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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 26 "For the MAA Short Course: " }
{TEXT 260 45 "An Introduction to the Mathematics of Biology" }}{PARA 
0 "" 0 "" {TEXT -1 31 "Presented Friday, March 8, 2002" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "By " }{TEXT 261 14 "R
on Shonkwiler" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Nsteps:=20; \+
Ntrials:=1000;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "seed:=rea
dlib(randomize)();" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "step:
=2*rand(0..1)-1; # output= -1 or 1 with 50/50 chance" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 104 "walk:=proc()\nlocal x,i;\n  x:=0;\n  for
 i from 1 to Nsteps do\n    x := x + step();\n  od;\n  RETURN(x);\nend
:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 28 "The simulation and \+
histogram" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "with(stats): wi
th(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "histo:= array
(-Nsteps..Nsteps):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "for \+
j from -Nsteps to Nsteps do\n  histo[j]:=0;\nod:\nfor j from 0 to Ntri
als-1 do\n  k:= walk();\n  histo[k]:=histo[k]+1;\nod:" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }{TEXT 257 21 "Display the histogram" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "data:=[seq(Weight(i-.5..i+.5,eval(h
isto[i]/Ntrials)),\n     i=-Nsteps..Nsteps)]:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 36 "hPlot := statplots[histogram](data):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "hPlot;" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(combinat):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "N:=20; # number of steps#" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "cltApprox:=r->sqrt(2/(Pi*N
))*exp(-(2*r-N)^2/(2*N));" }}}{PARA 0 "" 0 "" {TEXT 258 21 "Do the com
binatorial." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "exact:=proc(r
)\n  RETURN( evalf(binomial(N,r)/2^N));\nend:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 37 "ePts:=[seq([2*r-N,exact(r)],r=0..N)]:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ePlot:=plot(ePts,style=POINT
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "aPts:=[seq([2*r-N,eva
lf(cltApprox(r))],r=0..N)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
43 "aPlot:=plot(aPts,style=POINT,symbol=CROSS):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "display(\{aPlot,ePlot,hPlot\});" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "for r from 0 to N do\n printf(`m= \+
%d, exact= %f cltapprox= %f\\n`,\n  2*r-N,exact(r),evalf(cltApprox(r))
);\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }{TEXT 259 3 "AND" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "u:=(x,t)-> exp(-x^2/(2*t))/sqrt(2*3.1415926*t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "t:=1;\np1:=plot([(x,u(x,t))
,x=-10..10]):\nt:=2;\np2:=plot([(x,u(x,t)),x=-10..10]):\nt:=4;\np4:=pl
ot([(x,u(x,t)),x=-10..10]):\nt:=8;\np8:=plot([(x,u(x,t)),x=-10..10]):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plots[display](\{p1,p2,
p4,p8\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 \+
0" 1 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }