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\title{Calculus II Project 1: Density and Distribution Functions}

\maketitle
\begin{centering}

\noun{Due Date: Some Date }

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\vspace{.3in}

\textbf{Guidelines:} 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.

\vspace{.3in}

\textbf{Problem 1.} 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
($x$) in that population.

(a) A large middle class.

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

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 $x$-axis
is to the left of the vertical line. 

\vspace{.3in}

\textbf{Problem 2.} A study of grade point averages (GPAs) for students
at a certain university yields the following data: 

\vspace{0.3cm}


\begin{center}
\begin{tabular}{|c|c|}
\hline 
GPA (G) & Fraction of students with GPA$\leq G$\tabularnewline
\hline 
\hline 
$1.0$ & $.01$\tabularnewline
\hline 
$1.5$ & $.04$\tabularnewline
\hline 
$2.0$ & $.10$\tabularnewline
\hline 
$2.5$ & $.22$\tabularnewline
\hline 
$3.0$ & $.40$\tabularnewline
\hline 
$3.25$ & $.57$\tabularnewline
\hline 
$3.5$ & $.75$\tabularnewline
\hline 
$3.75$ & $.89$\tabularnewline
\hline 
$3.9$ & $.95$\tabularnewline
\hline 
\end{tabular}
\par\end{center}

\vspace{0.3cm}


In sketching the curves described below, use the same scale for the
horizontal ($G$) axis in both cases. 

(a) Plot the data above with $G$ on the horizontal axis. Sketch the
best curve you can through these points. Is this a density function
or a cumulative distribution function?

(b) If you think you drew a density function in part (a), sketch the
corresponding cumulative distribution function. If you think you drew
a cumulative distribution function, sketch the corresponding density
function.

\vspace{.3in}

\textbf{Problem 3.} Suppose we are interested in studying the life
of an electronic part. Let $x$ denote the lifetime (in years) of
the part and suppose that the density function of $x$ has the form
(for some constant $c$ )
\[
p(x)=\left\{ \begin{array}{rr}
cxe^{-x^{2}}, & \textrm{if }x\geq0\\
0, & \textrm{if }x<0
\end{array}\right.
\]


(a) Use the definition of density function to show that $c$ must
equal $2.$ 

(b) Sketch the graph of $p(x).$ 

(c) Find the cumulative distribution function of $p(x)$ and sketch
its graph.
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