#LyX 1.1 created this file. For more info see http://www.lyx.org/
\lyxformat 2.16
\textclass article
\language default
\inputencoding latin1
\fontscheme times
\graphics default
\paperfontsize 12
\spacing single 
\papersize Default
\paperpackage a4
\use_geometry 1
\use_amsmath 1
\paperorientation portrait
\leftmargin 1in
\topmargin 1in
\rightmargin 1in
\bottommargin 1in
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\defskip medskip
\quotes_language english
\quotes_times 2
\papercolumns 1
\papersides 1
\paperpagestyle default

\layout Section*

Math 107 Project : Probability and Distributions
\layout Standard


\noun on 
Due Date: Thursday, November 30, 2000 
\layout Standard


\latex latex 

\backslash 
vspace{.1in}
\layout Standard


\series bold 
Guidelines:
\series default 
 This project is a group project which is based on material found in the
 section on probability and more distribution functions in the text (pages
 416-423).
 Since this section will not be covered in lecture, you should first read
 this material before attempting the problems.
 Remember that part of your grade will be based on the quality of your written
 work.
 The paper you turn in should be a mix of equations, formulas and prose.
 Graphs may be copied from your calculator, but should be clearly labelled.
 Use complete sentences, good grammar, correct spelling and correct punctuation.
 You should write your answers in such a way that it can be read and understood
 by anyone who knows the material for this course.
 Finally, neatness counts, so the project should be neatly typed or written
 on good paper (not torn from a notebook).
 
\layout Standard


\series bold 
About Group Projects.

\series default 
 To get everyone involved and the group functioning smoothly, it is a good
 idea to meet as early as possible to arrange meeting times, etc.
 It might be helpful to bear in mind the there are at least four roles to
 played by various participants at various times: the chair, reporter, scheduler
 and scribe.
 The role of the chair is to try to get everyone involved and make sure
 everyone is understanding the ideas developed by the group.
 The reporter jots down the ideas of the group as they are discussed.
 The scheduler finds times and places where everyone in the group can meet,
 and finally, the scribe writes up the final report for the group.
 These jobs can be rotated on a per meeting basis if the group wishes.
 However, everyone should proofread the final draft and help in the other
 duties as they see fit.
 
\layout Standard

When the project is turned in, students will be asked to evaluate the level
 of participation by other group members by way of a project participation
 report to be filled out by each member individually and turned in to the
 recitation instructor.
\layout Standard

This project comes in the form of a memo from a division manager.
 
\layout Standard


\latex latex 

\backslash 
vspace{.1in}
\layout Section*

Intelligent Communications Corporation Standard Memo Form
\layout Standard

Date: 10/31/00
\layout Standard
\noindent 
To: Math analysis team
\layout Standard
\noindent 
From: J.
 Datapoint, Manager, statistics analysis team
\layout Standard
\noindent 
Subject: Reliability of our Talk Now line
\layout Standard


\latex latex 

\backslash 
vspace{.1in}
\layout Standard

As you know, our Talk Now two-way radio systems have been quite a success.
 Recently, however, we have had a large number of warranty returns within
 the one year warranty period.
 These returns seems to be localized at a small number of retail sites,
 and the question we need to answer is whether or not these warranty returns
 are reasonable.
 If not, we'll need to stop sales at these sites or send a team to track
 down the difficulties, which could be expensive.
 So management asked us to review our model, which we formulated during
 the development phase of the Talk Now project, and send information about
 the our model to you.
 Your role in this is to use calculus techniques (and only calculus techniques)
 to confirm our conclusions or draw new conclusions about this situation
 (we aren't going to tell you what ours were, so as not to bias your analysis).
 We would like you to write a report on this subject, intelligible to anyone
 with about two semesters of calculus under her/his belt.
 
\layout Standard


\series bold 
The Model.

