STAT 4813 Homework 2

hw2.tex

\documentclass{article}

\usepackage[margin=1in]{geometry}
\usepackage{amsmath}
\usepackage{amssymb}
\title{Homework 2}
\author{Blake Burkhart \vspace{\intextsep} \\
 Math 4813, Section 1}
\date{January 31, 2012}
\renewcommand{\theenumi}{\alph{enumi}}

\usepackage{fancyhdr}

\pagestyle{fancyplain}
\fancyhf{}
\renewcommand{\headrulewidth}{0pt}
\renewcommand{\footrulewidth}{0pt}
\rhead[]{Blake Burkhart\quad \thepage}

\fancypagestyle{plain}{ %
\fancyhf{} % remove everything
\renewcommand{\headrulewidth}{0pt} % remove lines as well
\renewcommand{\footrulewidth}{0pt}}



\begin{document}
\maketitle

\section*{Question 2-78}
A boiler has five identical relief valves. The probability that any particular valve will open on demand is .95. Assuming independent operation of the valves, calculate $P(\text{at least one valve opens})$ and \emph{P}(at least one valve fails to open).

\section*{Question 2-82}
Consider independently rolling two fair dice, one red and the other green. Let $A$ be the event that the red die shows 3 dots, $B$ be the event that the green die shows four dots, and $C$ be the event that the total number of dots showing on the two dice is 7. Are these events pairwise independent (i.e, are $A$ and $B$ events, are $A$ and $C$ independent, and are $B$ and $C$ independent)? Are the three events mutually independent?

\section*{Question 2-94}
A transmitter is sending a message by using a binary code, namely, a sequence of 0's and 1's. Each transmitted bit (0 or 1) must pass through three relays to reach the receiver. At each relay, the probability is .20 that the bit sent will be different from the bit received (a reversal). Assume that the relays operate independently of one another.

\begin{center} Transmitter $\rightarrow$ Relay 1 $\rightarrow$ Relay 2 $\rightarrow$ Relay 3 $\rightarrow$ Receiver \end{center}

\begin{enumerate}
\item If a 1 is sent from the transmitter, what is the probability that a 1 is sent by all three relays?
\item If a 1 is sent from the transmitter, what is the probability that a 1 is received by the receiver? [\emph{Hint:} The eight experimental outcomes can be displayed on a tree diagram with three generations of branches, one generation for each relay.]
\item Suppose 70\% of all bits sent from the transmitter are 1's. If a 1 is received by the receiver, what is the probability that a 1 was sent?
\end{enumerate}

\section*{Question 2-100}
One percent of all individuals in a certain population are carriers of a particular disease. A diagnostic test for this disease has a 90\% detection rate for carriers and a 5\% detection rate for noncarriers. Suppose the test is applied independently to two different blood samples from the same randomly selected individual.
\begin{enumerate}
\item What is the probability that both tests yield the same result?
\item If both tests are positive, what is the probability that the selected individual is a carrier?
\end{enumerate}

\section*{Question 3-12}
Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable $Y$ as the number of ticketed passengers who actually show up for the flight. The probability mass function of $Y$ appears in the accompanying table.

\begin{center}
	\begin{tabular}{c|cccccccccccc}
		   $y$ &  45 &  46 &  47 &  48 &  49 &  50 &  51 &  52 &  53 &  54  & 55\\ \hline
		$p(y)$ & .05 & .10 & .12 & .14 & .25 & .17 & .06 & .05 & .03 & .02 & .01
	\end{tabular}
\end{center}

\begin{enumerate}
\item What is the probability that the flight will accommodate all ticketed passengers who show up?
\item What is the probability that not all ticketed passengers who show up can be accommodated?
\item If you are the first person on the standby list, what is the probability that you will be able to take the flight? What is this probability if you are the third person on the standby list?
\end{enumerate}

\section*{Question 3-18}
Two fair six-sided dice are tossed independently. Let $M$ be the maximum of the two tosses (so $M(1, 5) = 5$, $M(3, 3) = 3$, etc.).
\begin{enumerate}
\item What is the pmf of $M$? [\emph{Hint:} First determine $p(1)$, then $p(2)$, and so on.]
\item Determine the cdf of $M$ and graph it.
\end{enumerate}

\section*{Question 3-30}
An individual who has automobile insurance from a certain company is randomly selected. Let $Y$ be the number of moving violations for which the individual was cited during the last 3 years. The pmf of $Y$ is

\begin{center}
	\begin{tabular}{l|cccc}
		$y$ & 0 & 1 & 2 & 3 \\ \hline
		$p(y)$ & .60 & .25 & .10 & .05
	\end{tabular}
\end{center}

\begin{enumerate}
\item Compute $E(Y)$.
\item Suppose an individual with $Y$ violations incurs a surcharge of $100Y^2$. Calculate the expected amount of the surcharge.
\end{enumerate}

\section*{Question 3-48}
When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is 5\%. Let $X$ = the number of defective boards in a random sample of size $n = 25$, so $X \tilde{}$ Bin(25, .05).
\begin{enumerate}
\item Determine $P(X \leq 2)$.
\item Determine $P(X \geq 5)$.
\item Determine $P(1 \leq X \leq 4)$.
\item What is the probability that none of the 25 boards is defective?
\item Calculate the expected value and standard deviation of $X$.
\end{enumerate}

\section*{Question 3-66}
An airport limousine can accommodate up to four passengers on any one trip. The company will accept a maximum of six reservations for a trip, and a passenger must have a reservation. From previous records, 20\% of all those making reservations do not appear for the trip. Answer the following questions, assuming independence wherever appropriate.
\begin{enumerate}
\item If six reservations are made, what is the probability that at least one individual with a reservation cannot be accommodated on the trip?
\item If six reservations are mde, what is the expected number of available places when the limousine departs?
\item Suppose the probability distribution of the number of reservations made is given in the accompanying table.
\begin{center}
	\begin{tabular}{l|cccc}
		Number of reservations & 3 & 4 & 5 & 6 \\ \hline
		Probability & .1 & .2 & .3 & .4
	\end{tabular}
\end{center}
Let $X$ denote the number of passengers on a randomly selected trip. Obtain the probability mass function of $X$.
\end{enumerate}

\end{document}