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Getting started" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "restart:with(linalg):with(st ats):with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definit ion for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition f or trace" }}}{EXCHG {PARA 262 "" 0 "" {TEXT 325 19 "2. Useful function s" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "rand();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"-\"3p'>uU" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 7 "Rand( )" }{TEXT -1 35 " generates a random number in [0,1]" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Rand:=proc()" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "evalf((rand()*10^(-12)),5);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 7 "Exampl e" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Rand();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&6@$!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 8 "Rands(n)" }{TEXT 261 2 " " }{TEXT -1 66 "generates an n-dimensional random vector with co mponents in [0,1]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Rands: =proc(n)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "local x,k;" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 16 "x:=vector(n,[]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "for k from 1 to n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "do x[k]:=Rand();od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalm(x) ;end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 7 "Example" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Rands(20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#76$ \"&jV$!\"&$\"&Eu%F)$\"&Ye&F)$\"&vY(F)$\"&i?$!\"'$\"&(HsF)$\"&J/'F)$\"& eX(F)$\"&\")f#F)$\"&35$F)$\"&=(zF)$\"&q\"RF2$\"&J%))F2$\"&]g*F)$\"&#H \")F)$\"&v`%F)$\"&.W'F)$\"&i?*F)$\"&0^*F)$\"&\\Y\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "convert(%,list);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#76$\"&jV$!\"&$\"&Eu%F&$\"&Ye&F&$\"&vY(F&$\"&i?$!\"'$\"& (HsF&$\"&J/'F&$\"&eX(F&$\"&\")f#F&$\"&35$F&$\"&=(zF&$\"&q\"RF/$\"&J%)) F/$\"&]g*F&$\"&#H\")F&$\"&v`%F&$\"&.W'F&$\"&i?*F&$\"&0^*F&$\"&\\Y\"F& " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "sort(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#76$\"&i?$!\"'$\"&q\"RF&$\"&J%))F&$\"&\\Y\"!\"&$\"& \")f#F-$\"&35$F-$\"&jV$F-$\"&v`%F-$\"&Eu%F-$\"&Ye&F-$\"&J/'F-$\"&.W'F- $\"&(HsF-$\"&eX(F-$\"&vY(F-$\"&=(zF-$\"&#H\")F-$\"&i?*F-$\"&0^*F-$\"&] g*F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 7 "Cum(x) " }{TEXT -1 52 "co mputes the vector of partial sums of elements of x" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Cum:=proc(x)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "local k,n,v;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "n:=vectdim( x);v:=array(1..n);v[1]:=x[1];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "fo r k from 2 to n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "do v[k]:=v[k-1]+ x[k] od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(v);" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 7 "Ex ample" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Cum([1,2,3,4,5,6,7 ,8,9,10]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7,\"\"\"\"\" $\"\"'\"#5\"#:\"#@\"#G\"#O\"#X\"#b" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 12 "Lookup(x,a) " }{TEXT -1 38 " finds the last occurrence of x[k ]>=a" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Lookup:=proc(x,a)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "local n,k,ka;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "n:=vectdim(x);ka:=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for k from 1 to n do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "if \+ a>=x[k] then ka:=k fi;od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "RETURN (ka);end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 8 "Examples" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Lookup([1,2,3,4,5,6,5,8],5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Lookup([1,2,3,4,5,6,7,8],4.99);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 10 "Maximum( x)" }{TEXT 265 1 " " }{TEXT -1 81 "returns a vector [g[1],g[2]], g[2]: =max(x[k]:k=1..vectdim(x)), g[1]:=argmax(x[k])" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "Maximum:=proc(x) local i,g;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "g:=array(1..2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "g[1]:=1; g[2]:=x[1]; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "for i \+ from 2 to vectdim(x) do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "if x[i] >g[2] then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "g[1]:=i; g[2]:=x[i]; \+ fi" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "od; [g[1],g[2]];" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 10 "Minimum(x)" }{TEXT 274 1 " " }{TEXT -1 81 "returns a vector [g[1],g[2 ]], g[2]:=min(x[k]:k=1..vectdim(x)), g[1]:=argmin(x[k])" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Minimum:=proc(x) local i,g;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "g:=array(1..2);" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "g[1]:=1; g[2]:=x[1]; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "for i from 2 to vectdim(x) do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "if x[i] " 0 "" {MPLTEXT 1 0 23 "g[1]:=i; g[2]:=x[i]; fi" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "od; \+ [g[1],g[2]];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 273 8 "Examples" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Maximum([1,2,-3,3,4,-3,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"&\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Minimum([1,2,-3,3,4,-3,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$ \"\"$!