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What happe ns as " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 11 " increases? 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W hat happens to the CDF as theta increases from 0.1 to 2.1?" }}{PARA 0 "" 0 "" {TEXT -1 115 "Note that the plot range is the same as above, x =0..4. 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Qn7g(F97$F_\\l$\"1CHY*o\"\\&p(F97$Fd\\l$\"1:X!e(Q.%y(F97$Fi\\l$\"1W@8S l3tyF97$F^]l$\"1:+ca*pa&zF97$Fc]l$\"1ArtS'>k.)F97$Fh]l$\"1LCm2L`8\")F9 7$F[^l$\"1<=)F97$F`^l$\"1MzcAQ%oD)F97$Fc^l$\"1nkn;@S@$)F97$Fh^ l$\"1E-foKg(Q)F97$F[_l$\"1bs.i%>&[%)F97$F^_l$\"1wL-#>>9^)F9F`_lFe`lF[a l-%&TITLEG6#Q9Increasing~theta~for~CDFFi`l" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "Sample Probability Calculations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We can calculate probabilities \+ for the Exp(" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 146 ") di stribution using either the probability distribution function (PDF) or the cumulative distribution function (CDF). We will denote the PDF b y " }{TEXT 284 1 "f" }{TEXT -1 16 " and the CDF by " }{TEXT 285 1 "F" }{TEXT -1 90 ". The CDF for the Exp(1) appears below, but feel free t o plot it for different values of " }{XPPEDIT 18 0 "theta;" "6#%&theta G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 9 "theta:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% &thetaG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot(Expon entialCDF(theta,x),x=0..4, title=\"CDF for Exp(1)\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$\"\"!F(7$$\" 1mmmm;')=()!#<$\"1b&4S')y&\\$)F,7$$\"1LLLe'40j\"!#;$\"1^#\\g5@X]\"F27$ $\"1nmm;6m$[#F2$\"1W$*3LqD*>#F27$$\"1nmm;yYULF2$\"1C7;\\!H7%GF27$$\"1L LLeF>(>%F2$\"1B$>P0(oFMF27$$\"1mmm\">K'*)\\F2$\"17cF[n&F27$$\"1nmm;c0T \"*F2$\"1%\\sAw\\7*fF27$$\"1+++I,Q+5!#:$\"1c#[Eu.EK'F27$$\"1+++]*3q3\" F_o$\"1]*))[gjxi'F27$$\"1+++q=\\q6F_o$\"1qe$Q\"o&y*oF27$$\"1nm;fBIY7F_ o$\"1wSi-&QV7(F27$$\"1LLLj$[kL\"F_o$\"1R^3tpAstF27$$\"1LLL`Q\"GT\"F_o$ \"1#[$3YtUlvF27$$\"1++]s]k,:F_o$\"1!QHQ.mBx(F27$$\"1LLL`dF!e\"F_o$\"1p e%G#p\"3%zF27$$\"1++]sgam;F_o$\"110$\\=;56)F27$$\"1++]F_o$\"1+BX[Q \"Q_)F27$$\"1nmmTc-)*>F_o$\"15%3Z_sRk)F27$$\"1mm;f`@'3#F_o$\"1U[ie'Q%e ()F27$$\"1++]nZ)H;#F_o$\"1!3&RWe=]))F27$$\"1mmmJy*eC#F_o$\"1M;X?_nT*)F 27$$\"1+++S^bJBF_o$\"1ZgEQWbG!*F27$$\"1+++0TN:CF_o$\"1gb(*3Ik1\"*F27$$ \"1++]7RV'\\#F_o$\"1%[4`c$GF_o$\"1`?