\documentstyle[12pt,twoside,amstex,eepic]{article}
\input{macros-course-lectures}

\parskip1.5ex
\parindent0pt

\begin{document}
\lecture{notes22}{Notes 22}{November 21, 1996}

\section{Random Variables}

\begin{defin}
  A {\em random variable} $R$ on sample space $S$ is a function $R:S
  \rightarrow \reals$.
\end{defin}

Note defined with respect to a sample space---associates a value with
each point.

\begin{xample}
  Let $S=$ outcomes of 3 coin tosses = $\{$HHH, HHT, HTH, HTT, THH, THT,
  TTH, TTT$\}$. 
  \begin{itemize}
  \item
    $R_H = $ number of H in outcome. So $R_H(HHH)=3,
    R_H(HHT)=2$. 
  \item $R_M(p) = $ 1 if all coins match, 0 otherwise. So $R_M(HHH)=1,
    R_M(HHT)=0$.  $R_M$ is an {\em indicator} or {\em characteristic}
    random variable: $1$ if event happens, $0$ otherwise.
  \end{itemize}
  Note {\bf functions}: $R_H:S\rightarrow \{0,1,2,3\},\;
  R_M:S\rightarrow \{0,1\}$
\end{xample}

\begin{defin}
An {\em indicator} (or {\em Bernoulli} or {\em characteristic} ) random 
variable is a random variable with range {0,1}.
\end{defin}

A random variable partitions $S$ into events (subsets of sample
points).  \\

``$R=x$'' corresponds to the event  $\{p\in S | R(p)=x\}$ \\

Note that indicators partition the sample space into two events. \\

\setlength{\unitlength}{0.00083333in}
\begin{picture}(3525,2583)(0,-10)
  \thicklines
  \path(12,2337)(2712,2337)(2712,12)
  (12,12)(12,2337)
  \path(12,162)(2487,2337)
  \path(12,162)(2487,2337)
  \path(12,1437)(1287,2337)
  \path(12,1437)(1287,2337)
  \path(1737,12)(2712,1287)
  \path(1737,12)(2712,1287)
  \put(312,2412){$R_H=3$}
  \put(1587,2412){$R_H=2$}
  \put(2787,1737){$R_H=1$}
  \put(2787,387){$R_H=0$}
  \put(2187,162){TTT}
  \put(162,1812){HHH}
  \put(162,837){HHT}
  \put(687,1362){THH}
  \put(1362,1962){HTH}
  \put(762,162){TTH}
  \put(1437,837){THT}
  \put(1887,1362){HTT}
\end{picture}

\begin{defin}
\[
Pr(R=x) \equiv \sum_{p\in S: R(p) = x} Pr(p)
\]
\end{defin}

For example, \\
$Pr(R_H=2) = Pr( \{ HHT, HTH, THH \}) = 3/8$ \\
$Pr(R_M=1) = Pr( \{ HHH, TTT \}) = 1/4 $ \\

We can also define the probability that a random variable is within some
range or in some subset of specific values. \\

\begin{xample}

$Pr(R_H \geq 2) = \sum_{i=2}^{\infty}{ Pr(R_H = i)} = 1/4$ 
 
or, for $A \subseteq \reals : Pr(R_H \in A) = \sum_{a \in A}{Pr(R=a)}$

\end{xample}

Our random variables will be {\em  discrete--range}, {\em finite},
or {\em countably infinite} (ie, as here or $R:S\rightarrow \naturals$).
There are also
{\em continuous} random variables with values in the reals; for these
sums must be replaced by integrals.

\section{Independence of Random Variables}

Recall, events $E_1$ and $E_2$ are independent if $Pr(E_1 \cap E_2) =
Pr(E_1)Pr(E_2)$.

\begin{defin}
  Random variables $R_1$ and $R_2$ are {\em independent} if 
$\forall x,y \in \reals$,
  $$Pr(R_1=x \wedge R_2=y) = Pr(R_1=x)Pr(R_2=y)$$
\end{defin}

The coins being fair and mutually independent (so each $P\in S$ has
probability $1/8$) does not mean $R_H$ and $R_M$ are independent,
because  \\
$Pr(R_H=2 \wedge R_M=1) = 0 \neq Pr(R_H=2)Pr(R_M=1) = \frac{3}{8} \cdot \frac{1}{4}$.

Intuition: knowing the value of one variable biases
the distribution of the other.

The picture, with area proportional to probability:\\

\setlength{\unitlength}{0.00083333in}
\begin{picture}(7079,3135)(0,-10)
  \thicklines
  \path(12,2889)(2412,2889)(2412,489)
  (12,489)(12,2889)
  \path(3612,2889)(6312,2889)(6312,489)
  (3612,489)(3612,2889)
  \path(612,2889)(612,489)
  \path(1287,2889)(1287,489)
  \path(12,1989)(2412,1989)
  \path(12,1689)(2412,1689)
  \path(12,1389)(2412,1389)
  \path(4512,2889)(4512,489)
  \path(3612,1989)(6312,1989)
  \path(3612,1389)(4512,1389)
  \path(4512,1089)(6312,1089)
  \put(30,2964){$R_1=1$}
  \put(1587,2964){$R_1=3$}
  \put(612,2964){$R_1=2$}
  \put(2487,2289){$R_2=6$}
  \put(2487,1764){$R_2=7$}
  \put(2487,789){$R_2=9$}
  \put(2487,1464){$R_2=8$}
  \put(3687,2964){$R_1=1$}
  \put(5037,2964){$R_1=2$}
  \put(6387,2364){$R_2=6$}
  \put(6387,1464){$R_2=7$}
  \put(6387,714){$R_2=8$}
  \put(312,39){$R_1, R_2$ independent}
  \put(3987,39){$R_1, R_2$ not independent}
\end{picture}

Note that independence of random variables covers {\em all pairs} of
``value events.''  

\begin{example}
  Suppose we roll two independent fair dice to get values $D_1$ and
  $D_2$.  Let $T=D_1+D_2$ be the sum of values.  Let $S=1$ if $T=7$
  and 0 otherwise.
  \begin{itemize}
  \item $T$ and $D_1$ are not independent
    since, for example, $\Pr(D_1=6 \mid T=1) = 0 \neq Pr(T=2) = 1/36$.
  \item $T$ and $S$ are not independent since knowing $T$ completely 
determines $S$
  \item $S$ and $D_1$ are independent.
  $Pr(S = 1 \mid D_1 = 1) = 1/6 = Pr(S=1)$ \\
  $Pr(S = 1 \mid D_1 = 2) = 1/6 = Pr(S=1)$ \\
  $Pr(S = 1 \mid D_1 = 6) = 1/6 = Pr(S=1)$ \\
  $Pr(S = 0 \mid D_1 = 1) = 5/6 = Pr(S=0)$ \\
  $Pr(S = 0 \mid D_1 = 2) = 5/6 = Pr(S=0)$ \\
  $Pr(S = 0 \mid D_1 = 6) = 5/6 = Pr(S=0)$ \\

  \end{itemize}
\end{example}

The definition of mutually independent random variables works in the same way.\\

\begin{defin}
Given a random variable $R$, the {\em probability distribution function (pdf)}
for $R$ is $f(x) \equiv Pr(R=x)$
\end{defin}

\begin{defin}
Given a random variable $R$, the {\em cumulative distribution function}
for $R$ is $F(x) \equiv Pr(R \leq x) = \sum_{y \leq x}{f(y)}$
\end{defin}

\begin{xample} For a {\em Bernoulli} random variable \\
$f(0) = p$ \hspace{1in} $F(0) = p$ \\
$f(1) = 1-p$ \hspace{1in} $F(1) = 1$ \\

For a {\em uniform} random variable $R:S \rightarrow [1,N]$ \\
$f(k) = \frac{1}{N}$ \hspace{1in} $F(k) = \frac{k}{N}$

\end{xample}

\section{The Binomial Distribution}

We now study what is perhaps the most common random variable
distribution in computer science: the binomial distribution.  

\begin{defin} 
  The {\em binomial distribution} is given by
  \[
   f_{n,p}(k) = \binom{n}{k}p^k(1-p)^{n-k}
  \]
\end{defin}

Sanity check: is this a probability distribution?
\begin{eqnarray*}
  \sum_k f_{n,p}(k) &= &\sum {n \choose k}p^k(1-p)^{n-k}\\
  &= &(p+(1-p))^n \quad \mbox{(binomial theorem)}\\
  &= &1
\end{eqnarray*}

A special case is the {\em unbiased binomial distribution} which is the 
binomial distribution with $p = 1/2$.
\[
 f_{n}(k) = \binom{n}{k}2^{-n}
\]

\begin{xample}
Given a system with n components each of which fails with probability $p$ 
find the probability distribution function for the number of failures. \\

Let $R$ = number of failures.  We wish to compute $f(k) = Pr(R=k)$ \\

{\bf solution} \\
\begin{enumerate}

\item The sample space $S$ consists of $2^{n}$ binary strings. 
(Say, a 0 corresponds to a failure)

\item The probability to get a sample point with $k$ failures is
$p^k(1-p)^{n-k}$

\item Identify the event of interest. $A$ = sample points with $k$ failures.