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 268 14 "Stat istics(x) " }{TEXT -1 7 "prints " }{TEXT 270 11 "statistics " }{TEXT -1 28 "of the elements of a vector " }{TEXT 269 1 "x" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Statistics:=proc(x)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "local n,k,s,ss,w,fl;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "n:=vectdim(x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "s:=sum('x[k ]','k'=1..n)/n;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ss:=sum('x[k]^2' ,'k'=1..n)/n;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "printf(\"%A %G\\n \",'Average',evalf(s));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "printf( \"%A %G\\n\",'Std_dev',evalf(sqrt(ss-s^2)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "w:=convert(x,list):w:=sort(w);fl:=floor(n/2);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "printf(\"%A %G\\n\",'Median',w[fl]) ;printf(\"%a %g\\n\",'Maximum',w[n]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "printf(\"%A %G\\n\",'Minimum',w[1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 271 7 "Example" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Statistics(Rands(100));" }}{PARA 6 "" 1 "" {TEXT -1 15 "Average .493814" }}{PARA 6 "" 1 "" {TEXT -1 15 "Std_dev .299507" }}{PARA 6 "" 1 "" {TEXT -1 14 "Median .5 2543 " }}{PARA 6 "" 1 "" {TEXT -1 15 "Maximum .99299 " }}{PARA 6 "" 1 "" {TEXT -1 15 "Minimum .005863" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Check standard deviation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sqrt(int((x-0.5)^2,x=0..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"+Y8v')G!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 275 15 "Repetitions(v) \+ " }{TEXT -1 20 "gives the number of " }{TEXT 277 11 "repetitions" } {TEXT -1 13 " in a vector " }{TEXT 276 1 "v" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Repetitions:=proc(v)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "local n,k,r,w;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " n:=vectdim(v);r:=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "w:=convert(v ,list);w:=sort(w);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for k from 1 \+ to n-1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "if w[k+1]=w[k] then r: =r+1;fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "eval(r);od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 278 8 "Examples" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Repetitions([1 ,-3,2,3,-3,4,5,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Repetitions([dog,cat,dog,cat ,dog]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 279 24 "3. Simulation with Maple" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 284 15 "Simulation(x,p)" }{TEXT -1 58 " gene rates one observation from the discrete distribution " }{TEXT 298 5 "[ x,p]" }{TEXT -1 7 ", with " }{TEXT 299 15 "p[i]=prob(x[i])" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Simulation:=proc(x,p)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "local k,cp;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "cp:=Cum(p);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "k:=Lookup(cp,R and());" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "RETURN(x[k+1]);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 280 18 "Simulations(x,p,n)" }{TEXT -1 7 " gives " }{TEXT 308 1 " n" }{TEXT -1 45 " observations from the discrete distribution " } {TEXT 309 6 "[x,p]," }{TEXT -1 6 " with " }{TEXT 310 15 "p[i]=prob(x[i ])" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Simulations:=proc(x,p ,n)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "local k,v;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "v:=vector(n,[]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "for k from 1 to n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "do v[k ]:=Simulation(x,p);od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalm(v); end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 313 7 "Example" }{TEXT -1 6 ": C oin" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Simulation([H,T],[0. 5,0.5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"HG" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 32 "Simulations([H,T],[0.5,0.5],20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#76%\"TG%\"HGF(F'F'F'F'F'F'F'F(F( F'F(F'F(F'F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "v:=Simu lations([0,1],[0.5,0.5],100);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"v G-%'vectorG6#7`q\"\"!\"\"\"F)F*F)F)F*F*F)F)F*F)F*F*F*F)F)F*F)F*F)F)F)F )F)F*F)F)F*F)F)F)F*F*F*F*F*F)F)F)F*F)F*F*F*F)F*F)F)F)F*F*F)F)F)F*F*F)F *F*F*F*F*F)F)F)F*F*F*F)F)F)F)F)F)F*F*F)F)F*F*F)F*F*F)F*F*F*F*F*F*F*F)F )F)F*F*F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Statistics (v);" }}{PARA 6 "" 1 "" {TEXT -1 15 "Average .49 " }}{PARA 6 "" 1 " " {TEXT -1 15 "Std_dev .4999 " }}{PARA 6 "" 1 "" {TEXT -1 15 "Median \+ 0 " }}{PARA 6 "" 1 "" {TEXT -1 16 "Maximum 1 " }}{PARA 6 " " 1 "" {TEXT -1 16 "Minimum 0 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 314 7 "Example" }{TEXT -1 5 ": Die" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Simulation([1,2,3,4,5,6],[1/6,1/6,1/6,1/6,1/6,1/6]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 60 "v:=Simulations([1,2,3,4,5,6],[1/6,1/6,1/6,1/6,1/6,1 /6],100):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Statistics(v); " }}{PARA 6 "" 1 "" {TEXT -1 16 "Average 3.29 " }}{PARA 6 "" 1 "" {TEXT -1 16 "Std_dev 1.626622" }}{PARA 6 "" 1 "" {TEXT -1 15 "Median 3 " }}{PARA 6 "" 1 "" {TEXT -1 16 "Maximum 6 " }}{PARA 6 " " 1 "" {TEXT -1 16 "Minimum 1 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 288 24 "ContSimulation(f, x, L) " }{TEXT -1 72 "generates one observat ion of a random variable with probability density " }{TEXT 289 1 "f" } {TEXT -1 6 ", with" }}{PARA 0 "" 0 "" {TEXT -1 14 "left endpoint " } {TEXT 290 2 "L " }{TEXT -1 9 "(for the " }{TEXT 293 6 "normal" }{TEXT -1 9 " density " }{TEXT 291 13 "L = -infinity" }{TEXT -1 1 "," }{TEXT 292 1 " " }{TEXT -1 8 "for the " }{TEXT 294 12 "exponential " }{TEXT -1 7 "density" }{TEXT 295 4 " L=0" }{TEXT -1 1 ")" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 27 "ContSimulation:=proc(f,x,L)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "local t,F;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "F:=int(f,x=L..t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "randomize(see d);solve(F=Rand(),t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 286 8 "Examples" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f:=1:ContSimulation(f,x,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++g3:!