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\begin{center}
{\Large Warm-Up Problems and Lecture Problems

March 24, 2003}
\end{center}

\begin{enumerate}

\item  \label{constant}
Suppose you have a continuous random variable $X$ that has a probability density function given as:
\[
f(t) = 
\begin{cases}
\frac{1}{10} & 4 \leq t \leq 14 \\
0 & \mbox{otherwise}
\end{cases}
\]
\begin{enumerate}
	\item  What are the two conditions a function has to satisfy in order to be a probability density function?  Show that $f(t)$ satisfies these.

\item  What is the probability that $10 \leq X \leq 15$?
\end{enumerate}

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\item  \label{car average}
Suppose you are a used car deal and you have sold 10 cars.  For these cars, they all came back in a certain number of weeks with problems.  Here is how many weeks it took each car to come back:

\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|}
\hline
\textbf{Car Number:} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
\textbf{Weeks:} & 50 & 102 & 243 & 12 & 323 & 72 & 2 & 45 & 21 & 13 \\
\hline
\end{tabular}

Find the average number of weeks before the car came back.

\newpage
\lecture

\item  Find the cumulative distribution function for problem~\ref{constant}.
	We will do this in steps.  We will let $F(x)$ denote the CDF.
	\begin{enumerate}
	\item  What is the definition of the CDF?

	\item  If $x<4$, what is $F(x)$?

	\item  If $4 \leq x \leq 14$, then what is $F(x)$?

	\item  If $x>14$ then what is $F(x)$?

	\item  What is the CDF, $F(x)$?
	\end{enumerate}

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\item  Suppose you want to model your used car failure with an exponential probability density function:
\[
f(t) = 
\begin{cases}
0 & t < 0 \\
c e^{-ct} & t \geq 0
\end{cases}
\]
\begin{enumerate}
\item  What should $c$ be using the data from problem~\ref{car average}?

\item  Using this exponential model, what is the probability that a car will fail within the first year of purchase?
\end{enumerate}


\end{enumerate}


\end{document}