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<DIV><FONT face=Arial size=2>OK, I joined TUG and got the disks.
Everything is installed. What I want to be able to do is translate stuff
like what follows into something resembling math. So far, I haven't a clue
about how to do this.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>% This file was created by EXP Version 5.1.</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><?xml:namespace prefix = o ns
= "urn:schemas-microsoft-com:office:office" /><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\documentclass{article}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\usepackage{exptex}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\usepackage{expthmi}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\newcounter{SEQequation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{document}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\title{}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\author{}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\date{}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\maketitle</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\ Averages \ and Variability</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>W. D. Markel, 1995</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>Theoretical Averages</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\hspace{0.5in}\=\kill</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{If you rolled a single die a large number of times and took the
average</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>of the numbers that came up, what would you expect this average to be? \
It</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>would appear to be reasonable, since there is no basis for believing that
any</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>number is any more likely to come up than any other number, that this
average</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>should be in the vicinity of $\frac{1+2+3+4+5+6}{6}=3.5$.}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\hspace{0.5in}\=\kill</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{Of course, if you actually rolled the die, say 50 times, and
averaged</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>the numbers that turned up, you would probably not get exactly 3.5,
although</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>you would probably get something close to it. \ As a matter of fact, if
you</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>were to roll the die 50 more times, you would probably get a
different</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>average than you did the first time.}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\hspace{0.5in}\=\kill</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{The number 3.5 is called the expected value or theoretical average
of</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>the variable which we might designate ``the spot showing on the die'',
and is</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>denoted by the symbol $\mu$ or $E(X)$ where $X$ is the variable in
question.</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\ In general, if we let $X$ represent the name of a variable,</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>$x_1,x_2,x_3,...,x_N$ be the specific values that the variable may
assume,</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>and $p_i$ be the probability associated with each $x_i$, then we
define</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>E(X)=\mu=x_1p_1+x_2p_2+\cdots+x_Np_N</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>In the example that we started with,
$x_1=1,x_2=2,x_3=3,x_4=4,x_5=5,x_6=6,$</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>and each $p_i=\frac{1}{6}.\,\,$So the 3.5 represents a kind of average; \
not</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>what you would necessarily get if you actually rolled the die, but what
you</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>would expect to average if you rolled the die a large number of times. \
\</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>(It's pretty boring to roll the die, say, 10000 times, but, if you did,
you</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>would get an average pretty close to 3.5. \ Fortunately, we can simulate
this</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>on a computer pretty easily and \ possibly convince you, if you
need</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>convincing, that this is the case.)</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\hspace{0.5in}\=\kill</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{Let's look at another example. \ Suppose that I flip a coin three
times</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>and let my variable, $X$, represent the number of heads that I obtain.
\</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>Notice that $X$ may assume the values 0,1,2, or 3. \ Furthermore, by
looking</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>at all the ways the coins can fall $(HHH,HHT,...)$ it is not hard to see
that</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>the number of heads occurs with the following
probabilities.}\\</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{$x_i\,\,0\,\,1\,\,2\,\,3$}\\</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{$p_i\,\,\frac{1}{8}\,\frac{3}{8}\,\frac{3}{8}\,\frac{1}{8}$}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>Therefore,</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>E(X)=0\cdot\frac{1}{8}+1\cdot\frac{3}{8}+2\cdot\frac{3}{8}+3\cdot%</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\frac{1}{8}=1\frac{1}{2}.</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>Does this mean that you are ever going to get 1$\frac{1}{2}$ heads when
you</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>perform this experiment? \ Of course not! \ What it does mean, however,
is</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>that if you repeated the experiment a large number of times (that
is,</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>flipping a coin three times and noting the number of heads) and averaged
the</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>results, this average should be in the neighborhood of
1$\frac{1}{2}$.</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\hspace{0.5in}\=\kill</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{The theoretical mean, or expected value, of a variable is used in
a</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>wide variety of applications, including playing games of chance, the
stock</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>market, and buying insurance. \ As a very simple example, suppose that
a</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>basketball team makes 40\% of its three-point shots and 50\% of its
two-point</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>shots. \ From strictly an expectation perspective, the expected value of
the</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>three-point shot is \ (.4)(3)=1.2 and of the two-point shot is (.5)(2)=1,
so</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>the team would be better off taking three pointers. \ (I am sure that you
can</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>see that, in a practical sense, there are other things to consider, but
let's</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>skip this.)}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\hspace{0.5in}\=\kill</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{The $E(X)$ notation is handy because it is general. \ That is, it
is</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>just as easy to associate meaning with, say, $E(2X+1)\,$or $E(X^2)$ as it
is</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>with $E(X)$. \ In the three flips of the coin example, we have
$X\,=0,1,2,3$,</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>hence, $2X+1=1,3,5,7$ and $X^2=0,1,4,9$. \ Then,</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>E(2X+1)=1\cdot\frac{1}{8}+3\cdot\frac{3}{8}+5\cdot\frac{3}{8}+7\cdot%</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\frac{1}{8}=4\,</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>(Note that 4=2(1.5)+1, 1.5 being the expected value of just plain $X$.