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "f:=exp(-x^2/2)/sqrt(2*Pi):ContSimulation(f,x,-infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+!)ee$>\"!\"*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "f:=5*exp(-5*x):ContSimulation(f,x,0);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+*H&\\&3&!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 280 28 "ContSimulations(f, x, L, n) " }{TEXT -1 10 "generates " }{TEXT 300 1 "n" }{TEXT -1 60 " observations of a random variable with probability density " }{TEXT 281 1 "f" }{TEXT -1 6 ", with" }}{PARA 0 "" 0 "" {TEXT -1 14 "left endpoint " }{TEXT 282 1 "L" }{TEXT 296 1 " " }{TEXT -1 9 "(for the " }{TEXT 285 6 "normal" }{TEXT -1 9 " density " }{TEXT 283 13 "L = -infinity" }{TEXT -1 1 "," }{TEXT 284 1 " " } {TEXT -1 27 "for the exponential density" }{TEXT 287 4 " L=0" }{TEXT -1 1 ")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ContSimulations: =proc(f,x,L,n)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "local k,v;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "v:=vector(n,[]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "for k from 1 to n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "do v[k]:=ContSimulation(f,x,L);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalm(v);convert(v,list );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 297 8 "Examples" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " f:=1:ContSimulations(f,x,0,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'$ \"+++SE!)!#5$\"+++q&y&F&$\"+++g9 " 0 "" {MPLTEXT 1 0 64 "f:=exp(-x^2/2)/sqrt(2*Pi):L:=-infin ity:ContSimulations(f,x,L,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'$!+ `+qs()!#5$\"+vR1OJF&$\"+H&HJ[\"!\"*$!+4E(p$\\F&$!+kTw77F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "f:=exp(-x^2/2)/sqrt(2*Pi):ContSimul ations(f,x,-infinity,100):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Statistics(%);" }}{PARA 6 "" 1 "" {TEXT -1 16 "Average -.060445" } }{PARA 6 "" 1 "" {TEXT -1 16 "Std_dev 1.028469" }}{PARA 6 "" 1 "" {TEXT -1 15 "Median -.097738" }}{PARA 6 "" 1 "" {TEXT -1 16 "Maximum 2 .559576" }}{PARA 6 "" 1 "" {TEXT -1 17 "Minimum -2.752523" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "f:=5*exp(-5*x):L:=0:ContSimulations (f,x,L,100):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Statistics( %);" }}{PARA 6 "" 1 "" {TEXT -1 15 "Average .190393" }}{PARA 6 "" 1 " " {TEXT -1 15 "Std_dev .194157" }}{PARA 6 "" 1 "" {TEXT -1 14 "Median \+ .139829" }}{PARA 6 "" 1 "" {TEXT -1 15 "Maximum .914928" }}{PARA 6 "" 1 "" {TEXT -1 15 "Minimum .002015" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f:=1:v:=ContSimulations(f,x,0,100):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Statistics(%);" }}{PARA 6 "" 1 "" {TEXT -1 15 "Average .509755" }}{PARA 6 "" 1 "" {TEXT -1 15 "Std_dev .296059 " }}{PARA 6 "" 1 "" {TEXT -1 14 "Median .49499 " }}{PARA 6 "" 1 "" {TEXT -1 15 "Maximum .9997 " }}{PARA 6 "" 1 "" {TEXT -1 15 "Minimum . 009105" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 301 30 "DiscreteSimulation(f, \+ x, L, U)" }{TEXT 315 1 " " }{TEXT 302 0 "" }{TEXT -1 55 "generates one observation from a discrete distribution " }{TEXT 303 4 "f(x)" }} {PARA 0 "" 0 "" {TEXT -1 12 "for integer " }{TEXT 306 1 "x" }{TEXT -1 5 " from" }{TEXT 304 3 " L " }{TEXT -1 3 "to " }{TEXT 305 1 "U" } {TEXT -1 2 ". " }{TEXT 307 1 "U" }{TEXT -1 38 " must be sufficiently l arge (possibly " }{TEXT 280 8 "infinity" }{TEXT -1 2 ")." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "DiscreteSimulation:=proc(f,x,L,U)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "local t,F,m;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "randomize(seed);t:=Rand();F:=0;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 45 "for m from L to U do F:=evalf(F+subs(x=m,f));" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "if F>t then RETURN(m);BREAK;fi;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 280 35 "DiscreteSimulations (f, x, L, U, n) " }{TEXT -1 54 "generates n observations from a discre te distribution " }{TEXT 282 4 "f(x)" }}{PARA 0 "" 0 "" {TEXT -1 12 "f or integer " }{TEXT 285 1 "x" }{TEXT -1 5 " from" }{TEXT 283 3 " L " } {TEXT -1 3 "to " }{TEXT 284 1 "U" }{TEXT -1 1 "." }{TEXT 312 2 " U" } {TEXT -1 38 " must be sufficiently large (possibly " }{TEXT 280 8 "inf inity" }{TEXT -1 2 ")." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "D iscreteSimulations:=proc(f,x,L,U,n)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "local v,k;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "v:=vector(n,[]); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for k from 1 to n do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "v[k]:=DiscreteSimulation(f,x,L,U);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalm(v);convert(v,list);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "e nd:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 311 8 "Examples" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "DiscreteSimulations((2.3^x)*exp(-2.3)/x!, x,0,infinity,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#76\"\"#\"\"$F$F%F %F$\"\"\"F$\"\"&F$F%F%F%F$F%F'F$F&F%F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Statistics(%);" }}{PARA 6 "" 1 "" {TEXT -1 16 "Averag e 2.6 " }}{PARA 6 "" 1 "" {TEXT -1 16 "Std_dev 1.019804" }}{PARA 6 "" 1 "" {TEXT -1 15 "Median 2 " }}{PARA 6 "" 1 "" {TEXT -1 16 "Maximum 5 " }}{PARA 6 "" 1 "" {TEXT -1 16 "Minimum 1 " }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "DiscreteSimulations((0.5)^( x-1)*0.5,x,1,100,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#76\"\"\"\"\"# \"\"%F$F$F%F%\"\"$F%F'F$F&F$F$F$F%F$\"\"'F'F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "DiscreteSimulations((0.5)^(x-1)*0.5,x,1,100,100) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Statistics(%);" }} {PARA 6 "" 1 "" {TEXT -1 16 "Average 2.02 " }}{PARA 6 "" 1 "" {TEXT -1 16 "Std_dev 1.529575" }}{PARA 6 "" 1 "" {TEXT -1 15 "Median 1 " }}{PARA 6 "" 1 "" {TEXT -1 17 "Maximum 10 " }}{PARA 6 " " 1 "" {TEXT -1 16 "Minimum 1 " }}}{EXCHG {PARA 261 "" 0 "" {TEXT 316 23 "4. The birthday problem" }}{PARA 0 "" 0 "" {TEXT -1 96 " Find smallest n such that the probability of a common birthday in a ra ndom sample of n is > 1/2" }}{PARA 0 "" 0 "" {TEXT -1 91 "It is easie r to compute the probability of the complementary event, i.e. no commo n birthday" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 25 "ProbCommonBirthday (n, N) " }{TEXT -1 55 "is the probability of a common birthday in a gr oup of n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ProbCommonBirth day:=proc(n,N)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "local k;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "evalf(1-product((N-k+1)/N,k=1..n));" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 320 8 "Examples" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Pr obCommonBirthday(23,365);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+VB(H2 &!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "What if there are 10000 d ays in a year?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "v:=vector (150,[]):for k from 1 to 150 do v[k]:=ProbCommonBirthday(k,10000);od: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "1+Lookup(v,0.5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"$>\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 318 28 "ProbDifferentBirthday(n, N) " }{TEXT -1 23 "is computed \+ recursively" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ProbDifferen tBirthday:=proc(n,N)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "if n=1 then 1 else" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "evalf(ProbDifferentBirth day(n-1,N)*(N-n+1)/N);fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 319 8 "Examples" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "1-ProbDifferentBirthday(23,365);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+PB(H2&!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "v:=vector(150,[]):for k from 1 to 150 do v[k]:=1-Prob DifferentBirthday(k,10000);od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "1+Lookup(v,0.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$>\"" }} }{EXCHG {PARA 0 "" 0 "" {TEXT 322 16 "Birthdays(n, N) " }{TEXT -1 10 " generates " }{TEXT 323 2 "n " }{TEXT -1 25 "birthdays in a year with \+ " }{TEXT 324 2 "N " }{TEXT -1 4 "days" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Birthdays:=proc(n,N)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "local v,k;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "v:=vector(n,[]) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for k from 1 to n do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "v[k]:=floor(Rand()*N);od;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "evalm(v);end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 321 7 "Example" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Bir thdays(10,365);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7,\"$- \"\"#G\"$a\"\"$g$\"$I#\"$)>\"$K\"\"$(G\"$a#\"#\"*" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 24 "convert(%,list):sort(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7,\"#G\"#\"*\"$-\"\"$K\"\"$a\"\"$)>\"$I#\"$a#\"$(G\"$ g$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Birthdays(23,365);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#79\"$L\"\"$j$\"$^\"\"#)) \"$l#F+\"$p#\"$)H\"$]\"\"$i#\"$r\"\"\"$\"$k\"\"#l\"$-#\"$6\"\"#P\"$R$ \"#p\"$;#\"$r#\"$U$\"#O" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " Repetitions(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 0 "" }{TEXT 326 28 "5. The St Petersburg Pa radox" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Two players," }{TEXT 328 3 " A " }{TEXT -1 3 "and" }{TEXT 329 2 " B" }{TEXT -1 30 ". A fair coin is tossed until " }{TEXT 327 2 "H " }{TEXT -1 8 "appears." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Simulations([H,T],[1/2,1/2], 20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#76%\"HG%\"TGF%F$F%F$F$F%F%F%F$ F%F%F%F$F%F$F%F$F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "If this hap pens on trial " }{TEXT 330 2 "n " }{TEXT -1 5 "then " }{TEXT 331 2 "B \+ " }{TEXT -1 5 "pays " }{TEXT 332 1 "A" }{TEXT -1 12 " the amount " } {TEXT 333 7 "2^(n-1)" }{TEXT -1 14 ". What is the " }{TEXT 336 5 "valu e" }{TEXT -1 4 " of " }}{PARA 0 "" 0 "" {TEXT -1 13 "the game for " } {TEXT 335 1 "A" }{TEXT -1 22 "? In other words, if " }{TEXT 337 2 "B \+ " }{TEXT -1 45 "were to sell tickets for this game, how much " }} {PARA 0 "" 0 "" {TEXT -1 7 "should " }{TEXT 334 1 "A" }{TEXT -1 25 " b e willing to pay for a " }{TEXT 338 6 "ticket" }{TEXT -1 26 " ? A reas onable answer is " }{TEXT 345 14 "expected value" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 95 "The probability of first H in trial n is \+ (1/2)^n, therefore the expected value of A's income is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "evalf(sum('2^(k-1)*(1/2)^k','k'=1.. infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "hence the paradox. " }{TEXT 339 17 "Danie l Bernoulli " }{TEXT -1 18 "(1738) proposed a " }{TEXT 340 29 "logarit hmic utility, C log(w)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 39 "Accordingly, the value of the game for " }{TEXT 343 2 "A " }{TEXT -1 7 "is his " }{TEXT 344 16 "expected utility" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "evalf(sum('C*log(2^(k-1))*(1/2)^k','k'=1..infinity ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"CG$\"+1=ZJp!