#LyX 1.1 created this file. For more info see http://www.lyx.org/
\lyxformat 218
\textclass article
\language english
\inputencoding latin1
\fontscheme times
\graphics default
\paperfontsize 12
\spacing single 
\papersize Default
\paperpackage a4
\use_geometry 0
\use_amsmath 1
\paperorientation portrait
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\defskip medskip
\quotes_language english
\quotes_times 2
\papercolumns 1
\papersides 1
\paperpagestyle default

\layout Title

Math 107 Project 1: Density and Distribution Functions
\layout Standard


\latex latex 

\backslash 
begin{centering}
\layout Standard


\noun on 
Due Date: Thursday, October 19, 2000 
\layout Standard


\latex latex 

\backslash 
end{centering}
\layout Standard


\latex latex 

\backslash 
vspace{.3in}
\layout Standard


\series bold 
Guidelines:
\series default 
 This project consists of three independent problems.
 These problems concern the material found in the section on distribution
 functions section in the text (pages 409-415).
 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.
 Also, your document should be reasonably self-contained.
\layout Standard


\latex latex 

\backslash 
vspace{.3in}
\layout Standard


\series bold 
Problem 1.

\series default 
 For each of the following descriptions of populations, sketch the graph
 of a density function and a cumulative distribution function which could
 reasonably represent the distribution of income (
\begin_inset Formula \( x \)
\end_inset 

) in that population.
\layout Standard

(a) A large middle class.
\layout Standard

(b) Few rich people but large middle and lower classes.
\layout Standard

In each graph of a density function draw a vertical line in such a way that
 about half the total area between the curve and 
\begin_inset Formula \( x \)
\end_inset 

-axis is to the left of the vertical line.
 
\layout Standard


\latex latex 

\backslash 
vspace{.3in}
\layout Standard


\series bold 
Problem 2.

\series default 
 A study of grade point averages (GPAs) for students at a certain university
 yields the following data: 
\layout Standard
\added_space_top 0.3cm \added_space_bottom 0.3cm \align center 

\begin_inset  Tabular
<lyxtabular version="2" rows="10" columns="2">
<features rotate="false" islongtable="false" endhead="0" endfirsthead="0" endfoot="0" endlastfoot="0">
<column alignment="center" valignment="top" leftline="true" rightline="false" width="" special="">
<column alignment="center" valignment="top" leftline="true" rightline="true" width="" special="">
<row topline="true" bottomline="true" newpage="false">
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
\begin_inset Text

\layout Standard

GPA (G)
\end_inset 
</cell>
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
\begin_inset Text

\layout Standard

Fraction of students with GPA
\begin_inset Formula \( \leq G \)
\end_inset 


\end_inset 
</cell>
</row>
<row topline="true" bottomline="false" newpage="false">
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
\begin_inset Text

\layout Standard


\begin_inset Formula \( 1.0 \)
\end_inset 


\end_inset 
</cell>
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
\begin_inset Text

\layout Standard


\begin_inset Formula \( .01 \)
\end_inset 


\end_inset 
</cell>
</row>
<row topline="true" bottomline="false" newpage="false">
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
\begin_inset Text

\layout Standard


\begin_inset Formula \( 1.5 \)
\end_inset 


\end_inset 
</cell>
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
\begin_inset Text

\layout Standard


\begin_inset Formula \( .04 \)
\end_inset 


\end_inset 
</cell>
</row>
<row topline="true" bottomline="false" newpage="false">
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
\begin_inset Text

\layout Standard


\begin_inset Formula \( 2.0 \)
\end_inset 


\end_inset 
</cell>
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
\begin_inset Text

\layout Standard


\begin_inset Formula \( .10 \)
\end_inset 


\end_inset 
</cell>
</row>
<row topline="true" bottomline="false" newpage="false">
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
\begin_inset Text

\layout Standard


\begin_inset Formula \( 2.5 \)
\end_inset 


\end_inset 
</cell>
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
\begin_inset Text

\layout Standard


\begin_inset Formula \( .22 \)
\end_inset 


\end_inset 
</cell>
</row>
<row topline="true" bottomline="false" newpage="false">
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
\begin_inset Text

\layout Standard


\begin_inset Formula \( 3.0 \)
\end_inset 


\end_inset 
</cell>
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
\begin_inset Text

\layout Standard


\begin_inset Formula \( .40 \)
\end_inset 


\end_inset 
</cell>
</row>
<row topline="true" bottomline="false" newpage="false">
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
\begin_inset Text

\layout Standard


\begin_inset Formula \( 3.25 \)
\end_inset 


\end_inset 
</cell>
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
\begin_inset Text

\layout Standard


\begin_inset Formula \( .57 \)
\end_inset 


\end_inset 
</cell>
</row>
<row topline="true" bottomline="false" newpage="false">
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
\begin_inset Text

\layout Standard


\begin_inset Formula \( 3.5 \)
\end_inset 


\end_inset 
</cell>
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
\begin_inset Text

\layout Standard


\begin_inset Formula \( .75 \)
\end_inset 


\end_inset 
</cell>
</row>
<row topline="true" bottomline="false" newpage="false">
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
\begin_inset Text

\layout Standard


\begin_inset Formula \( 3.75 \)
\end_inset 


\end_inset 
</cell>
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
\begin_inset Text

\layout Standard


\begin_inset Formula \( .89 \)
\end_inset 


\end_inset 
</cell>
</row>
<row topline="true" bottomline="true" newpage="false">
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
\begin_inset Text

\layout Standard


\begin_inset Formula \( 3.9 \)
\end_inset 


\end_inset 
</cell>
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
\begin_inset Text

\layout Standard


\begin_inset Formula \( .95 \)
\end_inset 


\end_inset 
</cell>
</row>
</lyxtabular>

\end_inset 


\layout Standard

In sketching the curves described below, use the same scale for the horizontal
 (
\begin_inset Formula \( G \)
\end_inset 

) axis in both cases.
 
\layout Standard

(a) Plot the data above with 
\begin_inset Formula \( G \)
\end_inset 

 on the horizontal axis.
 Sketch the best curve you can through these points.
 Is this a density function or a cumulative distribution function?
\layout Standard

(b) If you think you drew a density function in part (a), sketch the correspondi
ng cumulative distribution function.
 If you think you drew a cumulative distribution function, sketch the correspond
ing density function.
\layout Standard


\latex latex 

\backslash 
vspace{.3in}
\layout Standard


\series bold 
Problem 3.

\series default 
 Suppose we are interested in studying the life of an electronic part.
 Let 
\begin_inset Formula \( x \)
\end_inset 

 denote the lifetime (in years) of the part and suppose that the density
 function of 
\begin_inset Formula \( x \)
\end_inset 

 has the form (for some constant 
\begin_inset Formula \( c \)
\end_inset 

 )
\begin_inset Formula \[
p(x)=\left\{ \begin{array}{rr}
cxe^{-x^{2}}, & \textrm{if }x\geq 0\\
0, & \textrm{if }x<0
\end{array}\right. \]

\end_inset 


\layout Standard

(a) Use the definition of density function to show that 
\begin_inset Formula \( c \)
\end_inset 

 must equal 
\begin_inset Formula \( 2. \)
\end_inset 

 
\layout Standard

(b) Sketch the graph of 
\begin_inset Formula \( p(x). \)
\end_inset 

 
\layout Standard

(c) Find the cumulative distribution function of 
\begin_inset Formula \( p(x) \)
\end_inset 

 and sketch its graph.
\the_end