\item Compute the probability of the event.

\[
 f(k) = Pr(R=k) = Pr(E) = \sum_{A}{p^k(1-p)^{n=k}} 
 = |A|p^k(1-p)^{n-k}
 = \binom{n}{k}p^k(1-p)^{n-k}
\]

\end{enumerate}

We get a binomial distribution. In general, we get a binomial distribution 
whenever we have $n$ independent Bernoulli events with the same distribution.

\end{xample}


\subsection{Asymptotics}

It's hard to see the qualitative behavior of the binomial distribution
for large $k$ and $n$.

Lets consider case $p=1/2$: uniform distribution.  Each outcome has
probability
$$f_{n,1/2}(k) = \binom{n}{k}\left(\frac{1}{2}\right)^n$$

Suppose I flip $100$ coins
\begin{itemize}
\item what is most likely number of heads?
\item what probability of exactly $50$ heads?
\item what probability of exactly $25$ heads
\item what probability less than $25$ heads?
\item what probability of exactly $25$ heads, given at most $25$?
\end{itemize}
vote: $1$, $.5$, $.1$, $.01$, $.000001$

\subsubsection{The central term}

Where is $f_{n,p}(k)$ maximized?
\begin{align*}
f_{n,p}(k)/f_{n.p}(k-1) &= \frac{n-k+1)p}{k(1-p)}\\
&= 1+\frac{(n+1)p-k}{k(1-p)}
\end{align*}
Therefore, $f_{n,p}(k)$ increases until $k=np$, decreases after.

What is the value at the central term?  Let's consider $p=1/2$, so the
central term is at $k=n/2$:
\begin{equation*}
\begin{aligned}
Pr(\frac{n}{2} \mbox{heads}) = 
\binom{n}{n/2}2^{-n} \\
 &= \frac{n!}{\frac{n}{2}!\frac{n}{2}!2^{n}} \\ 
 &= \frac{\sqrt{2\pi n}\left(\frac{n}{e}\right)^ne^{a_n}}{\left(\sqrt{\pi
    n}\left(\frac{n}{2e}\right)^{n/2}e^{a_{n}/2}\right)^2 2^n} \\
 &= \frac{\sqrt{2\pi n}}{\pi n} \frac{n^n (2e)^n}{e^n n^n 2^n} 
\frac{e^{a_n}}{e^{2a_{n/2}}} \\
 &= \sqrt{\frac{2}{\pi n}}e^{a_n - 2a_{n/2}}
\end{aligned}
\end{equation*}

Recall that $\frac{1}{12n+1} \leq a_n \leq \frac{1}{12n}$ \\
so $\frac{1}{12n+1}-{2}{6n} \leq a_n - 2a_{n/2} \leq \frac{1}{12n} - \frac{2}{6n+1}$. \
Therefore $e^{a_n - 2a_{n/2}} \rightarrow 1$. \\
So $Pr(\frac{n}{2} heads) \sim \sqrt{\frac{2}{\pi n}} = 
\Theta(\frac{1}{\sqrt{n}})$ \\

For $n = 100$ we have \\
$.079589176 \leq Pr(\frac{n}{2}heads) \leq .079589671$ \\
So Stirling's formula gives very tight bounds. \\

\subsubsection{The tails}

In general, for $k = \alpha n, \alpha \leq 1/2$, 
\begin{equation*}
\begin{aligned}
 Pr(\alpha \mbox{n heads}) \\ 
 &= \binom{n}{k}2^{-n} \\
 &= \frac{n!}{(\alpha n)!(1-\alpha n)! 2^n} \\     
 &= \sqrt{\frac{1}{2\pi \alpha(1-\alpha)n}} 2^{n(H(\alpha)-1)}
\end{aligned}
\end{equation*}

where $H(\alpha) = \alpha\log(1/\alpha)+(1-\alpha)\log(\frac{1}{1-\alpha})$ is
the {\em entropy function.}  It arises in the study of how much
information is carried in a binary string with a fraction $\alpha$ of
the bits set to one, and thus in the study of error correction over
faulty communication links.