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 341 14 "Petersburg(n) " }{TEXT -1 23 "simulates \+ the game for " }{TEXT 342 1 "n" }{TEXT -1 49 " trials (a truncated gam e) and returns A's income" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Petersburg:=proc(n)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "local v, k,x,p,temp;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "x:=[H,T]:p:=[1/2,1/2 ]:v:=array(1..n):temp:=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for k \+ from 1 to n do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "v[k]:=Simulation( x,p);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "if v[k]=H then break;fi;od ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eval(2^(k-1));" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Petersburg(20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "v:=array(1..1000):for k from 1 to 1000 do v[k]:=Petersburg(20);od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Statistics(v);" }}{PARA 6 "" 1 "" {TEXT -1 17 "Averag e 16.581 " }}{PARA 6 "" 1 "" {TEXT -1 18 "Std_dev 174.642393" }} {PARA 6 "" 1 "" {TEXT -1 15 "Median 2 " }}{PARA 6 "" 1 "" {TEXT -1 19 "Maximum 4096 " }}{PARA 6 "" 1 "" {TEXT -1 16 "Minimum 1 \+ " }}}{EXCHG {PARA 264 "" 0 "" {TEXT 346 25 "6. The Three Doors Pu zzle" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 347 14 "ThreeDoors(n) " }{TEXT -1 69 "simulates the probability of winning for the strategy of no-swi tching" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ThreeDoors:=proc( n)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "local k,p,q,d,w;w:=0;#w = num ber of wins in n trials" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for k fr om 1 to n do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "p:=Simulation([1,2, 3],[1/3,1/3,1/3]);#door with prize" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "q:=Simulation([1,2,3],[1/3,1/3,1/3]);#door chosen by cotestant" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "d:=p+Simulation([1,2],[1/2,1/2]) mo d 3;#empty door openned" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "if p=q t hen w:=w+1 fi;od; #contestant stays with original choice" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "printf(\"%A %G\\n\",'Frequency_of_wins',evalf (w/n));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 348 7 "Example" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ThreeDoors(100);" }}{PARA 6 "" 1 "" {TEXT -1 25 "Frequency_of_wins .33 " }}}{EXCHG {PARA 265 "" 0 "" {TEXT 350 21 "7. The Buffon Neee dle" }}{PARA 0 "" 0 "" {TEXT -1 79 "A needle of length L is dropped in to a plane marked by parallel lines, D apart." }}{PARA 0 "" 0 "" {TEXT -1 55 "The probability that the needle will cross any line is " }{TEXT 349 8 "2*L/Pi*D" }}{PARA 0 "" 0 "" {TEXT -1 82 "The following f unction simulates n needles, and records the frequency of crossings" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Buffon:=proc(L,D,n)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "local k,c,x;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "c:=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for k from 1 to n do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "x:=evalf(D*Rand()+L*c os(2*Pi*Rand()));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "if (x>D or x<0 ) then c:=c+1 fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "lprint('Frequency_of_crossings');" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "lprint('Simulated_value',evalf(c/n));" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "lprint('Theoretical_value',evalf(2* L/(Pi*D)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 356 7 "Example" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Buffon(1,1,100);" }}{PARA 6 "" 1 "" {TEXT -1 22 "Freq uency_of_crossings" }}{PARA 6 "" 1 "" {TEXT -1 29 "Simulated_value . 5800000000" }}{PARA 6 "" 1 "" {TEXT -1 31 "Theoretical_value .636619 7722" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 351 11 "Buffons(n) " }{TEXT -1 13 "displays the " }{TEXT 352 26 "theoretical probabilities " }{TEXT -1 25 "(thick red line) and the " }{TEXT 353 21 "simulated frequencies " }}{PARA 0 "" 0 "" {TEXT -1 3 "of " }{TEXT 354 13 "Buffon(L,1,n)" } {TEXT -1 5 " for " }{TEXT 355 13 "L=0.1(0.1)1.2" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 16 "Buffons:=proc(n)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "local m,L,k,c,x,V,W;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "V:=array(1..24);W:=array(1..24);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for m from 1 to 24 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "L: =evalf(m/20);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "W[m]:=[L,evalf(2*L /Pi)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "c:=0;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "for k from 1 to n do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "x:=evalf(Rand()+L*cos(2*Pi*Rand()));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "if (x>1 or x<0) then c:=c+1 fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "V[m]:=[L,eva lf(c/n)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "V:=pointplot(V,color=black,connect=true):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "W:=pointplot(W,color=red,connect=true,thi ckness=3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "display(V,W);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Buffons(100);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6$-%'CURVESG6$7:7$$\"+++++]!#6$\"+++++gF*7$$\"+++++5! #5$\"+++++qF*7$$\"+++++:F0F.7$$\"+++++?F0F.7$$\"+++++DF0$\"+++++;F07$$ \"+++++IF0$\"+++++EF07$$\"+++++NF0$\"+++++GF07$$\"+++++SF0F:7$$\"+++++ XF0F?7$$F)F0$\"+++++KF07$$\"+++++bF0$\"+++++HF07$$F,F0$\"+++++VF07$$\" +++++lF0$\"+++++QF07$$F2F0$\"+++++`F07$$\"+++++vF0$\"+++++WF07$$\"++++ +!)F0FI7$$\"+++++&)F0$\"+++++_F07$$\"+++++!*F0Fio7$$\"+++++&*F0$\"++++ +kF07$$\"\"\"\"\"!FX7$$\"++++]5!\"*$\"+++++hF07$$\"+++++6FjpFap7$$\"++ ++]6Fjp$\"+++++nF07$$\"+++++7FjpFfn-%'COLOURG6&%$RGBGFfpFfpFfp-F$6%7:7 $F($\"+h))4$=$F*7$F.