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\begin{document}

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

March 26, 2003}
\end{center}

\begin{enumerate}

\item  True or false (properties about probability density functions).
	Make sure you justify your answer.
	If the answer if ``false,'' try to turn the statement into a true statement.
	\begin{enumerate}
	\item  If $f(t)$ is a probabilty density function, then $f(t)>0$ for all $t$.
\bigskip

	\item  If $f(t)$ is a probability density function, then $\int_0^\infty f(t) \; dt =1$.
\bigskip

	\item  If $f(t)$ is a pdf, it is possible that $\int_0^{\infty} f(t) \; dt =0$.
\bigskip

	\item  If $F(t)$ is a cumulative distribution function, then $\lim_{t \ra \infty} F(t)=0$.
\bigskip

	\item  If $F(t)$ is a cdf, then $\lim_{t \ra -\infty} F(t) = 0$.
\bigskip

	\item  If $f(t)$ is a pdf, then $\lim_{t \ra \infty} f(t) = 1$.
\bigskip

	\item  Suppose $X$ is a continuous random variable that we model with an exponential distribution.
		Then $P(X \leq 0) = 0$.
\bigskip

	\item  Suppose $X$ is a continuous random variable that we model with an exponential distribution.
		Then $P(X \geq 0) = 1$.
\bigskip

	\item  Suppose $X$ is a continuous random variable that we model with an exponential distribution.
		Suppose also that the mean of this random variable is $10$.
		Then the pdf representing $X$ is
		\[
		f(t) = \begin{cases}
		0 & t<0 \\
		10e^{-10t} & t \geq 0
		\end{cases}
		\]
\bigskip

	\item  If $f(t)$ is a pdf, then the mean of the distribution is given by 
		$\int_{-\infty}^{\infty} f(t)\;dt$.
\bigskip

	\end{enumerate}

\newpage
\lecture

\item  \label{limits} 
	Find the limits of the following sequences.  
	(Hint, if you can't figure this out algebraically, try plugging in large values of $n$ into the formula.)
	\begin{enumerate}
	\item  \label{lim1} 
		$a_n = \frac{n}{n+1}$
\bigskip

	\item  \label{lim2}
		$a_n = \left(\frac{2}{3}\right)^n$
\bigskip

	\item  \label{lim3}
		$a_n = \left(-\frac{2}{3}\right)^n$
\bigskip

	\item  \label{lim4}
		$a_n = (-1)^n$
\bigskip

	\item  \label{lim8}
		$a_n = 3n^2$
\bigskip

	\item  \label{lim5}
		$a_n = \frac{n}{n^2+1}$
\bigskip

	\item  \label{lim6}
		$a_n = \frac{\ln n}{n}$
\bigskip

	\item  \label{lim7}
		$a_n = \sqrt[n]{n}$
\bigskip
	
	\end{enumerate}

\item  Determine which of the sequences from problem~\ref{limits} are increasing, decreasing,
	monotonic or none of these.
\bigskip

\item  Determine which of the sequences from problem~\ref{limits} are bounded above, bounded below
	or bounded.






\end{enumerate}


\end{document}