                                % GNUPLOT: LaTeX picture
\setlength{\unitlength}{0.240900pt}
\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi
\sbox{\plotpoint}{\rule[-0.200pt]{0.400pt}{0.400pt}}%
\begin{picture}(1500,900)(0,0)
  \font\gnuplot=cmr10 at 10pt
  \gnuplot
  \sbox{\plotpoint}{\rule[-0.200pt]{0.400pt}{0.400pt}}%
  \put(220.0,113.0){\rule[-0.200pt]{292.934pt}{0.400pt}}
  \put(220.0,113.0){\rule[-0.200pt]{0.400pt}{184.048pt}}
  \put(220.0,113.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
  \put(198,113){\makebox(0,0)[r]{0}}
  \put(1416.0,113.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
  \put(220.0,266.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
  \put(198,266){\makebox(0,0)[r]{0.2}}
  \put(1416.0,266.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
  \put(220.0,419.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
  \put(198,419){\makebox(0,0)[r]{0.4}}
  \put(1416.0,419.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
  \put(220.0,571.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
  \put(198,571){\makebox(0,0)[r]{0.6}}
  \put(1416.0,571.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
  \put(220.0,724.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
  \put(198,724){\makebox(0,0)[r]{0.8}}
  \put(1416.0,724.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
  \put(220.0,877.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
  \put(198,877){\makebox(0,0)[r]{1}}
  \put(1416.0,877.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
  \put(220.0,113.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
  \put(220,68){\makebox(0,0){0}}
  \put(220.0,857.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
  \put(463.0,113.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
  \put(463,68){\makebox(0,0){0.2}}
  \put(463.0,857.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
  \put(706.0,113.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
  \put(706,68){\makebox(0,0){0.4}}
  \put(706.0,857.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
  \put(950.0,113.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
  \put(950,68){\makebox(0,0){0.6}}
  \put(950.0,857.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
  \put(1193.0,113.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
  \put(1193,68){\makebox(0,0){0.8}}
  \put(1193.0,857.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
  \put(1436.0,113.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
  \put(1436,68){\makebox(0,0){1}}
  \put(1436.0,857.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
  \put(220.0,113.0){\rule[-0.200pt]{292.934pt}{0.400pt}}
  \put(1436.0,113.0){\rule[-0.200pt]{0.400pt}{184.048pt}}
  \put(220.0,877.0){\rule[-0.200pt]{292.934pt}{0.400pt}}
  \put(45,495){\makebox(0,0){$H(\alpha)$}}
  \put(828,23){\makebox(0,0){$\alpha$}}
  \put(220.0,113.0){\rule[-0.200pt]{0.400pt}{184.048pt}}
  \put(232,175){\usebox{\plotpoint}}
  \multiput(232.58,175.00)(0.493,1.845){23}{\rule{0.119pt}{1.546pt}}
  \multiput(231.17,175.00)(13.000,43.791){2}{\rule{0.400pt}{0.773pt}}
  \multiput(245.58,222.00)(0.492,1.746){21}{\rule{0.119pt}{1.467pt}}
  \multiput(244.17,222.00)(12.000,37.956){2}{\rule{0.400pt}{0.733pt}}
  \multiput(257.58,263.00)(0.492,1.573){21}{\rule{0.119pt}{1.333pt}}
  \multiput(256.17,263.00)(12.000,34.233){2}{\rule{0.400pt}{0.667pt}}
  \multiput(269.58,300.00)(0.492,1.401){21}{\rule{0.119pt}{1.200pt}}
  \multiput(268.17,300.00)(12.000,30.509){2}{\rule{0.400pt}{0.600pt}}
  \multiput(281.58,333.00)(0.493,1.250){23}{\rule{0.119pt}{1.085pt}}
  \multiput(280.17,333.00)(13.000,29.749){2}{\rule{0.400pt}{0.542pt}}
  \multiput(294.58,365.00)(0.492,1.272){21}{\rule{0.119pt}{1.100pt}}
  \multiput(293.17,365.00)(12.000,27.717){2}{\rule{0.400pt}{0.550pt}}
  \multiput(306.58,395.00)(0.492,1.142){21}{\rule{0.119pt}{1.000pt}}
  \multiput(305.17,395.00)(12.000,24.924){2}{\rule{0.400pt}{0.500pt}}
  \multiput(318.58,422.00)(0.493,1.052){23}{\rule{0.119pt}{0.931pt}}
  \multiput(317.17,422.00)(13.000,25.068){2}{\rule{0.400pt}{0.465pt}}
  \multiput(331.58,449.00)(0.492,1.056){21}{\rule{0.119pt}{0.933pt}}
  \multiput(330.17,449.00)(12.000,23.063){2}{\rule{0.400pt}{0.467pt}}
  \multiput(343.58,474.00)(0.492,0.970){21}{\rule{0.119pt}{0.867pt}}
  \multiput(342.17,474.00)(12.000,21.201){2}{\rule{0.400pt}{0.433pt}}
  \multiput(355.58,497.00)(0.492,0.970){21}{\rule{0.119pt}{0.867pt}}
  \multiput(354.17,497.00)(12.000,21.201){2}{\rule{0.400pt}{0.433pt}}
  \multiput(367.58,520.00)(0.493,0.853){23}{\rule{0.119pt}{0.777pt}}
  \multiput(366.17,520.00)(13.000,20.387){2}{\rule{0.400pt}{0.388pt}}
  \multiput(380.58,542.00)(0.492,0.841){21}{\rule{0.119pt}{0.767pt}}
  \multiput(379.17,542.00)(12.000,18.409){2}{\rule{0.400pt}{0.383pt}}
  \multiput(392.58,562.00)(0.492,0.841){21}{\rule{0.119pt}{0.767pt}}
  \multiput(391.17,562.00)(12.000,18.409){2}{\rule{0.400pt}{0.383pt}}
  \multiput(404.58,582.00)(0.493,0.734){23}{\rule{0.119pt}{0.685pt}}
  \multiput(403.17,582.00)(13.000,17.579){2}{\rule{0.400pt}{0.342pt}}
  \multiput(417.58,601.00)(0.492,0.712){21}{\rule{0.119pt}{0.667pt}}
  \multiput(416.17,601.00)(12.000,15.616){2}{\rule{0.400pt}{0.333pt}}
  \multiput(429.58,618.00)(0.492,0.755){21}{\rule{0.119pt}{0.700pt}}
  \multiput(428.17,618.00)(12.000,16.547){2}{\rule{0.400pt}{0.350pt}}
  \multiput(441.58,636.00)(0.492,0.669){21}{\rule{0.119pt}{0.633pt}}
  \multiput(440.17,636.00)(12.000,14.685){2}{\rule{0.400pt}{0.317pt}}
  \multiput(453.58,652.00)(0.493,0.616){23}{\rule{0.119pt}{0.592pt}}
  \multiput(452.17,652.00)(13.000,14.771){2}{\rule{0.400pt}{0.296pt}}
  \multiput(466.58,668.00)(0.492,0.625){21}{\rule{0.119pt}{0.600pt}}
  \multiput(465.17,668.00)(12.000,13.755){2}{\rule{0.400pt}{0.300pt}}
  \multiput(478.58,683.00)(0.492,0.582){21}{\rule{0.119pt}{0.567pt}}
  \multiput(477.17,683.00)(12.000,12.824){2}{\rule{0.400pt}{0.283pt}}
  \multiput(490.00,697.58)(0.497,0.493){23}{\rule{0.500pt}{0.119pt}}
  \multiput(490.00,696.17)(11.962,13.000){2}{\rule{0.250pt}{0.400pt}}
  \multiput(503.58,710.00)(0.492,0.539){21}{\rule{0.119pt}{0.533pt}}
  \multiput(502.17,710.00)(12.000,11.893){2}{\rule{0.400pt}{0.267pt}}
  \multiput(515.58,723.00)(0.492,0.539){21}{\rule{0.119pt}{0.533pt}}
  \multiput(514.17,723.00)(12.000,11.893){2}{\rule{0.400pt}{0.267pt}}
  \multiput(527.00,736.58)(0.496,0.492){21}{\rule{0.500pt}{0.119pt}}
  \multiput(527.00,735.17)(10.962,12.000){2}{\rule{0.250pt}{0.400pt}}
  \multiput(539.00,748.58)(0.590,0.492){19}{\rule{0.573pt}{0.118pt}}
  \multiput(539.00,747.17)(11.811,11.000){2}{\rule{0.286pt}{0.400pt}}
  \multiput(552.00,759.58)(0.600,0.491){17}{\rule{0.580pt}{0.118pt}}
  \multiput(552.00,758.17)(10.796,10.000){2}{\rule{0.290pt}{0.400pt}}
  \multiput(564.00,769.58)(0.543,0.492){19}{\rule{0.536pt}{0.118pt}}
  \multiput(564.00,768.17)(10.887,11.000){2}{\rule{0.268pt}{0.400pt}}
  \multiput(576.00,780.59)(0.669,0.489){15}{\rule{0.633pt}{0.118pt}}
  \multiput(576.00,779.17)(10.685,9.000){2}{\rule{0.317pt}{0.400pt}}
  \multiput(588.00,789.59)(0.728,0.489){15}{\rule{0.678pt}{0.118pt}}
  \multiput(588.00,788.17)(11.593,9.000){2}{\rule{0.339pt}{0.400pt}}
  \multiput(601.00,798.59)(0.669,0.489){15}{\rule{0.633pt}{0.118pt}}
  \multiput(601.00,797.17)(10.685,9.000){2}{\rule{0.317pt}{0.400pt}}
  \multiput(613.00,807.59)(0.758,0.488){13}{\rule{0.700pt}{0.117pt}}
  \multiput(613.00,806.17)(10.547,8.000){2}{\rule{0.350pt}{0.400pt}}
  \multiput(625.00,815.59)(0.950,0.485){11}{\rule{0.843pt}{0.117pt}}
  \multiput(625.00,814.17)(11.251,7.000){2}{\rule{0.421pt}{0.400pt}}
  \multiput(638.00,822.59)(0.874,0.485){11}{\rule{0.786pt}{0.117pt}}
  \multiput(638.00,821.17)(10.369,7.000){2}{\rule{0.393pt}{0.400pt}}
  \multiput(650.00,829.59)(1.033,0.482){9}{\rule{0.900pt}{0.116pt}}
  \multiput(650.00,828.17)(10.132,6.000){2}{\rule{0.450pt}{0.400pt}}
  \multiput(662.00,835.59)(1.033,0.482){9}{\rule{0.900pt}{0.116pt}}
  \multiput(662.00,834.17)(10.132,6.000){2}{\rule{0.450pt}{0.400pt}}
  \multiput(674.00,841.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
  \multiput(674.00,840.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
  \multiput(687.00,847.59)(1.267,0.477){7}{\rule{1.060pt}{0.115pt}}
  \multiput(687.00,846.17)(9.800,5.000){2}{\rule{0.530pt}{0.400pt}}
  \multiput(699.00,852.59)(1.267,0.477){7}{\rule{1.060pt}{0.115pt}}
  \multiput(699.00,851.17)(9.800,5.000){2}{\rule{0.530pt}{0.400pt}}
  \multiput(711.00,857.60)(1.797,0.468){5}{\rule{1.400pt}{0.113pt}}
  \multiput(711.00,856.17)(10.094,4.000){2}{\rule{0.700pt}{0.400pt}}
  \multiput(724.00,861.61)(2.472,0.447){3}{\rule{1.700pt}{0.108pt}}
  \multiput(724.00,860.17)(8.472,3.000){2}{\rule{0.850pt}{0.400pt}}
  \multiput(736.00,864.61)(2.472,0.447){3}{\rule{1.700pt}{0.108pt}}
  \multiput(736.00,863.17)(8.472,3.000){2}{\rule{0.850pt}{0.400pt}}
  \multiput(748.00,867.61)(2.472,0.447){3}{\rule{1.700pt}{0.108pt}}
  \multiput(748.00,866.17)(8.472,3.000){2}{\rule{0.850pt}{0.400pt}}
  \put(760,870.17){\rule{2.700pt}{0.400pt}}
  \multiput(760.00,869.17)(7.396,2.000){2}{\rule{1.350pt}{0.400pt}}
  \put(773,872.17){\rule{2.500pt}{0.400pt}}
  \multiput(773.00,871.17)(6.811,2.000){2}{\rule{1.250pt}{0.400pt}}
  \put(785,874.17){\rule{2.500pt}{0.400pt}}
  \multiput(785.00,873.17)(6.811,2.000){2}{\rule{1.250pt}{0.400pt}}
  \put(810,875.67){\rule{2.891pt}{0.400pt}}
  \multiput(810.00,875.17)(6.000,1.000){2}{\rule{1.445pt}{0.400pt}}
  \put(797.0,876.0){\rule[-0.200pt]{3.132pt}{0.400pt}}
  \put(834,875.67){\rule{2.891pt}{0.400pt}}
  \multiput(834.00,876.17)(6.000,-1.000){2}{\rule{1.445pt}{0.400pt}}
  \put(822.0,877.0){\rule[-0.200pt]{2.891pt}{0.400pt}}
  \put(859,874.17){\rule{2.500pt}{0.400pt}}
  \multiput(859.00,875.17)(6.811,-2.000){2}{\rule{1.250pt}{0.400pt}}
  \put(871,872.17){\rule{2.500pt}{0.400pt}}
  \multiput(871.00,873.17)(6.811,-2.000){2}{\rule{1.250pt}{0.400pt}}
  \put(883,870.17){\rule{2.700pt}{0.400pt}}
  \multiput(883.00,871.17)(7.396,-2.000){2}{\rule{1.350pt}{0.400pt}}
  \multiput(896.00,868.95)(2.472,-0.447){3}{\rule{1.700pt}{0.108pt}}
  \multiput(896.00,869.17)(8.472,-3.000){2}{\rule{0.850pt}{0.400pt}}
  \multiput(908.00,865.95)(2.472,-0.447){3}{\rule{1.700pt}{0.108pt}}
  \multiput(908.00,866.17)(8.472,-3.000){2}{\rule{0.850pt}{0.400pt}}
  \multiput(920.00,862.95)(2.472,-0.447){3}{\rule{1.700pt}{0.108pt}}
  \multiput(920.00,863.17)(8.472,-3.000){2}{\rule{0.850pt}{0.400pt}}
  \multiput(932.00,859.94)(1.797,-0.468){5}{\rule{1.400pt}{0.113pt}}
  \multiput(932.00,860.17)(10.094,-4.000){2}{\rule{0.700pt}{0.400pt}}
  \multiput(945.00,855.93)(1.267,-0.477){7}{\rule{1.060pt}{0.115pt}}
  \multiput(945.00,856.17)(9.800,-5.000){2}{\rule{0.530pt}{0.400pt}}
  \multiput(957.00,850.93)(1.267,-0.477){7}{\rule{1.060pt}{0.115pt}}
  \multiput(957.00,851.17)(9.800,-5.000){2}{\rule{0.530pt}{0.400pt}}
  \multiput(969.00,845.93)(1.123,-0.482){9}{\rule{0.967pt}{0.116pt}}
  \multiput(969.00,846.17)(10.994,-6.000){2}{\rule{0.483pt}{0.400pt}}
  \multiput(982.00,839.93)(1.033,-0.482){9}{\rule{0.900pt}{0.116pt}}
  \multiput(982.00,840.17)(10.132,-6.000){2}{\rule{0.450pt}{0.400pt}}
  \multiput(994.00,833.93)(1.033,-0.482){9}{\rule{0.900pt}{0.116pt}}
  \multiput(994.00,834.17)(10.132,-6.000){2}{\rule{0.450pt}{0.400pt}}
  \multiput(1006.00,827.93)(0.874,-0.485){11}{\rule{0.786pt}{0.117pt}}
  \multiput(1006.00,828.17)(10.369,-7.000){2}{\rule{0.393pt}{0.400pt}}
  \multiput(1018.00,820.93)(0.950,-0.485){11}{\rule{0.843pt}{0.117pt}}
  \multiput(1018.00,821.17)(11.251,-7.000){2}{\rule{0.421pt}{0.400pt}}
  \multiput(1031.00,813.93)(0.758,-0.488){13}{\rule{0.700pt}{0.117pt}}
  \multiput(1031.00,814.17)(10.547,-8.000){2}{\rule{0.350pt}{0.400pt}}
  \multiput(1043.00,805.93)(0.669,-0.489){15}{\rule{0.633pt}{0.118pt}}
  \multiput(1043.00,806.17)(10.685,-9.000){2}{\rule{0.317pt}{0.400pt}}
  \multiput(1055.00,796.93)(0.728,-0.489){15}{\rule{0.678pt}{0.118pt}}
  \multiput(1055.00,797.17)(11.593,-9.000){2}{\rule{0.339pt}{0.400pt}}
  \multiput(1068.00,787.93)(0.669,-0.489){15}{\rule{0.633pt}{0.118pt}}
  \multiput(1068.00,788.17)(10.685,-9.000){2}{\rule{0.317pt}{0.400pt}}
  \multiput(1080.00,778.92)(0.543,-0.492){19}{\rule{0.536pt}{0.118pt}}
  \multiput(1080.00,779.17)(10.887,-11.000){2}{\rule{0.268pt}{0.400pt}}
  \multiput(1092.00,767.92)(0.600,-0.491){17}{\rule{0.580pt}{0.118pt}}
  \multiput(1092.00,768.17)(10.796,-10.000){2}{\rule{0.290pt}{0.400pt}}
  \multiput(1104.00,757.92)(0.590,-0.492){19}{\rule{0.573pt}{0.118pt}}
  \multiput(1104.00,758.17)(11.811,-11.000){2}{\rule{0.286pt}{0.400pt}}
  \multiput(1117.00,746.92)(0.496,-0.492){21}{\rule{0.500pt}{0.119pt}}
  \multiput(1117.00,747.17)(10.962,-12.000){2}{\rule{0.250pt}{0.400pt}}
  \multiput(1129.58,733.79)(0.492,-0.539){21}{\rule{0.119pt}{0.533pt}}
  \multiput(1128.17,734.89)(12.000,-11.893){2}{\rule{0.400pt}{0.267pt}}
  \multiput(1141.58,720.79)(0.492,-0.539){21}{\rule{0.119pt}{0.533pt}}
  \multiput(1140.17,721.89)(12.000,-11.893){2}{\rule{0.400pt}{0.267pt}}
  \multiput(1153.00,708.92)(0.497,-0.493){23}{\rule{0.500pt}{0.119pt}}
  \multiput(1153.00,709.17)(11.962,-13.000){2}{\rule{0.250pt}{0.400pt}}
  \multiput(1166.58,694.65)(0.492,-0.582){21}{\rule{0.119pt}{0.567pt}}
  \multiput(1165.17,695.82)(12.000,-12.824){2}{\rule{0.400pt}{0.283pt}}
  \multiput(1178.58,680.51)(0.492,-0.625){21}{\rule{0.119pt}{0.600pt}}
  \multiput(1177.17,681.75)(12.000,-13.755){2}{\rule{0.400pt}{0.300pt}}
  \multiput(1190.58,665.54)(0.493,-0.616){23}{\rule{0.119pt}{0.592pt}}
  \multiput(1189.17,666.77)(13.000,-14.771){2}{\rule{0.400pt}{0.296pt}}
  \multiput(1203.58,649.37)(0.492,-0.669){21}{\rule{0.119pt}{0.633pt}}
  \multiput(1202.17,650.69)(12.000,-14.685){2}{\rule{0.400pt}{0.317pt}}
  \multiput(1215.58,633.09)(0.492,-0.755){21}{\rule{0.119pt}{0.700pt}}
  \multiput(1214.17,634.55)(12.000,-16.547){2}{\rule{0.400pt}{0.350pt}}
  \multiput(1227.58,615.23)(0.492,-0.712){21}{\rule{0.119pt}{0.667pt}}
  \multiput(1226.17,616.62)(12.000,-15.616){2}{\rule{0.400pt}{0.333pt}}
  \multiput(1239.58,598.16)(0.493,-0.734){23}{\rule{0.119pt}{0.685pt}}
  \multiput(1238.17,599.58)(13.000,-17.579){2}{\rule{0.400pt}{0.342pt}}
  \multiput(1252.58,578.82)(0.492,-0.841){21}{\rule{0.119pt}{0.767pt}}
  \multiput(1251.17,580.41)(12.000,-18.409){2}{\rule{0.400pt}{0.383pt}}
  \multiput(1264.58,558.82)(0.492,-0.841){21}{\rule{0.119pt}{0.767pt}}
  \multiput(1263.17,560.41)(12.000,-18.409){2}{\rule{0.400pt}{0.383pt}}
  \multiput(1276.58,538.77)(0.493,-0.853){23}{\rule{0.119pt}{0.777pt}}
  \multiput(1275.17,540.39)(13.000,-20.387){2}{\rule{0.400pt}{0.388pt}}
  \multiput(1289.58,516.40)(0.492,-0.970){21}{\rule{0.119pt}{0.867pt}}
  \multiput(1288.17,518.20)(12.000,-21.201){2}{\rule{0.400pt}{0.433pt}}
  \multiput(1301.58,493.40)(0.492,-0.970){21}{\rule{0.119pt}{0.867pt}}
  \multiput(1300.17,495.20)(12.000,-21.201){2}{\rule{0.400pt}{0.433pt}}
  \multiput(1313.58,470.13)(0.492,-1.056){21}{\rule{0.119pt}{0.933pt}}
  \multiput(1312.17,472.06)(12.000,-23.063){2}{\rule{0.400pt}{0.467pt}}
  \multiput(1325.58,445.14)(0.493,-1.052){23}{\rule{0.119pt}{0.931pt}}
  \multiput(1324.17,447.07)(13.000,-25.068){2}{\rule{0.400pt}{0.465pt}}
  \multiput(1338.58,417.85)(0.492,-1.142){21}{\rule{0.119pt}{1.000pt}}
  \multiput(1337.17,419.92)(12.000,-24.924){2}{\rule{0.400pt}{0.500pt}}
  \multiput(1350.58,390.43)(0.492,-1.272){21}{\rule{0.119pt}{1.100pt}}
  \multiput(1349.17,392.72)(12.000,-27.717){2}{\rule{0.400pt}{0.550pt}}
  \multiput(1362.58,360.50)(0.493,-1.250){23}{\rule{0.119pt}{1.085pt}}
  \multiput(1361.17,362.75)(13.000,-29.749){2}{\rule{0.400pt}{0.542pt}}
  \multiput(1375.58,328.02)(0.492,-1.401){21}{\rule{0.119pt}{1.200pt}}
  \multiput(1374.17,330.51)(12.000,-30.509){2}{\rule{0.400pt}{0.600pt}}
  \multiput(1387.58,294.47)(0.492,-1.573){21}{\rule{0.119pt}{1.333pt}}
  \multiput(1386.17,297.23)(12.000,-34.233){2}{\rule{0.400pt}{0.667pt}}
  \multiput(1399.58,256.91)(0.492,-1.746){21}{\rule{0.119pt}{1.467pt}}
  \multiput(1398.17,259.96)(12.000,-37.956){2}{\rule{0.400pt}{0.733pt}}
  \multiput(1411.58,215.58)(0.493,-1.845){23}{\rule{0.119pt}{1.546pt}}
  \multiput(1410.17,218.79)(13.000,-43.791){2}{\rule{0.400pt}{0.773pt}}
  \put(846.0,876.0){\rule[-0.200pt]{3.132pt}{0.400pt}}
\end{picture}