*3\"3/6%*F27$$\"1+++qfaw$Q$y*F27$$\"1++]2%)38RF_o$\"18l%p'H@+)*F27$$\"\"%F($\"1eE66O%o\")*F 2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%&TITLEG6#Q/CDF~for~Exp(1)6\"-%+AXES LABELSG6$Q\"xFa[l%!G-%%VIEWG6$;F(Fbz%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f:=x->ExponentialPDF(theta,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%/ExponentialPDFG6$%&th etaG9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "F:=x->Expon entialCDF(theta,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGR6#%\"xG6 \"6$%)operatorG%&arrowGF(-%/ExponentialCDFG6$%&thetaG9$F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "a:=plot(f(x),x=0..4):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "b:=plot(f(x),x=0..1.5,color= yellow,filled=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dis play([a,b]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6& -%'CURVESG6$7S7$\"\"!$\"\"\"F(7$$\"1mmmm;')=()!#<$\"1X!*f8@/l\"*!#;7$$ \"1LLLe'40j\"F1$\"1\\2&R*)ya\\)F17$$\"1nmm;6m$[#F1$\"1c1\"p'Hu+yF17$$ \"1nmm;yYULF1$\"1w(Q3&4xerF17$$\"1LLLeF>(>%F1$\"1x1GYHJslF17$$\"1mmm\" >K'*)\\F1$\"1)QCFao$\"1+xa^h=w9F17$$\"1nmmTc-)*>Fao$\"1!f\"Hvu-c8F17$$ \"1mm;f`@'3#Fao$\"1e^PT8cT7F17$$\"1++]nZ)H;#Fao$\"1@\\gbT\")\\6F17$$\" 1mmmJy*eC#Fao$\"1n$[&zZKe5F17$$\"1+++S^bJBFao$\"1O&RthbWr*F.7$$\"1+++0 TN:CFao$\"1-WC5*pN$*)F.7$$\"1++]7RV'\\#Fao$\"1o^!pMCyB)F.7$$\"1+++:#fk e#Fao$\"1?cDuShGvF.7$$\"1LLL`4NnEFao$\"1!yh()3#fVpF.7$$\"1+++],s`FFao$ \"1%Q7\"z\"[!pjF.7$$\"1mm;zM)>$GFao$\"1u%z5*=f*)eF.7$$\"1+++qfaF.7$$\"\"%F($\"1= M()))QcJ=F.-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%)POLYGONSG6U7&7$F(F(F'7$$ \"1+++DJdpKF.$\"1ByGh*H$y'*F17$Fe[lF(7&Fi[lFd[l7$$\"1++v=7T9hF.$\"1d1] am(oS*F17$F\\\\lF(7&F`\\lF[\\l7$$\"1++](=HPJ*F.$\"1U<>%4%o5\"*F17$Fc\\ lF(7&Fg\\lFb\\l7$$\"1++DJaU`7F1$\"1q516m%>#))F17$Fj\\lF(7&F^]lFi\\l7$$ \"1+]P%GZRd\"F1$\"1J@(R\\nOa)F17$Fa]lF(7&Fe]lF`]l7$$\"1+](=276(=F1$\"1 C)*39]^$H)F17$Fh]lF(7&F\\^lFg]l7$$\"1+](o**3)y@F1$\"17Rh*=7A/)F17$F_^l F(7&Fc^lF^^l7$$\"1+](ofHq\\#F1$\"1,tgE:K!z(F17$Ff^lF(7&Fj^lFe^l7$$\"1+ ]Pf'HU\"GF1$\"1aGS<04ZvF17$F]_lF(7&Fa_lF\\_l7$$\"1++]7*309$F1$\"1%4*y? &=[I(F17$Fd_lF(7&Fh_lFc_l7$$\"1++Dce*yU$F1$\"1rS^`a(y4(F17$F[`lF(7&F_` lFj_l7$$\"1++]([D9v$F1$\"1%pfa88>(oF17$Fb`lF(7&Ff`lFa`l7$$\"1++]iNGwSF 1$\"1>U&**fgAl'F17$Fi`lF(7&F]alFh`l7$$\"1++]7XM*Q%F1$\"1_1M!RIsW'F17$F `alF(7&FdalF_al7$$\"1+](o%QjtYF1$\"1b(3HcAegF17$F^blF(7&FbblF]bl7$$\"1+++]>0)H&F1$\"1-oQSk> ()eF17$FeblF(7&FiblFdbl7$$\"1+](=-p6j&F1$\"1\")HeGVJ%p&F17$F\\clF(7&F` clF[cl7$$\"1+++vS.EfF1$\"1iLN=-')GbF17$FcclF(7&FgclFbcl7$$\"1+](=xZ&\\ iF1$\"17b$4NcGN&F17$FjclF(7&F^dlFicl7$$\"1+]i:$4wb'F1$\"13XrI&p/>&F17$ FadlF(7&FedlF`dl7$$\"1++v=#R!zoF1$\"1f%3J9&GE]F17$FhdlF(7&F\\elFgdl7$$ \"1+]P4A@urF1$\"1iLaB54!)[F17$F_elF(7&FcelF^el7$$\"1++Dchf#\\(F1$\"12D c]T;FZF17$FfelF(7&FjelFeel7$$\"1+](of2L#yF1$\"1F]=7#)QtXF17$F]flF(7&Fa flF\\fl7$$\"1+]7yG>6\")F1$\"1%Quc?/OW%F17$FdflF(7&FhflFcfl7$$\"1++voo6 A%)F1$\"1vL@+xc2VF17$F[glF(7&F_glFjfl7$$\"1*****\\xJLu)F1$\"1fa_`3SrTF 17$FbglF(7&FfglFagl7$$\"1++v$*ydd!*F1$\"1^V%oVaB/%F17$FiglF(7&F]hlFhgl 7$$\"1+](=1\"Fao$\"1'oz9$F17$Fa[mF(7&Fe[mF`[m7$$\"1++D \"3\"o'=\"Fao$\"1\\o6riK_IF17$Fh[mF(7&F\\\\mFg[m7$$\"1+voz;)*=7Fao$\"1 _;D*e4`&HF17$F_\\mF(7&Fc\\mF^\\m7$$\"1+++&*44]7Fao$\"1kIUvsykGF17$Ff\\ mF(7&Fj\\mFe\\m7$$\"1+]7jZ!>G\"Fao$\"12`7$4#3vFF17$F]]mF(7&Fa]mF\\]m7$ $\"1+v=(4bMJ\"Fao$\"1\\\\G0T*))o#F17$Fd]mF(7&Fh]mFc]m7$$\"1++]xlWU8Fao $\"1*H1*e#e?h#F17$F[^mF(7&F_^mFj]m7$$\"1+]i&3ucP\"Fao$\"16&*)p<#pEDF17 $Fb^mF(7&Ff^mFa^m7$$\"1+++lJR09Fao$\"1K&fn01FX#F17$Fi^mF(7&F]_mFh^m7$$ \"1+v=-*zqV\"Fao$\"1N,M?i?wBF17$F`_mF(7&Fd_mF__m7$$\"1+D\"G:3uY\"Fao$ \"1LYe)*=A0BF17$Fg_mF(7&F[`mFf_m7$$\"1+++++++:Fao$\"1)H%[,;IJAF17$F^`m F(7\"-%&STYLEG6#%,PATCHNOGRIDG-Fiz6&F[[l$\"*++++\"!\")Fj`mF(-%+AXESLAB ELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fdz%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 109 "Find the value of the yellow shaded \+ area, the area under the Exponential(1) density curve to the left of 1 .5." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(F(1.5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +*R)pox!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "The value of this area can also be found by using the int egration command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 26 "evalf(int(f(x),x=0..1.5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*R)pox!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "a:=plot(f(x),x=0..4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "b:=plot(f(x),x=2..3,color=yellow,filled=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display([a,b]);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "Find the value of the yellow shaded area, the area under the Exponential(1) density \+ curve between 2 and 3." }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$\"\"!$\"\"\"F(7$$\"1mmmm;')=()!#<$\"1X! *f8@/l\"*!