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

April 7, 2003}
\end{center}

\begin{enumerate}

\item  Consider the probability density function below:
	\[
	f(x) = \begin{cases}
	0 & x < 0 \\
	\frac{x}{2} & 0 \leq x \leq 2 \\
	0 & x>2
	\end{cases}
	\]
	\begin{enumerate}
	\item  Show that $f(x)$ is a probablity density function.
\inch

	\item  Find the mean, $\mu$, of the random variable.
\inch

	\item  Find the median of the random variable.
	\\Hint: the median is the number $m$ so that $\int_{-\infty}^m f(x) \; dx = 0.5$.
\inch
	\end{enumerate}
	
\item  Suppose a sequence is given recursively as:
	\[
	\begin{cases}
	a_1 = 1 & \\
	a_{n+1} = \frac{1}{4-a_n} & \mbox{ if } n >1
	\end{cases}
	\]
	Suppose we know that $\lim_{n \ra \infty} =L$, find $L$.

\newpage
\lecture

\item  Lets find an estimate for $S=\sum_{n=1}^\infty \frac{1}{n^3}$.
	\begin{enumerate}
	\item  Using your calculator, find $s_{10}$.
\inch
	\item  Find $\int_{10}^\infty \frac{1}{x^3} \; dx$.
\inch
	\item  Find $\int_{11}^\infty \frac{1}{x^3} \; dx$.
\inch
	\item  How does $R_{10}$ relate to the previous integrals?
\inch
	\item  What inequalities can we set up for the sume $S=\sum_{n=1}^\infty \frac{1}{n^3}$?
\inch	
	\item  What is a good estimate for the sum and at most, how far off can you be?
		
	\end{enumerate}


\end{enumerate}


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