\begin{lemma}
  For $\alpha$ a constant $\ne 1/2$, $f_{n,1/2}(\alpha n)$ is
  exponentially small in $n$.
\end{lemma}

In other words, as $n$ increases, the likely values become
concentrated more and more tightly around $\alpha = 1/2$.

\begin{example}

$$ \binom{n}{n/4} \sim \sqrt{\frac{8}{3\pi n}}2^{0.811n} $$

$$ f_{n,1/2}(n/4) \sim \sqrt{\frac{8}{3\pi n}}2^{-0.189n} \;\;
\text{exponetially small} $$

Returning to our beginning questions, 
$$ Pr(\mbox{25 heads}) = f_{100,1/2}(25) \approx 1.913 \times 10^{-7} $$

\end{example}

\subsubsection{Bounding Sums with Geometric Series}
For a random variable $R$, the {\em cumulative distribution function}
$F_R(k) = \Pr(R \le k)$.

Since $f_{n,1/2}(k)$ is increasing with $k<n/2$, we can bound the
probability $F_R(25) \le 25\cdot f_{n,1/2}(25)$.  But we can do better.

Let $k = \alpha n$ where $\alpha \le 1/2$. Then
\begin{align*}
Pr(R\leq \alpha n)
  &= \sum_{i=0}^{\alpha n} Pr(R=i) \\
  &= \frac{1}{2^n}\left[ \binom{n}{\alpha n}+\binom{n}{\alpha n - 1}
  \cdots \binom{n}{0} \right]
\end{align*}

To upper bound the sum with a geometric series, we first compute the size of
successive terms. \\
\begin{align*}
\frac{\binom{n}{r-1}}{\binom{n}{r}} &=
\frac{\frac{n!}{(r-1)!(n-r+1)!}}{\frac{n!}{r!(n-r)!}}\\
&= \frac{r}{n-r+1}\\
&\leq \frac{\alpha n}{n- \alpha n + 1}\\
&\leq \frac{\alpha}{1-\alpha} \\
\intertext{
(Note that we used $r \leq \alpha n$.)
Therefore,}
\binom{n}{\alpha n - r} &\le
\left(\frac{\alpha}{1-\alpha}\right)^r\binom{n}{\alpha n}
\end{align*}

Plugging in, we find
\begin{align*}
Pr[R\leq \alpha n]
  &\leq 2^{-n}\left[ \binom{n}{\alpha n} +
  \frac{\alpha}{1-\alpha}\binom{n}{\alpha n} +
  \left(\frac{\alpha}{1-\alpha}\right)^2\binom{n}{\alpha n}+ \cdots
    + \left(\frac{\alpha}{1-\alpha}\right)^{\alpha n}\binom{n}{\alpha
      n} \right] \\
    &\leq 2^{-n}\binom{n}{\alpha n}\frac{1}{1-\frac{\alpha}{1-\alpha}}
    \;\; (\alpha < 1/2 \Rightarrow \frac{\alpha}{1-\alpha}<1) \\
    &= 2^{-n}\binom{n}{\alpha n}\frac{1-\alpha}{1-2\alpha}
\end{align*}

Notice that $\frac{1-\alpha}{1-2\alpha}$ is constant!.
$$ Pr(\leq 25\mbox{ heads}) \; (n=100, \alpha=1/4) \; = (1.913\times
10^{-7})\frac{3/4}{1/2}  $$

For $\beta = o(\sqrt{n}), f(n/2-\beta\sqrt{n}) \sim \sqrt{\frac{2}{\pi
    n}}e^{-2\beta^2}$.

Furthermore,
\begin{align*}
\Pr(\le 24\mbox{ heads}) &\le \frac{\alpha}{1-\alpha}\Pr(\le 25\mbox{
  heads})\\
\intertext{so that}
\Pr(\mbox{exactly $25$ heads } \mid \mbox{at most $25$ heads}) &> 1/2
\end{align*}

\subsubsection{Limiting Distribution}

As $n \rightarrow \infty$, $f_{n,p}$ approaches ``Bell curve'' or
``normal distribution'' (continuous). (Main difference is in the
tails, which are exponentially small anyway.)