#;7$$\"1LLLe'40j\"F1$\"1\\2&R*)ya\\)F17$$\"1nmm;6m$[#F1$\"1c 1\"p'Hu+yF17$$\"1nmm;yYULF1$\"1w(Q3&4xerF17$$\"1LLLeF>(>%F1$\"1x1GYHJs lF17$$\"1mmm\">K'*)\\F1$\"1)QCFao$\"1+xa^h=w9F17$$\"1nmmTc-)*>Fao$\"1! f\"Hvu-c8F17$$\"1mm;f`@'3#Fao$\"1e^PT8cT7F17$$\"1++]nZ)H;#Fao$\"1@\\gb T\")\\6F17$$\"1mmmJy*eC#Fao$\"1n$[&zZKe5F17$$\"1+++S^bJBFao$\"1O&RthbW r*F.7$$\"1+++0TN:CFao$\"1-WC5*pN$*)F.7$$\"1++]7RV'\\#Fao$\"1o^!pMCyB)F .7$$\"1+++:#fke#Fao$\"1?cDuShGvF.7$$\"1LLL`4NnEFao$\"1!yh()3#fVpF.7$$ \"1+++],s`FFao$\"1%Q7\"z\"[!pjF.7$$\"1mm;zM)>$GFao$\"1u%z5*=f*)eF.7$$ \"1+++qfaF. 7$$\"\"%F($\"1=M()))QcJ=F.-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%)POLYGONSG 6U7&7$$\"\"#F(F(7$Fd[l$\"1FhOKGN`8F17$$\"1nm;arz@?Fao$\"1?\"H&oF\"F17$F]^lF( 7&Fa^lF\\^l7$$\"1+]7LRDX@Fao$\"1_rlLJQq6F17$Fd^lF(7&Fh^lFc^l7$$\"1n;zR 'ok;#Fao$\"1w!*o>`\"e9\"F17$F[_lF(7&F__lFj^l7$$\"1+]i5`h(=#Fao$\"19msP &R=7\"F17$Fb_lF(7&Ff_lFa_l7$$\"1LL$3En$4AFao$\"17)oP#3q(4\"F17$Fi_lF(7 &F]`lFh_l7$$\"1nmT!RE&GAFao$\"17ko=+(o2\"F17$F``lF(7&Fd`lF_`l7$$\"1++] K]4]AFao$\"10)fo3#*Q0\"F17$Fg`lF(7&F[alFf`l7$$\"1++]PAvrAFao$\"1E#f668 8.\"F17$F^alF(7&FbalF]al7$$\"1++]nHi#H#Fao$\"1Rvl7>,55F17$FealF(7&Fial Fdal7$$\"1n;z*ev:J#Fao$\"1'>\\(oz\\5**F.7$F\\blF(7&F`blF[bl7$$\"1LL$34 7TL#Fao$\"1JwWhzk*o*F.7$FcblF(7&FgblFbbl7$$\"1LLLjM?`BFao$\"1*4xE%RT1& *F.7$FjblF(7&F^clFibl7$$\"1+]7o7TvBFao$\"1R88=Vi(H*F.7$FaclF(7&FeclF`c l7$$\"1LLLQ*o]R#Fao$\"1;6n%QRm6*F.7$FhclF(7&F\\dlFgcl7$$\"1+]7=lj;CFao $\"1!p*zD27A*)F.7$F_dlF(7&FcdlF^dl7$$\"1+]PaRWb&)F.7$F]elF(7&FaelF\\el7$$ \"1L$eR\"3GyCFao$\"1@MMg>t)Q)F.7$FdelF(7&FhelFcel7$$\"1nmT5k]*\\#Fao$ \"1qC]#\\_D@)F.7$F[flF(7&F_flFjel7$$\"1m;zRQb@DFao$\"15>A\\#oM.)F.7$Fb flF(7&FfflFafl7$$\"1+](=>Y2a#Fao$\"1Qx\\Fsv!)yF.7$FiflF(7&F]glFhfl7$$ \"1nm\"zXu9c#Fao$\"1#etY=%3>xF.7$F`glF(7&FdglF_gl7$$\"1+++&y))Ge#Fao$ \"1V@C3DabvF.7$FgglF(7&F[hlFfgl7$$\"1++DE&QQg#Fao$\"1\\5#>O-*)R(F.7$F^ hlF(7&FbhlF]hl7$$\"1+]7y%3Ti#Fao$\"1?\"zeuO/D(F.7$FehlF(7&FihlFdhl7$$ \"1++v.[hYEFao$\"1P.[b'y!*3(F.7$F\\ilF(7&F`ilF[il7$$\"1LLLQx$om#Fao$\" 1kr'*el:ZpF.7$FcilF(7&FgilFbil7$$\"1++]P+V)o#Fao$\"1A.2g$f()z'F.7$Fjil F(7&F^jlFiil7$$\"1n;zpe*zq#Fao$\"1SMC&))Gqm'F.7$FajlF(7&FejlF`jl7$$\"1 ++]#\\'QHFFao$\"1l&)zY9$f_'F.7$FhjlF(7&F\\[mFgjl7$$\"1L$e9S8&\\FFao$\" 1WI.)f(*eR'F.7$F_[mF(7&Fc[mF^[m7$$\"1+]i?=bqFFao$\"1@cU4OuiiF.7$Ff[mF( 7&Fj[mFe[m7$$\"1LL$3s?6z#Fao$\"1()R&RkT_8'F.7$F]\\mF(7&Fa\\mF\\\\m7$$ \"1+]7`Wl7GFao$\"1!*o/)HRX+'F.7$Fd\\mF(7&Fh\\mFc\\m7$$\"1nmm'*RRLGFao$ \"1XEuu.H\")eF.7$F[]mF(7&F_]mFj\\m7$$\"1nmTvJgaGFao$\"1iuy@l'yv&F.7$Fb ]mF(7&Ff]mFa]m7$$\"1L$e9tOc(GFao$\"1O#)o+H-QcF.7$Fi]mF(7&F]^mFh]m7$$\" 1+++&Qk\\*GFao$\"1)>a/Y*4IbF.7$F`^mF(7&Fd^mF_^m7$$\"1LL3dg6>&F.7$Fe_mF(7&Fi_mFd_m7$$\"1+](= 5s#yHFao$\"1`T`du1)3&F.7$F\\`mF(7&F``mF[`m7$$\"\"$F($\"1$R'yOoqy\\F.7$ Fc`mF(7\"-%&STYLEG6#%,PATCHNOGRIDG-Fiz6&F[[l$\"*++++\"!