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<H1><A name=SECTION002100000000000000000>Random Number Generation</A> </H1>Read 
the following web site pages: <A 
href="http://www.maa.org/mathland/mathland_4_22.html" 
name=tex2html4>http://www.maa.org/mathland/mathland_4_22.html</A>
<P><A href="http://stat.fsu.edu/~geo/diehard.html" 
name=tex2html5>http://stat.fsu.edu/<IMG height=52 alt=$\~$ 
src="Random Number Generation.files/img268.png" width=337 align=bottom 
border=0>geo/diehard.html</A> 
<P>And read chapter 7 of Numerical Recipes. 
<P><B>Motivation</B>: <BR>Random numbers are used: <BR>1- To solve problems that 
are too complex (stochastic algorithms for NP complete problems) 
<P>2- For optimization (simulated annealing, genetic algorithms). 
<P>3- System Simulation (physics and biology, econometrics...) 
<P>4- Non-parametric tests, bootstrap, permutation tests. 
<P>5- Quadrature methods using Monte Carlo. 
<P>Solving a deterministic problem by rewriting it as equivalent to a problem 
with a stochastic component. 
<P>Little Example as a reminder of what Monte-Carlo is: <BR>
<P></P>
<DIV align=center><!-- MATH \begin{displaymath}\theta=\int_0^1 f(t)dt\end{displaymath} --><IMG 
height=37 alt="\begin{displaymath}\theta=\int_0^1 f(t)dt\end{displaymath}" 
src="Random Number Generation.files/img269.png" width=71 border=0> </DIV><BR 
clear=all>
<P></P>
<P>If this exists it can be seen as <IMG height=37 alt=$E(f(X))$ 
src="Random Number Generation.files/img270.png" width=42 align=middle border=0> 
where <IMG height=17 alt=$X$ src="Random Number Generation.files/img12.png" 
width=22 align=bottom border=0> has a uniform distribution on <IMG height=65 
alt=$[0,1]$ src="Random Number Generation.files/img271.png" width=347 
align=middle border=0>. So we approximate it with <BR>
<P></P>
<DIV align=center><!-- MATH \begin{displaymath}\hat{\theta}= \frac{1}{N}\sum_{i=1}^N f(X_i)\end{displaymath} --><IMG 
height=18 
alt="\begin{displaymath}\hat{\theta}= \frac{1}{N}\sum_{i=1}^N f(X_i)\end{displaymath}" 
src="Random Number Generation.files/img272.png" width=48 border=0> </DIV><BR 
clear=all>
<P></P>for a sample of size <IMG height=17 alt=$N$ 
src="Random Number Generation.files/img76.png" width=22 align=bottom border=0> 
of uniform random variables. 
<P>Most methods, even if they need non-uniform random variates, as in 
simulation, still use uniform random variables as their starting points. 
<P>So we will spend some time explaining the basics of random numbers. 
<P>The machine being deterministic, there is definitely a problem, we will never 
get purely random numbers, they will only look so. 
<P>(We already saw that there are some pretty bad ones, when we did the 
multidimensional visualization of the <B>IBM randu</B> generated numbers with 
xgobi). 
<P>Also beware of <IMG height=31 alt=$srand$ 
src="Random Number Generation.files/img273.png" width=383 align=bottom 
border=0>, the basic unix one is bad. 
<P>Every time I reconnect to matlab, it provides the same random number the 
first time I ask for one: <PRE>&gt;&gt; rand
ans =    0.9501
</PRE>
<P>So it is quite predictable, and I could prove I am psychic. 
<P><BR>
<HR>
<!--Table of Child-Links--><A name=CHILD_LINKS><STRONG>Subsections</STRONG></A> 
<UL>
  <LI><A 
  href="http://www-stat.stanford.edu/~susan/courses/b494/index/node48.html" 
  name=tex2html773>Congruential Methods</A> 
  <LI><A 
  href="http://www-stat.stanford.edu/~susan/courses/b494/index/node49.html" 
  name=tex2html774>Other `Better?' Methods</A> </LI></UL><!--End of Table of Child-Links-->
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<ADDRESS>Susan Holmes 2002-01-12 </ADDRESS></BODY></HTML>
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