$\"+Ax>mjF*7$F4$\"+$e'H\\&*F*7$F7$\"+W&RKF\"F07$F: $\"+J%\\:f\"F07$F?$\"+<$f)4>F07$FD$\"+.#p\"GAF07$FI$\"+*3zka#F07$FL$\" +v*)ykGF07$FO$FarF07$FS$\"+Z(39]$F07$FX$\"+L'=(>QF07$Ffn$\"+>&G!QTF07$ F[o$\"+0%QjX%F07$F_o$\"+#H[Yx%F07$Fdo$\"+y\"eH4&F07$Fgo$\"+k!o7T&F07$F \\p$\"+]zdHdF07$F_p$\"+Oy)y/'F07$Fdp$FdrF07$Fhp$\"+3w]%o'F07$F^q$\"+% \\ " 0 "" {MPLTEXT 1 0 28 "DailyProfit:=proc(x,d,c,r,s)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "evalf(r*min(x,d)+s*max(x-d,0)-c*x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 378 7 "Exam ple" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "DailyProfit(9,6,.2,. 4,.1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"*!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 357 34 "DailyProfits(x, d, p, c, r, s, n) " } {TEXT -1 40 "simulates daily profits for a period of " }{TEXT 358 2 "n " }{TEXT -1 5 "days," }}{PARA 0 "" 0 "" {TEXT -1 8 "stock = " }{TEXT 359 1 "x" }{TEXT -1 11 ", demand = " }{TEXT 360 1 "d" }{TEXT -1 39 " ( vector of values) with probabilities " }{TEXT 361 1 "p" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "DailyProfits:=proc(x,d,p,c,r,s,n)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "local k,temp,dapr;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "dapr:=array(1..n);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for k from 1 to n do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "temp:=Simulation(d,p);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "dap r[k]:=DailyProfit(x,temp,c,r,s);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalm(dapr);od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 379 7 "Example" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "d:=[6,7,8,9,10,11,12,13,14]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "p:=[0.05,0.1,0.15,0.2,0.25,0.15,0.05,0.03, 0.02] : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "DailyProfits(10,d,p,0 .2,0.4,0.1,100);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7`q$\" # " 0 "" {MPLTEXT 1 0 14 "Statist ics(%);" }}{PARA 6 "" 1 "" {TEXT -1 16 "Average 1.7 " }}{PARA 6 " " 1 "" {TEXT -1 15 "Std_dev .367423" }}{PARA 6 "" 1 "" {TEXT -1 15 "Me dian 1.7 " }}{PARA 6 "" 1 "" {TEXT -1 16 "Maximum 2.0 " }} {PARA 6 "" 1 "" {TEXT -1 15 "Minimum .8 " }}}{EXCHG {PARA 0 "" 0 " " {TEXT 372 40 "AverageDailyProfit(x, d, p, c, r, s, n) " }{TEXT -1 50 "computes the average daily profit for a period of " }{TEXT 373 2 " n " }{TEXT -1 5 "days," }}{PARA 0 "" 0 "" {TEXT -1 8 "stock = " } {TEXT 374 1 "x" }{TEXT -1 11 ", demand = " }{TEXT 375 1 "d" }{TEXT -1 39 " (vector of values) with probabilities " }{TEXT 376 1 "p" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "AverageDailyProfit:=proc(x,d ,p,c,r,s,n)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "local k,temp,cump;cu mp:=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for k from 1 to n do" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "temp:=DailyProfit(x,Simulation(d,p) ,c,r,s);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "cump:=cump+temp;od:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(cump/n);" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 377 7 "Example " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "d:=[6,7,8,9,10,11,12,13 ,14]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "p:=[0.05,0.1,0.15, 0.2,0.25,0.15,0.05,0.03, 0.02]: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "AverageDailyProfit(7,d,p,0.2,0.4,0.1,100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++z8!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "for k from 6 to 14 do lprint('stock',k,'average_daily _profit'," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "AverageDailyProfit(k,d ,p,0.2,0.4,0.1,100));od:" }}{PARA 6 "" 1 "" {TEXT -1 46 "stock 6 a verage_daily_profit 1.200000000" }}{PARA 6 "" 1 "" {TEXT -1 46 "stoc k 7 average_daily_profit 1.376000000" }}{PARA 6 "" 1 "" {TEXT -1 46 "stock 8 average_daily_profit 1.552000000" }}{PARA 6 "" 1 "" {TEXT -1 46 "stock 9 average_daily_profit 1.653000000" }} {PARA 6 "" 1 "" {TEXT -1 47 "stock 10 average_daily_profit 1.739 000000" }}{PARA 6 "" 1 "" {TEXT -1 47 "stock 11 average_daily_prof it 1.666000000" }}{PARA 6 "" 1 "" {TEXT -1 47 "stock 12 average_ daily_profit 1.503000000" }}{PARA 6 "" 1 "" {TEXT -1 47 "stock 13 \+ average_daily_profit 1.523000000" }}{PARA 6 "" 1 "" {TEXT -1 47 "s tock 14 average_daily_profit 1.420000000" }}}{EXCHG {PARA 267 " " 0 "" {TEXT 380 24 "9. An Investment Problem" }}{PARA 0 "" 0 "" {TEXT -1 5 "Your " }{TEXT 381 6 "wealth" }{TEXT -1 14 " initially is \+ " }{TEXT 382 2 "w0" }{TEXT -1 22 ". In any year you can " }{TEXT 390 7 "invest " }{TEXT -1 29 "any part of your wealth in a " }{TEXT 389 7 "project" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 42 "and deposit t he rest in a bank. The bank's" }{TEXT 383 21 " annual interest rate" } {TEXT -1 4 " is " }{TEXT 384 2 "i%" }{TEXT -1 6 ". The " }{TEXT 387 14 "rate of return" }{TEXT -1 8 " of the " }}{PARA 0 "" 0 "" {TEXT 388 18 "investment project" }{TEXT -1 25 " is random, assuming the " } {TEXT 385 8 "values r" }{TEXT -1 6 " with " }{TEXT 386 15 "probabiliti es p" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 29 "Your strategy is \+ to invest a " }{TEXT 391 11 "fraction f " }{TEXT -1 55 "of your wealth at any year. The wealth after n years is" }}{PARA 0 "" 0 "" {TEXT -1 12 "simulated by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Wealth: =proc(f,i,r,p,w0,n)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "local k,v;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "v:=array(0..n);v[0]:=w0;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for k from 1 to n do" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 60 "v[k]:=evalf(f*v[k-1]*Simulation(r,p)+(1-f)*(1+ i/100)*v[k-1])" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "RETURN(v[n]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 395 7 "Example" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "r:=[0,1,2]:p:=[0.3,0.2,0.5]:Wealth( 0.5,8,r,p,1000,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++k!