Here is a plot of the ``Bell curve'':

                                % GNUPLOT: LaTeX picture
\setlength{\unitlength}{0.240900pt}
\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi
\sbox{\plotpoint}{\rule[-0.200pt]{0.400pt}{0.400pt}}%
\begin{picture}(1500,900)(0,0)
  \font\gnuplot=cmr10 at 10pt
  \gnuplot
  \sbox{\plotpoint}{\rule[-0.200pt]{0.400pt}{0.400pt}}%
  \put(176.0,68.0){\rule[-0.200pt]{303.534pt}{0.400pt}}
  \put(806.0,68.0){\rule[-0.200pt]{0.400pt}{194.888pt}}
  \put(176,68){\usebox{\plotpoint}}
  \put(240,67.67){\rule{2.891pt}{0.400pt}}
  \multiput(240.00,67.17)(6.000,1.000){2}{\rule{1.445pt}{0.400pt}}
  \put(176.0,68.0){\rule[-0.200pt]{15.418pt}{0.400pt}}
  \put(278,68.67){\rule{3.132pt}{0.400pt}}
  \multiput(278.00,68.17)(6.500,1.000){2}{\rule{1.566pt}{0.400pt}}
  \put(291,69.67){\rule{2.891pt}{0.400pt}}
  \multiput(291.00,69.17)(6.000,1.000){2}{\rule{1.445pt}{0.400pt}}
  \put(252.0,69.0){\rule[-0.200pt]{6.263pt}{0.400pt}}
  \put(316,71.17){\rule{2.700pt}{0.400pt}}
  \multiput(316.00,70.17)(7.396,2.000){2}{\rule{1.350pt}{0.400pt}}
  \put(329,72.67){\rule{2.891pt}{0.400pt}}
  \multiput(329.00,72.17)(6.000,1.000){2}{\rule{1.445pt}{0.400pt}}
  \put(341,74.17){\rule{2.700pt}{0.400pt}}
  \multiput(341.00,73.17)(7.396,2.000){2}{\rule{1.350pt}{0.400pt}}
  \put(354,76.17){\rule{2.700pt}{0.400pt}}
  \multiput(354.00,75.17)(7.396,2.000){2}{\rule{1.350pt}{0.400pt}}
  \multiput(367.00,78.61)(2.695,0.447){3}{\rule{1.833pt}{0.108pt}}
  \multiput(367.00,77.17)(9.195,3.000){2}{\rule{0.917pt}{0.400pt}}
  \multiput(380.00,81.60)(1.651,0.468){5}{\rule{1.300pt}{0.113pt}}
  \multiput(380.00,80.17)(9.302,4.000){2}{\rule{0.650pt}{0.400pt}}
  \multiput(392.00,85.60)(1.797,0.468){5}{\rule{1.400pt}{0.113pt}}
  \multiput(392.00,84.17)(10.094,4.000){2}{\rule{0.700pt}{0.400pt}}
  \multiput(405.00,89.59)(1.378,0.477){7}{\rule{1.140pt}{0.115pt}}
  \multiput(405.00,88.17)(10.634,5.000){2}{\rule{0.570pt}{0.400pt}}
  \multiput(418.00,94.59)(0.950,0.485){11}{\rule{0.843pt}{0.117pt}}
  \multiput(418.00,93.17)(11.251,7.000){2}{\rule{0.421pt}{0.400pt}}
  \multiput(431.00,101.59)(0.758,0.488){13}{\rule{0.700pt}{0.117pt}}
  \multiput(431.00,100.17)(10.547,8.000){2}{\rule{0.350pt}{0.400pt}}
  \multiput(443.00,109.59)(0.728,0.489){15}{\rule{0.678pt}{0.118pt}}
  \multiput(443.00,108.17)(11.593,9.000){2}{\rule{0.339pt}{0.400pt}}
  \multiput(456.00,118.58)(0.590,0.492){19}{\rule{0.573pt}{0.118pt}}
  \multiput(456.00,117.17)(11.811,11.000){2}{\rule{0.286pt}{0.400pt}}
  \multiput(469.58,129.00)(0.492,0.539){21}{\rule{0.119pt}{0.533pt}}
  \multiput(468.17,129.00)(12.000,11.893){2}{\rule{0.400pt}{0.267pt}}
  \multiput(481.58,142.00)(0.493,0.576){23}{\rule{0.119pt}{0.562pt}}
  \multiput(480.17,142.00)(13.000,13.834){2}{\rule{0.400pt}{0.281pt}}
  \multiput(494.58,157.00)(0.493,0.655){23}{\rule{0.119pt}{0.623pt}}
  \multiput(493.17,157.00)(13.000,15.707){2}{\rule{0.400pt}{0.312pt}}
  \multiput(507.58,174.00)(0.493,0.774){23}{\rule{0.119pt}{0.715pt}}
  \multiput(506.17,174.00)(13.000,18.515){2}{\rule{0.400pt}{0.358pt}}
  \multiput(520.58,194.00)(0.492,0.927){21}{\rule{0.119pt}{0.833pt}}
  \multiput(519.17,194.00)(12.000,20.270){2}{\rule{0.400pt}{0.417pt}}
  \multiput(532.58,216.00)(0.493,0.972){23}{\rule{0.119pt}{0.869pt}}
  \multiput(531.17,216.00)(13.000,23.196){2}{\rule{0.400pt}{0.435pt}}
  \multiput(545.58,241.00)(0.493,1.052){23}{\rule{0.119pt}{0.931pt}}
  \multiput(544.17,241.00)(13.000,25.068){2}{\rule{0.400pt}{0.465pt}}
  \multiput(558.58,268.00)(0.493,1.171){23}{\rule{0.119pt}{1.023pt}}
  \multiput(557.17,268.00)(13.000,27.877){2}{\rule{0.400pt}{0.512pt}}
  \multiput(571.58,298.00)(0.492,1.401){21}{\rule{0.119pt}{1.200pt}}
  \multiput(570.17,298.00)(12.000,30.509){2}{\rule{0.400pt}{0.600pt}}
  \multiput(583.58,331.00)(0.493,1.369){23}{\rule{0.119pt}{1.177pt}}
  \multiput(582.17,331.00)(13.000,32.557){2}{\rule{0.400pt}{0.588pt}}
  \multiput(596.58,366.00)(0.493,1.448){23}{\rule{0.119pt}{1.238pt}}
  \multiput(595.17,366.00)(13.000,34.430){2}{\rule{0.400pt}{0.619pt}}
  \multiput(609.58,403.00)(0.492,1.659){21}{\rule{0.119pt}{1.400pt}}
  \multiput(608.17,403.00)(12.000,36.094){2}{\rule{0.400pt}{0.700pt}}
  \multiput(621.58,442.00)(0.493,1.607){23}{\rule{0.119pt}{1.362pt}}
  \multiput(620.17,442.00)(13.000,38.174){2}{\rule{0.400pt}{0.681pt}}
  \multiput(634.58,483.00)(0.493,1.607){23}{\rule{0.119pt}{1.362pt}}
  \multiput(633.17,483.00)(13.000,38.174){2}{\rule{0.400pt}{0.681pt}}
  \multiput(647.58,524.00)(0.493,1.646){23}{\rule{0.119pt}{1.392pt}}
  \multiput(646.17,524.00)(13.000,39.110){2}{\rule{0.400pt}{0.696pt}}
  \multiput(660.58,566.00)(0.492,1.789){21}{\rule{0.119pt}{1.500pt}}
  \multiput(659.17,566.00)(12.000,38.887){2}{\rule{0.400pt}{0.750pt}}
  \multiput(672.58,608.00)(0.493,1.607){23}{\rule{0.119pt}{1.362pt}}
  \multiput(671.17,608.00)(13.000,38.174){2}{\rule{0.400pt}{0.681pt}}
  \multiput(685.58,649.00)(0.493,1.567){23}{\rule{0.119pt}{1.331pt}}
  \multiput(684.17,649.00)(13.000,37.238){2}{\rule{0.400pt}{0.665pt}}
  \multiput(698.58,689.00)(0.493,1.488){23}{\rule{0.119pt}{1.269pt}}
  \multiput(697.17,689.00)(13.000,35.366){2}{\rule{0.400pt}{0.635pt}}
  \multiput(711.58,727.00)(0.492,1.444){21}{\rule{0.119pt}{1.233pt}}
  \multiput(710.17,727.00)(12.000,31.440){2}{\rule{0.400pt}{0.617pt}}
  \multiput(723.58,761.00)(0.493,1.250){23}{\rule{0.119pt}{1.085pt}}
  \multiput(722.17,761.00)(13.000,29.749){2}{\rule{0.400pt}{0.542pt}}
  \multiput(736.58,793.00)(0.493,1.052){23}{\rule{0.119pt}{0.931pt}}
  \multiput(735.17,793.00)(13.000,25.068){2}{\rule{0.400pt}{0.465pt}}
  \multiput(749.58,820.00)(0.492,0.927){21}{\rule{0.119pt}{0.833pt}}
  \multiput(748.17,820.00)(12.000,20.270){2}{\rule{0.400pt}{0.417pt}}
  \multiput(761.58,842.00)(0.493,0.655){23}{\rule{0.119pt}{0.623pt}}
  \multiput(760.17,842.00)(13.000,15.707){2}{\rule{0.400pt}{0.312pt}}
  \multiput(774.00,859.58)(0.539,0.492){21}{\rule{0.533pt}{0.119pt}}
  \multiput(774.00,858.17)(11.893,12.000){2}{\rule{0.267pt}{0.400pt}}
  \multiput(787.00,871.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