\</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>Hmmm. \ I leave it to the reader to convince himself/herself
that</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>$E(X^2)=3.\,\,$Now try these problems.</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>Problem set 1</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\hspace{0.5in}\=\kill</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{1. \ Show that the probabilities in the ``three flips of a
coin''</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>problem are correct. \ Also, show that $E(X^2)\,$is really equal to
3.}\\</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{2. \ If a lottery ticket cost \$1 and there are three prizes,
worth,</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>respectively, \$10, \$50, and \$100, and the probabilities of winning
each</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>are, respectively, 5\%, 1\%, and .1\%, what is the expected value of
the</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>lottery? \ ans: \$.10}\\</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{3. \ Sometimes the theoretical mean may be obtained by means of
a</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>shortcut. \ One of the more obvious cases involves a binomial variable. \
A</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>binomial experiment consists of any situation where each ``trial'' can
have</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>exactly \ two outcomes. \ Examples are flipping a coin, the results of
a</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>two-way election, or guessing the right answer on a multiple choice test.
\</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>The variable, $X$, represents the number of times we get the desired
result</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>(success) out of $n$ trials. \ If the probability of success on any of
these</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>trials is $p$ then $E(X)=np$.}\\</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{a) \ If you were guessing at answers on a 20 question multiple
choice</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>test, in which there were 5 choices per question, about how many
questions</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>should you get correct? ans: 4}\\</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{b) \ In the example involving the three flips of a coin, note
that</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>$n=3$ and $p=\frac{1}{2}$. \ Does the short-cut give the same answer as
the</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>longer method which we used?}\\</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{4. \ Some other theoretical means may be deduced by common sense. \
For</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>example, how many times, on the average, would you expect to roll a die
until</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>you got a 6? \ How many times would you expect to flip a coin until you
got a</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>head? \ (If not convinced, try doing these things a few times!) \
Now</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>generalize. \ If $\,p$ represents the probability of success on a
single</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>trial and $X$ represents the number of trials until a success is
obtained,</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>what is $E(X)$?}\\</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{$\bigstar 5$. \ Show that the following properties are true
for</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>expected values.}\\</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{a) \ $E(c)=c,\,$where c is any constant}\\</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{b) \ $E(cX)=cE(X)$}\\</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{c) \ $E(X+Y)=E(X)+E(Y)$}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>Sample Averages</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\hspace{0.5in}\=\kill</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{A very fundamental idea in statistics concerns whether a set of
numbers</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>represents a population or a sample. \ A population consists of all the
items</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>under consideration (and is kind of like a universal set) while a
population</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>consists of just part of the items (like a subset). \ A population, in
fact,</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>can consist of all of the possible outcomes of rolling a single die. \
In</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>this instance, a sample would consist of the actual outcomes when you
rolled</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>the die a certain number of times. \ Of course, we can compute averages
of</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>samples, just like averages are always computed. \ The thing to remember
is</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>that these sample averages are variable while the population average,
$\mu$</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>or $E(X)$ is considered constant. \ We call the sample average
$\overline{x}$</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>and it is defined by the familiar formula,</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\overline{x}=\frac{x_1+x_2+x_3+\cdots+x_n}{n}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>It is helpful to provide a little more general formula, in fact, which
allows</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>for each $x_i\,$to occur more than once. \ Thus, if $x_i\,$occurs