\series default 
 We know that the density function for the lifetime 
\begin_inset Formula \( x \)
\end_inset 

 of a Talk Now system is given by 
\begin_inset Formula \[
p(x)=Ce^{-x/b}\]

\end_inset 

 where 
\begin_inset Formula \( b \)
\end_inset 

 is a positive number, called the 
\emph on 
reliability factor
\emph default 
 of the density function, and 
\begin_inset Formula \( C \)
\end_inset 

 is a positive constant that is defined by the requirement that 
\begin_inset Formula \( p(x) \)
\end_inset 

 be a density function.
 Of course, the value of 
\begin_inset Formula \( C \)
\end_inset 

 depends on the reliability factor 
\begin_inset Formula \( b. \)
\end_inset 

 This type of density function is fairly typical for electronic devices
 like the Talk Now system.
 What about reliability factors? We know that this varies with retail sites,
 and depends on other conditions, like the factory that manufactured it,
 general climate at the site, quality of the installation, etc.
 To the best of our knowledge, the reliability factor 
\begin_inset Formula \( b \)
\end_inset 

 for systems at a given site is a property of that site such that the density
 function 
\begin_inset Formula \( q(b) \)
\end_inset 

 for 
\begin_inset Formula \( b \)
\end_inset 

 is a normal distribution with mean 
\begin_inset Formula \( \mu =2 \)
\end_inset 

 and standard deviation 
\begin_inset Formula \( \sigma =0.4. \)
\end_inset 

 For all practical purposes 
\begin_inset Formula \( q(b) \)
\end_inset 

 is zero outside the interval 
\begin_inset Formula \( [\frac{1}{2},\frac{7}{2}] \)
\end_inset 

 (we've never seen reliability factors outside this range).
\layout Standard


\series bold 
The Problem.

\series default 
 Here is the information we need from you: (1) management wants to know
 the reliability factor 
\begin_inset Formula \( b_{0} \)
\end_inset 

 for which the probability of a smaller reliability factor occurring is
 at most 
\begin_inset Formula \( .05. \)
\end_inset 

 (2) Most importantly, with this reliability factor, what is the probability
 that a TNS will fail within the first year? Management wants to use this
 information to decide what retail sites are doing an unacceptable job.
 (3) Management recently found a site where the warranty returns were at
 
\begin_inset Formula \( 60\%. \)
\end_inset 

 They think that this is so unlikely that fraud or gross incompetence is
 suspected.
 What do you think? (4) Finally, our department would like you to compute
 the mean value for the lifetime of a TNS using this reliability factor.
 We'd like a few nice sketches of the density functions and cumulative distribut
ion functions in question as well.
 
\layout Standard


\series bold 
Some Suggestions.

\series default 
 Here are a few suggestions we have for you.
 
\layout Standard

(a) You will need the value of the constant 
\begin_inset Formula \( C \)
\end_inset 

 of the density function 
\begin_inset Formula \( p(x) \)
\end_inset 

 and the cumulative distribution function 
\begin_inset Formula \( P(x) \)
\end_inset 

 corresponding to 
\begin_inset Formula \( p(x). \)
\end_inset 

 You should be able to compute these exactly and determine a formula for
 the constant 
\begin_inset Formula \( C \)
\end_inset 

 in terms of the reliability factor 
\begin_inset Formula \( b. \)
\end_inset 

 
\layout Standard

(b) You should also be able to compute the mean value of the lifetime and
 the probability of failure within one year exactly.
\layout Standard

(c) You probably won't be able to compute the cumulative distribution function
 
\begin_inset Formula \( Q(b) \)
\end_inset 

 corresponding to the density function 
\begin_inset Formula \( q(b), \)
\end_inset 

 but you can use your calculator and numerical techniques to approximate
 
\begin_inset Formula \( Q(b) \)
\end_inset 

 for any particular 
\begin_inset Formula \( b. \)
\end_inset 

 Trial and error will help you find (approximately) the reliability factor
 
\begin_inset Formula \( b_{0} \)
\end_inset 

 that we're looking for.
 
\layout Standard

(d) Once you've found this reliability factor, plot the resulting density
 function 
\begin_inset Formula \( p(x) \)
\end_inset 

 and cumulative distribution function 
\begin_inset Formula \( P(x). \)
\end_inset 

 Also plot the density function 
\begin_inset Formula \( q(b) \)
\end_inset 

 and (approximately) the cumulative distribution function 
\begin_inset Formula \( Q(b). \)
\end_inset 

 
\the_end