  \multiput(787.00,870.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
  \put(303.0,71.0){\rule[-0.200pt]{3.132pt}{0.400pt}}
  \multiput(812.00,875.93)(1.123,-0.482){9}{\rule{0.967pt}{0.116pt}}
  \multiput(812.00,876.17)(10.994,-6.000){2}{\rule{0.483pt}{0.400pt}}
  \multiput(825.00,869.92)(0.539,-0.492){21}{\rule{0.533pt}{0.119pt}}
  \multiput(825.00,870.17)(11.893,-12.000){2}{\rule{0.267pt}{0.400pt}}
  \multiput(838.58,856.41)(0.493,-0.655){23}{\rule{0.119pt}{0.623pt}}
  \multiput(837.17,857.71)(13.000,-15.707){2}{\rule{0.400pt}{0.312pt}}
  \multiput(851.58,838.54)(0.492,-0.927){21}{\rule{0.119pt}{0.833pt}}
  \multiput(850.17,840.27)(12.000,-20.270){2}{\rule{0.400pt}{0.417pt}}
  \multiput(863.58,816.14)(0.493,-1.052){23}{\rule{0.119pt}{0.931pt}}
  \multiput(862.17,818.07)(13.000,-25.068){2}{\rule{0.400pt}{0.465pt}}
  \multiput(876.58,788.50)(0.493,-1.250){23}{\rule{0.119pt}{1.085pt}}
  \multiput(875.17,790.75)(13.000,-29.749){2}{\rule{0.400pt}{0.542pt}}
  \multiput(889.58,755.88)(0.492,-1.444){21}{\rule{0.119pt}{1.233pt}}
  \multiput(888.17,758.44)(12.000,-31.440){2}{\rule{0.400pt}{0.617pt}}
  \multiput(901.58,721.73)(0.493,-1.488){23}{\rule{0.119pt}{1.269pt}}
  \multiput(900.17,724.37)(13.000,-35.366){2}{\rule{0.400pt}{0.635pt}}
  \multiput(914.58,683.48)(0.493,-1.567){23}{\rule{0.119pt}{1.331pt}}
  \multiput(913.17,686.24)(13.000,-37.238){2}{\rule{0.400pt}{0.665pt}}
  \multiput(927.58,643.35)(0.493,-1.607){23}{\rule{0.119pt}{1.362pt}}
  \multiput(926.17,646.17)(13.000,-38.174){2}{\rule{0.400pt}{0.681pt}}
  \multiput(940.58,601.77)(0.492,-1.789){21}{\rule{0.119pt}{1.500pt}}
  \multiput(939.17,604.89)(12.000,-38.887){2}{\rule{0.400pt}{0.750pt}}
  \multiput(952.58,560.22)(0.493,-1.646){23}{\rule{0.119pt}{1.392pt}}
  \multiput(951.17,563.11)(13.000,-39.110){2}{\rule{0.400pt}{0.696pt}}
  \multiput(965.58,518.35)(0.493,-1.607){23}{\rule{0.119pt}{1.362pt}}
  \multiput(964.17,521.17)(13.000,-38.174){2}{\rule{0.400pt}{0.681pt}}
  \multiput(978.58,477.35)(0.493,-1.607){23}{\rule{0.119pt}{1.362pt}}
  \multiput(977.17,480.17)(13.000,-38.174){2}{\rule{0.400pt}{0.681pt}}
  \multiput(991.58,436.19)(0.492,-1.659){21}{\rule{0.119pt}{1.400pt}}
  \multiput(990.17,439.09)(12.000,-36.094){2}{\rule{0.400pt}{0.700pt}}
  \multiput(1003.58,397.86)(0.493,-1.448){23}{\rule{0.119pt}{1.238pt}}
  \multiput(1002.17,400.43)(13.000,-34.430){2}{\rule{0.400pt}{0.619pt}}
  \multiput(1016.58,361.11)(0.493,-1.369){23}{\rule{0.119pt}{1.177pt}}
  \multiput(1015.17,363.56)(13.000,-32.557){2}{\rule{0.400pt}{0.588pt}}
  \multiput(1029.58,326.02)(0.492,-1.401){21}{\rule{0.119pt}{1.200pt}}
  \multiput(1028.17,328.51)(12.000,-30.509){2}{\rule{0.400pt}{0.600pt}}
  \multiput(1041.58,293.75)(0.493,-1.171){23}{\rule{0.119pt}{1.023pt}}
  \multiput(1040.17,295.88)(13.000,-27.877){2}{\rule{0.400pt}{0.512pt}}
  \multiput(1054.58,264.14)(0.493,-1.052){23}{\rule{0.119pt}{0.931pt}}
  \multiput(1053.17,266.07)(13.000,-25.068){2}{\rule{0.400pt}{0.465pt}}
  \multiput(1067.58,237.39)(0.493,-0.972){23}{\rule{0.119pt}{0.869pt}}
  \multiput(1066.17,239.20)(13.000,-23.196){2}{\rule{0.400pt}{0.435pt}}
  \multiput(1080.58,212.54)(0.492,-0.927){21}{\rule{0.119pt}{0.833pt}}
  \multiput(1079.17,214.27)(12.000,-20.270){2}{\rule{0.400pt}{0.417pt}}
  \multiput(1092.58,191.03)(0.493,-0.774){23}{\rule{0.119pt}{0.715pt}}
  \multiput(1091.17,192.52)(13.000,-18.515){2}{\rule{0.400pt}{0.358pt}}
  \multiput(1105.58,171.41)(0.493,-0.655){23}{\rule{0.119pt}{0.623pt}}
  \multiput(1104.17,172.71)(13.000,-15.707){2}{\rule{0.400pt}{0.312pt}}
  \multiput(1118.58,154.67)(0.493,-0.576){23}{\rule{0.119pt}{0.562pt}}
  \multiput(1117.17,155.83)(13.000,-13.834){2}{\rule{0.400pt}{0.281pt}}
  \multiput(1131.58,139.79)(0.492,-0.539){21}{\rule{0.119pt}{0.533pt}}
  \multiput(1130.17,140.89)(12.000,-11.893){2}{\rule{0.400pt}{0.267pt}}
  \multiput(1143.00,127.92)(0.590,-0.492){19}{\rule{0.573pt}{0.118pt}}
  \multiput(1143.00,128.17)(11.811,-11.000){2}{\rule{0.286pt}{0.400pt}}
  \multiput(1156.00,116.93)(0.728,-0.489){15}{\rule{0.678pt}{0.118pt}}
  \multiput(1156.00,117.17)(11.593,-9.000){2}{\rule{0.339pt}{0.400pt}}
  \multiput(1169.00,107.93)(0.758,-0.488){13}{\rule{0.700pt}{0.117pt}}
  \multiput(1169.00,108.17)(10.547,-8.000){2}{\rule{0.350pt}{0.400pt}}
  \multiput(1181.00,99.93)(0.950,-0.485){11}{\rule{0.843pt}{0.117pt}}
  \multiput(1181.00,100.17)(11.251,-7.000){2}{\rule{0.421pt}{0.400pt}}
  \multiput(1194.00,92.93)(1.378,-0.477){7}{\rule{1.140pt}{0.115pt}}
  \multiput(1194.00,93.17)(10.634,-5.000){2}{\rule{0.570pt}{0.400pt}}
  \multiput(1207.00,87.94)(1.797,-0.468){5}{\rule{1.400pt}{0.113pt}}
  \multiput(1207.00,88.17)(10.094,-4.000){2}{\rule{0.700pt}{0.400pt}}
  \multiput(1220.00,83.94)(1.651,-0.468){5}{\rule{1.300pt}{0.113pt}}
  \multiput(1220.00,84.17)(9.302,-4.000){2}{\rule{0.650pt}{0.400pt}}
  \multiput(1232.00,79.95)(2.695,-0.447){3}{\rule{1.833pt}{0.108pt}}
  \multiput(1232.00,80.17)(9.195,-3.000){2}{\rule{0.917pt}{0.400pt}}
  \put(1245,76.17){\rule{2.700pt}{0.400pt}}
  \multiput(1245.00,77.17)(7.396,-2.000){2}{\rule{1.350pt}{0.400pt}}
  \put(1258,74.17){\rule{2.700pt}{0.400pt}}
  \multiput(1258.00,75.17)(7.396,-2.000){2}{\rule{1.350pt}{0.400pt}}
  \put(1271,72.67){\rule{2.891pt}{0.400pt}}
  \multiput(1271.00,73.17)(6.000,-1.000){2}{\rule{1.445pt}{0.400pt}}
  \put(1283,71.17){\rule{2.700pt}{0.400pt}}
  \multiput(1283.00,72.17)(7.396,-2.000){2}{\rule{1.350pt}{0.400pt}}
  \put(800.0,877.0){\rule[-0.200pt]{2.891pt}{0.400pt}}
  \put(1309,69.67){\rule{2.891pt}{0.400pt}}
  \multiput(1309.00,70.17)(6.000,-1.000){2}{\rule{1.445pt}{0.400pt}}
  \put(1321,68.67){\rule{3.132pt}{0.400pt}}
  \multiput(1321.00,69.17)(6.500,-1.000){2}{\rule{1.566pt}{0.400pt}}
  \put(1296.0,71.0){\rule[-0.200pt]{3.132pt}{0.400pt}}
  \put(1360,67.67){\rule{2.891pt}{0.400pt}}
  \multiput(1360.00,68.17)(6.000,-1.000){2}{\rule{1.445pt}{0.400pt}}
  \put(1334.0,69.0){\rule[-0.200pt]{6.263pt}{0.400pt}}
  \put(1372.0,68.0){\rule[-0.200pt]{15.418pt}{0.400pt}}
\end{picture}