\\%!\"( " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 392 25 "Policies(i, r, p, w0, n) " } {TEXT -1 23 "simulates 11 values of " }{TEXT 393 26 "Wealth(f, i, r, p , w0, n) " }{TEXT -1 4 "for " }{TEXT 394 14 "f=0,0.1,..,1.0" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Policies:=proc(i,r,p,w0,n)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "local f,m;lprint('Return_after',n ,'years');" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "for m from 0 to 10 do f:=evalf(m/10);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "printf(\"%A %6G %-6s %12G\\n\",'Fraction_invested',f,'Return',evalf(Wealth(f,i,r,p,w0 ,n),9));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "r :=[0,1,2]:p:=[0.3,0.2,0.5]:Policies(5,r,p,1000,10);" }}{PARA 6 "" 1 " " {TEXT -1 25 "Return_after 10 years" }}{PARA 6 "" 1 "" {TEXT -1 46 "Fraction_invested 0 Return 1628.89462 " }}{PARA 6 "" 1 "" {TEXT -1 45 "Fraction_invested .1 Return 2177.37801 " }}{PARA 6 "" 1 "" {TEXT -1 45 "Fraction_invested .2 Return 696.58499 " }} {PARA 6 "" 1 "" {TEXT -1 45 "Fraction_invested .3 Return 1084.09 468 " }}{PARA 6 "" 1 "" {TEXT -1 45 "Fraction_invested .4 Return \+ 5886.65316 " }}{PARA 6 "" 1 "" {TEXT -1 45 "Fraction_invested .5 \+ Return 221.105608" }}{PARA 6 "" 1 "" {TEXT -1 45 "Fraction_invested .6 Return 541.512446" }}{PARA 6 "" 1 "" {TEXT -1 45 "Fraction_ invested .7 Return 477.732039" }}{PARA 6 "" 1 "" {TEXT -1 45 "F raction_invested .8 Return 38.158511" }}{PARA 6 "" 1 "" {TEXT -1 45 "Fraction_invested .9 Return 18302.7018 " }}{PARA 6 "" 1 " " {TEXT -1 46 "Fraction_invested 1 Return 0 " }}} {EXCHG {PARA 268 "" 0 "" {TEXT 400 20 "10. Queueing Systems" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 396 23 "ArrivalTimes(lambda,T) " }{TEXT -1 48 "simulates the inter-arrival times, with density " }{TEXT 404 1 "f" }{TEXT -1 9 ", during " }{TEXT 397 5 "[0,T]" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "ArrivalTimes:=proc(f,t,L,T)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "local a,v,n;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "a:=0;v:=vector(1,[0]);t:='t';" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "w hile a " 0 "" {MPLTEXT 1 0 34 "a:=evalf(a+ContSimul ation(f,t,L));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "if a " 0 "" {MPLTEXT 1 0 12 "evalm( v);od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 398 8 "Examples" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f:=5*exp(-5*t):ArrivalTimes(f,t,0,4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7#71\"\"!$\"+ZK3&\\#!#5$\"+CU\"o`(F+$\"+/&> s$))F+$\"+K6,*\\*F+$\"+Pp!\\N\"!\"*$\"+Fr,j:F4$\"+w@2(y\"F4$\"+>&o[$>F 4$\"+)=)[)4#F4$\"+gkz-@F4$\"+L'4O'GF4$\"+?K?uHF4$\"+!H[I4$F4$\"+C\\FJN F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f:=1:ArrivalTimes(f,t ,0,10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7#72\"\"!$\"+++ !e9)!#5$\"+++zL6!\"*$\"++!oh<\"F.$\"++!=kl\"F.$\"++!3e0#F.$\"++!yC#GF. $\"++!)RPMF.$\"++![5'RF.$\"++!)RT\\F.$\"+++iC]F.$\"+++&4#eF.$\"+++NMlF .$\"+++$>\\(F.$\"+++hX#)F.$\"+++8]\"*F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 401 20 "NextBus(f, t, L, T) " }{TEXT -1 58 "simulates the waitin g time of a person arriving at random " }}{PARA 0 "" 0 "" {TEXT -1 81 "in the bus station, if the bus inter-arrivals times are distributed w ith density " }{TEXT 405 1 "f" }}{PARA 0 "" 0 "" {TEXT -1 4 "and " } {TEXT 403 1 "T" }{TEXT -1 15 " is the horizon" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "NextBus:=proc(f,t,L,T)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "local v,k,s;t:='t';" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "v:=ArrivalTimes(f,t,L,T);v:=convert(v,vector);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "s:=evalf((T/2)*Rand());" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "k:=Lookup(v,s);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalf(v[k+1]-s);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "LastBus:=proc(f,t,L,T)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "local v,k,s;t:='t';" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "v:=ArrivalTimes(f,t,L,T);v:=convert(v,vector); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s:=evalf(T*Rand());" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "k:=Lookup(v,s);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(s-v[k]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "e nd:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 406 9 "Examples:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f:=2*exp(-2*t):NextBus(f,t,0,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*jBQ\"))!\"*" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 33 "f:=2*exp(-2*t):LastBus(f,t,0,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*5z(o@!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 407 22 "The Inspection Paradox" }}{PARA 0 "" 0 "" {TEXT -1 8 "Th e sum " }{TEXT 410 21 "average waiting time " }{TEXT -1 2 "+ " }{TEXT 411 18 "average spent time" }{TEXT -1 21 " is greater than the " } {TEXT 412 26 "average inter-arrival time" }}{PARA 0 "" 0 "" {TEXT -1 23 "We illustrate this for " }{TEXT 408 11 "exponential" }{TEXT -1 5 " and " }{TEXT 409 7 "uniform" }{TEXT -1 9 " arrivals" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f:=2*exp(-2*t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "n:=20:Next:=array(1..n):for m from 1 to n do Nex t[m]:=NextBus(f,t,0,5);od:evalm(Next):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "n:=20:Last:=array(1..n):for m from 1 to n do Last[m]: =LastBus(f,t,0,5);od:evalm(Last):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Statistics(Next);" }}{PARA 6 "" 1 "" {TEXT -1 15 "Ave rage .445881" }}{PARA 6 "" 1 "" {TEXT -1 15 "Std_dev .687783" }}{PARA 6 "" 1 "" {TEXT -1 14 "Median .238755" }}{PARA 6 "" 1 "" {TEXT -1 16 " Maximum 3.210872" }}{PARA 6 "" 1 "" {TEXT -1 15 "Minimum .002033" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Statistics(Last);" }}{PARA 6 "" 1 "" {TEXT -1 15 "Average .496974" }}{PARA 