$f_i$</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>times, </FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\overline{x}=\frac{x_1f_1+x_2f_2+\cdots+x_nf_n}{m}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>where $m=f_1+f_2+\cdots+f_n$.</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\hspace{0.5in}\=\kill</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{For example, if I average my class grades in which there are 6 A's
(4</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>points), 8 B's (3 points), 10 C's (2 points), 2 D's (1 point), and 1 F
(0</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>points), my average is</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\overline{x}=\frac{4\cdot 6+3\cdot 8+2\cdot 10+1\cdot 2+0\cdot
1}{27}=2.59</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>You may observe the similarity between (1), the theoretical mean, and (5)
the</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>sample mean if \ we write 5 as</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\overline{x}=x_1\cdot\frac{f_1}{m}+x_2\cdot\frac{f_2}{m}+\cdots+x_n\cdot%</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\frac{f_n}{m}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>and think of the $\frac{f_i}{m}\,$terms as ``probabilities.''</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\hspace{0.5in}\=\kill</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{The point is, both kinds of averages--population and
sample--represent,</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>essentially, the same concept. \ In fact, sometimes it is not even
important</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>whether the data represent a population or a sample, which is why
the</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>notations $\mu$ and $\overline{x}$ are often used
interchangibly.}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>Variance and Standard Deviation</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\hspace{0.5in}\=\kill</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{There are several ways to measure the ``spread'' or the variability
of</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>data, the most common of which are the variance$\,$and the
standard</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>deviation. \ (Actually, the standard deviation is no big deal, since it
is</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>just the square root of the variance.) \ The theoretical variance,
denoted by</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>the symbol $\sigma^2$is defined symbolically as</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\sigma^2=E\left[(X-\mu)^2\right]</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>Remembering that </FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{equation*}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>E(Anything)=Anything_1\cdot p_1+\cdots+Anything_n\cdot
p_{n\text{,}}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{equation*}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>this notation means that </FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\sigma^2=(x_1-\mu)^2p_1+(x_2-\mu)^2p_2+\cdots+(x_n-\mu)^2p_n</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>In terms of our example with the three coins, </FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\sigma^2\text{=}(0-1\frac{1}{2})^2\cdot\frac{1}{8}+(1-1\frac{1}{2})^2\cdot%</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\frac{3}{8}+(2-1\frac{1}{2})^2\cdot\frac{3}{8}+(3-1\frac{1}{2})^2\cdot%</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\frac{1}{8}=\frac{3}{4}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>As stated previously, the standard deviation, denoted by $\sigma$
(without</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>the square) is simply the square root of the variance. \ That is, in this
case</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\sigma=\sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{2}=.866</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{equation}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>The standard deviation is the measure most often referred to, because it
is</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>in the same units as the original data. \ (For example, if the data were
in</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>feet, the variance would be in square feet and the standard deviation
in</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>feet.)</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\begin{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\hspace{0.5in}\=\kill</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\LTab{You might well ask ``.866 what?'' \ What is a standard
deviation,</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>anyway? \ Roughly it measures the average amount of deviation from the
mean.</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\ That is, I could make the semi-accurate statement that, as I repeat
the</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>coin flipping experiment, I could expect the number of heads to deviate
from</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>1.5 by about .866. \ Again, this is only approximately true, and
sometimes</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>it's not very true at all! \ About all that can really be said is that
the</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>bigger the standard deviation the more spread there is.}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><FONT face="Courier New"
size=2>\end{tabbing}</FONT></P>
<P class=MsoPlainText style="MARGIN: 0in 0in 0pt"><o:p><FONT face="Courier New"
size=2> </FONT></o:p></P></DIV></BODY></HTML>