In fact, the binomial distribution is the discrete analog to the normal 
distribution. \\
So far, we have been focussing on the unbiased binomial distribution 
($p = 1/2$). It's not too hard to extend the results to the general binomial 
distribution.  \\
$f_p(\alpha n) \leq \frac{1}{\sqrt{2 \pi \alpha (1-\alpha)n}}
2^{ \alpha log(\frac{p}{\alpha})+(1- \alpha) log(\frac{1-p}{1- \alpha})n}$ \\
The maximum occurs at $\alpha = p$ so we have \\
$f_p(\alpha n) \leq \frac{1}{\sqrt{2 \pi p (1-p)n}}$ \\
Similarilly, \\
$F_p(\alpha n) \leq \frac{1- \alpha}{1 - \alpha/p} f_p(\alpha n)$ \\

\section{Applications}

\subsection{Polling}

We want to estimate the fraction $p$ of Democrats in the country by
conducting a poll, that is, we will sample $n$ people uniformly at
random, with replication, from the entire population. Let $D$ be the
number of Democrats in the sample we obtain. Then the intuition is
that $D$ should be a reasonably close approximation to $np$, since
each person is a Democrat with probability $p$ , and we sample $n$
times.\\

\begin{theorem}

  Given $\epsilon > 0$. Let $\delta$ be the sum of the probabilities
  of getting $\epsilon n$ too many democrats and $\epsilon n$ too few
  democrats in our sample, that is\\

  $$\delta = Pr(D < (p+\epsilon)n)+ Pr(D> (p-\epsilon)n)$$

  Then, with probability $1-\delta$, $ p-\epsilon < D < p+\epsilon$,
  that is, our poll will give a margin of error of $\pm \epsilon$, and
  be correct with probability $1-\delta$.
\end{theorem}

Given $\epsilon$, how do we compute $\delta$ ? We use the fact that
$D$ is binomially distributed. This allows us to apply the formulas
that we derived in the previous section. In particular, we have that\\ 

$$\delta = F_p((p-\epsilon)n)+ F_{1-p}((1-p-\epsilon)n)$$

The only problem with this is that we don't know $p$ yet. This is not
a serious problem, however. We have the following fact:\\ 

\begin{fact}
For any $\epsilon>0$, $\delta$ is maximized by setting $p=1/2$.
\end{fact}

Given this fact, we may set $p=1/2$ for the sake of analysis. Hence we
have that\\

\begin{math}
\delta \leq F_{1/2}((1/2-\epsilon)n)+ F_{1/2}((1/2-\epsilon)n) \\
\Rightarrow \delta \leq 2F_{1/2}((1/2-\epsilon)n)
\end{math}

Now we can just plug in values for $\epsilon$. Say we want an error
margin of $\pm .05$. In this case, we have that \\

\begin{math}
\delta \leq F_{1/2}(.45n) = 2 \frac{1-.45}{1-.90} \frac{1}{\sqrt{2
    \pi \cdot .45 \cdot .55n}} 2^{H(.45)-1)n} \\ 
\Rightarrow \delta \leq 11 \cdot \frac{.802}{\sqrt{n}} 2^{-.0072 n}
\end{math}

Plugging in for some values of $n$, we get:\\
\begin{eqnarray*}
n=1000 \Rightarrow & \delta \approx .00186 \\
n=700 \Rightarrow & \delta \approx .0100 \\
n=400 \Rightarrow & \delta \approx .05949 \\
\end{eqnarray*}

We conclude that, by sampling $700$ people, there is a $99\%$ chance
that the result of the poll will be within $5\%$ of the correct value. 
That does not, of course, exclude the possibility that we are really
unlucky in constructing our sample.
\subsection{Counting Fish}

There is a lake with $1$ million fish in it. We conduct an experiment
on the fish in two phases. In phase 1 we catch $A$ fish, tag them and
release them back into the lake. In phase 2 we repeatedly catch and
release the fish - we do this $B$ times.  What is the minimum total
number ($A+B$) of fish we should catch to make sure that we catch a
tagged fish (during phase 2) with probability greater than 99\%.

The probability we don't catch even one tagged fish during phase 2 is
$(1-\frac{A}{10^6})^B$. We want this probability to be less than $0.01$.
Now,
\[ (1-\frac{A}{10^6})^B < e^{-(\frac{AB}{10^6})}\].
Hence if we assure that 
\[ e^{-(\frac{AB}{10^6})} < 0.01 \]
we are done.
This implies that $AB > 4.6*10^6$. $A+B$ is minimized when $A=B$ for a given $AB$. Hence the minimum total number of fish we should catch is $2*\sqrt{4.6*10^6} \sim 4292$

\subsection{Leader Election}

There are $n$ persons in a room. They wish to pick one of themselves
as their leader. They wish to do this in a fair and democratic way, so
that each and everyone has the same chance to be the leader. The
scheme they employ is for everyone to toss a coin. If exactly one
person tosses a head that person is elected the leader. If no persons
or more than one person tosses heads then they repeat the entire
process. If the coins they use have probability $p$ of coming up heads
then what should $p$ be to maximize the probability of selecting a
leader in a given round.

If $n$ coins are tossed then the probability of having exactly one
head is ${n\choose 1}p(1-p)^{n-1}$. Differentiating w.r.t. $p$ and
equating to $0$ we find the maximum occurs when $p = 1/n$. Hence, they
should use coins so that the probability of coming up heads is
$1/n$. When they use such coins then the probability of selecting a
leader in a given round is ${n\choose 1}(1/n)(1-(1/n))^{n-1} \sim
1/e.$

\section{Summary}
Unbiased $(p=1/2)$ Binomial Distribution: $$ f(\frac{n}{2}) \sim
\sqrt{\frac{2}{\pi n}} = \Theta(\frac{1}{\sqrt{n}}) $$ $$ f(\alpha n)
\sim \sqrt{\frac{2}{\pi
n}}\sqrt{\frac{1}{4\alpha(1-\alpha)}}\frac{1}{2^{n(1-H(\alpha))}}
\;\; (\alpha <1/2) $$
General $(p \leq 1/2)$ Binomial Distribution:
$$ f(pn) \sim \sqrt{\frac{1}{2 \pi p(1-p)n}} = \Theta(\frac{1}{\sqrt{pn}}) $$
$$ f(\alpha n) \sim \sqrt{\frac{1}{2 \pi \alpha (1-\alpha)
    n}}2^{n(\alpha\log \frac{p}{\alpha} +
  (1-\alpha)\log\frac{p}{1-\alpha})} $$

\end{document}
