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\title{Stat 471: Lecture 10\\
Monte-Carlo Integral.}
% For single author (just remove % characters)
\author{Moo K. Chung\\
mchung@stat.wisc.edu}
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\begin{enumerate}
\item The constant $\pi$ is usually computed by summing up infinite series, i.e.$$\pi=4\Big(1-\frac{1}{3}+ \frac{1}{5} - \frac{1}{7} + \cdots\Big).$$
There is a slightly different way to estimate the digits of $\pi$ based on a computer simulation. Let $A$ be the area of a quarter circle with unit radius bounded by a unit square $B$. Suppose we throw darts at the square at random. Define binary random variable $X=1$ if a dart hits $A$ and $X=0$ if it hits $A^{c}$. Note that $$\mathbb{E} X = \int_{A} \;dx dy = \frac{\pi}{4}.$$ Note that the sample mean $\bar X_n=\sum_{i=1}^n X_i/n$ converges to $\mathbb{E} X$ as $n \to \infty$. So
$$\pi= 4\lim_{n \to \infty} \bar X_n.$$
\begin{verbatim}
x=rand(100000,1);y=rand(100000,1);
rsquare=x.^2+y.^2;
accept=find(rsquare <1);
>> accept'
ans =
     2 3 4 5 6 7 8 10 11 15 ...
plot(x(accept),y(accept),'.')
>>4*size(accept,1)/1000000
ans =
    3.1336
\end{verbatim}

\begin{figure}
\centering
\includegraphics[scale=0.32]{lecture10-1.eps}
\end{figure}

\item There are many Monte-Carlo techniques for evaluating
integrals. The above method is based on generating specific random
numbers. In general if we know how to generate random variable $X
\sim f$, we can generate $g(X)$. So
$$\mathbb{E} g(X) = \int g(x)f(x) \; dx.$$
$\mathbb{E}g(X)$ can be approximated by the sample mean $\bar
g(X)=\sum_{i=1}^n g(X_i)/n$. What if you want to compute $\int
g(x) \; dx$ ? Note that
$$\int g(x) \; dx = \int \frac{g(x)}{f(x)} f(x)\; dx =
\mathbb{E}\frac{g(X)}{f(X)}.$$ So we generate random numbers
$g(X)/f(X)$.

\item Empirical distribution. Estimate the cumulative distribution
of the standard normal using the Monte-Carlo integration method.
$$\Phi(x)=\int_{-\infty}^x \frac{e^{-x^2/2}}{\sqrt{2\pi}} \; dx.$$
Let $g$ be a index function defined as\\ $g(y) =
\mathbb{I}_{x}(y)=0$ if $y \leq x$ and 1 if $y > x$. Then
$\Phi(x)=\mathbb{E}\mathbb{I}_x(Z),$ where $Z \sim N(0,1)$. For
given $x$, $\mathbb{E}\mathbb{I}_x(Z)$ is estimated by the
proportion of the random numbers that is less than $x$. This is
true for an arbitrary cumulative distribution function and we call
this Monte-Carlo estimate as the empirical (cumulative)
distribution.


\end{enumerate}
 Interesting paper by John Von Neumann (1951): Various techniques
used in connection with random digits. Proceedings of Symposium on
¡°Monte Carlo Method¡± (held June. July, 1949, in Los Angeles),
chapter 13. Summary written by G. E. Forsythe, Journal of Research
of the National Bureau of Standards {\bf 12} 36-38. Next lecture
topic is about variance reduction in MC.

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