6 "" 1 "" {TEXT -1 15 "Std_dev .49451 " }}{PARA 6 "" 1 "" {TEXT -1 14 "Median .27702 " }} {PARA 6 "" 1 "" {TEXT -1 16 "Maximum 1.8105 " }}{PARA 6 "" 1 "" {TEXT -1 15 "Minimum .000627" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "n:=20:Next:=a rray(1..n):for m from 1 to n do Next[m]:=NextBus(f,t,0,5); od:evalm(Ne xt):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "n:=20:Last:=array(1 ..n):for m from 1 to n do Last[m]:=LastBus(f,t,0,5); od:evalm(Last):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Statistics(Next);" }} {PARA 6 "" 1 "" {TEXT -1 15 "Average .361926" }}{PARA 6 "" 1 "" {TEXT -1 15 "Std_dev .158856" }}{PARA 6 "" 1 "" {TEXT -1 14 "Median .350185 " }}{PARA 6 "" 1 "" {TEXT -1 15 "Maximum .649225" }}{PARA 6 "" 1 "" {TEXT -1 15 "Minimum .045585" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Statistics(Last);" }}{PARA 6 "" 1 "" {TEXT -1 15 "Average .47733 2" }}{PARA 6 "" 1 "" {TEXT -1 15 "Std_dev .27861 " }}{PARA 6 "" 1 "" {TEXT -1 14 "Median .44598 " }}{PARA 6 "" 1 "" {TEXT -1 15 "Maximum .9 5741 " }}{PARA 6 "" 1 "" {TEXT -1 15 "Minimum .030015" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 399 21 "Queue(lambda, mu, T) " }{TEXT -1 13 "sim ulates an " }{TEXT 402 6 "M/M/1 " }{TEXT -1 46 "queue (1 server, Poiss on arrivals and service)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Queue:=proc(lambda,mu,T)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "local \+ a,b,c,f,g,k,w,cumw,cumb,maxw,server;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "f:=lambda*exp(-lambda*t);g:=mu*exp(-mu*t):#arrival & service den sities" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "a:=0;c:=0;w:=0;cumw:=0;ma xw:=0;cumb:=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "k:=1;" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 44 "printf(\"%-6s %6s %6s %12s %8s\\n\",'Custome r'," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "'Arrival','Waiting','Service ','Server');" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "printf(\"%-6s %6s % 9s %9s %12s\\n\",'No','Time','Time','End','Status');" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "while a " 0 "" {MPLTEXT 1 0 51 "a:=evalf(a+ContSimulation(f,t,0),5);#a=arrival time" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "if c \+ " 0 "" {MPLTEXT 1 0 52 "else w:=evalf(c-a);server:='busy' fi;#w=waitin g time" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "cumw:=evalf(cumw+w,5);max w:=max(maxw,w);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "c:=max(a,c);#c=t ime service can start,b=service duration" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "b:=ContSimulation(g,t,0);cumb:=cumb+b;c:=evalf(c+b,5) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "printf(\"%3g %9G %9G %9G %6s\\ n\",k,a,w,c,server);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "k:=k+1;od; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "lprint();" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "lprint(Server_efficiency);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "printf(\"%a %g %a %g\\n\",'Theoretical',evalf(lambda/ mu)," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "'simulated',evalf(cumb/T,5) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "lprint('Average_waiting_time' );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "printf(\"%a %0g %a %g\\n\",'T heoretical',evalf(lambda/(mu*(mu-lambda)))," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "'simulated',evalf(cumw/k));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "lprint('Max_waiting_time',evalf(maxw));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 14 "Queue(2,5,80);" }}{PARA 6 "" 1 "" {TEXT -1 46 "Custo mer Arrival Waiting Service Server" }}{PARA 6 "" 1 "" {TEXT -1 46 "No Time Time End Status" }}{PARA 6 "" 1 "" {TEXT -1 45 "1 .080525 0 .30488 free" }}{PARA 6 "" 1 "" {TEXT -1 45 "2 2.0234 0 2.0932 free" }} {PARA 6 "" 1 "" {TEXT -1 45 "3 2.0810 .0122 2.1053 \+ busy" }}{PARA 6 "" 1 "" {TEXT -1 45 "4 2.1312 0 2.1 785 free" }}{PARA 6 "" 1 "" {TEXT -1 45 "5 2.4489 0 \+ 2.5789 free" }}{PARA 6 "" 1 "" {TEXT -1 45 "6 3.1283 \+ 0 3.1942 free" }}{PARA 6 "" 1 "" {TEXT -1 45 "7 \+ 3.6150 0 3.7131 free" }}{PARA 6 "" 1 "" {TEXT 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{TEXT -1 47 "1 26 72.441 0 72.606 free" }}{PARA 6 "" 1 "" {TEXT -1 47 "127 72.556 .050 72.787 busy" }}{PARA 6 "" 1 "" {TEXT -1 47 "128 72.906 0 72.934 free " }}{PARA 6 "" 1 "" {TEXT -1 47 "129 73.224 0 73.294 free" }}{PARA 6 "" 1 "" {TEXT -1 47 "130 74.029 0 \+ 74.217 free" }}{PARA 6 "" 1 "" {TEXT -1 47 "131 74.160 .057 74.358 busy" }}{PARA 6 "" 1 "" {TEXT -1 47 "132 \+ 74.381 0 74.705 free" }}{PARA 6 "" 1 "" {TEXT -1 47 "133 75.151 0 75.654 free" }}{PARA 6 "" 1 " " {TEXT -1 47 "134 75.305 .349 75.891 busy" }} {PARA 6 "" 1 "" {TEXT -1 47 "135 75.922 0 75.962 \+ free" }}{PARA 6 "" 1 "" {TEXT -1 47 "136 76.121 0 \+ 76.185 free" }}{PARA 6 "" 1 "" {TEXT -1 47 "137 76.699 \+ 0 76.730 free" }}{PARA 6 "" 1 "" {TEXT -1 47 "138 \+ 77.308 0 77.723 free" }}{PARA 6 "" 1 "" {TEXT -1 47 "1 39 77.523 .200 78.097 busy" }}{PARA 6 "" 1 "" {TEXT -1 47 "140 77.564 .533 78.150 busy" }}{PARA 6 "" 1 "" {TEXT -1 47 "141 77.613 .537 78.232 busy " }}{PARA 6 "" 1 "" {TEXT -1 47 "142 77.700 .532 78.524 busy" }}{PARA 6 "" 1 "" {TEXT -1 47 "143 78.767 0 \+ 78.800 free" }}{PARA 6 "" 1 "" {TEXT -1 47 "144 78.876 0 79.205 free" }}{PARA 6 "" 1 "" {TEXT -1 47 "145 \+ 79.096 .109 79.602 busy" }}{PARA 6 "" 1 "" {TEXT -1 47 "146 79.166 .436 79.891 busy" }}{PARA 6 "" 1 " " {TEXT -1 47 "147 79.445 .446 80.074 busy" }} {PARA 6 "" 1 "" {TEXT -1 47 "148 80.278 0 81.119 \+ free" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 17 "Server_efficiency" }}{PARA 6 "" 1 "" {TEXT -1 37 "Theoretical .4 \+ simulated .41711 " }}{PARA 6 "" 1 "" {TEXT -1 20 "Average_waiting_tim e" }}{PARA 6 "" 1 "" {TEXT -1 37 "Theoretical .133333 simulated .14108 1" }}{PARA 6 "" 1 "" {TEXT -1 24 "Max_waiting_time 1.251" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}}{MARK "55 0 0" 15 }{VIEWOPTS 1 1 0 1 1 1803 }