ball from urn $2$ is one of the three selected, and let Yi = $0$ otherwise. The number of white balls in the trio can now be written as ${鈭�}_{1}^{5} X_{i} + {鈭�}_{1}^{8} Y_{i}$

The best linear predictor of $Y$ with respect to $X_{1}$ and $X_{2}$ is equal to $a + b X_{1} + c X_{2}$ , where $a$ , $b$ , and $c$ are chosen to minimize $E [{(Y - (a + b X_{1} + c X_{2}))}^{2}]$ Determine $a$ , $b$ , and $c$ .

If $X_{1}, X_{2}, 鈥�, X_{n}$ are independent and identically distributed random variables having uniform distributions over $(0, 1)$ , find

(a) $E [\max (X_{1}, 鈥�, X_{n})]$ ;

(b) $E [\min (X_{1}, 鈥�, X_{n})]$ .

See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Geometry

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.