\")F_amF(-%+AXE SLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fdz%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 93 "Using the CDF, we find the area to th e left of 3 and then subtract the area to the left of 2." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "area 1:=F(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&area1G$\"+;$H@]*!#5" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "area2:=F(2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&area2G$\"+orkY')!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "area_between:=area1-area2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-area_betweenG$\"*[@[b)!#5" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Using the integration c ommand, we simply integrate the ExponentialPDF from 2 to 3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "e valf(int(f(x),x=2..3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+$[@[b)! #6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "a:=plot(f(x),x=0..4,l abels=[\" \",\" \"]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "b: =plot(f(x),x=1.38..4,color=yellow,filled=true,labels=[\" \",\" \"]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "c:=textplot([1.35,-.04,\" x\"],color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "d:=tex tplot([2,0.08,\".25\"],color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "display([a,b,c,d]);" }{TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Find the value of " } {TEXT 287 1 "x" }{TEXT -1 3 ". " }{TEXT 288 1 "x" }{TEXT -1 60 " is k nown as the third quartile for the Exp(1) distribution." }}{PARA 13 " " 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6$7S7$\"\"!$\"\" \"F(7$$\"1mmmm;')=()!#<$\"1X!*f8@/l\"*!#;7$$\"1LLLe'40j\"F1$\"1\\2&R*) ya\\)F17$$\"1nmm;6m$[#F1$\"1c1\"p'Hu+yF17$$\"1nmm;yYULF1$\"1w(Q3&4xerF 17$$\"1LLLeF>(>%F1$\"1x1GYHJslF17$$\"1mmm\">K'*)\\F1$\"1)QCFao$\"1+xa^h =w9F17$$\"1nmmTc-)*>Fao$\"1!f\"Hvu-c8F17$$\"1mm;f`@'3#Fao$\"1e^PT8cT7F 17$$\"1++]nZ)H;#Fao$\"1@\\gbT\")\\6F17$$\"1mmmJy*eC#Fao$\"1n$[&zZKe5F1 7$$\"1+++S^bJBFao$\"1O&RthbWr*F.7$$\"1+++0TN:CFao$\"1-WC5*pN$*)F.7$$\" 1++]7RV'\\#Fao$\"1o^!pMCyB)F.7$$\"1+++:#fke#Fao$\"1?cDuShGvF.7$$\"1LLL `4NnEFao$\"1!yh()3#fVpF.7$$\"1+++],s`FFao$\"1%Q7\"z\"[!pjF.7$$\"1mm;zM )>$GFao$\"1u%z5*=f*)eF.7$$\"1+++qfaF.7$$\"\"%F($\"1=M()))QcJ=F.-%'COLOURG6&%$RGBG $\"#5!\"\"F(F(-%)POLYGONSG6U7&7$$\"1++++++!Q\"FaoF(7$Fd[l$\"1lvfIby:DF 17$$\"1nm\"Ra3rV\"Fao$\"1y(fE;QhP#F17$Fj[lF(7&F^\\lFi[l7$$\"1L3i#Q)z'[ \"Fao$\"1`ie)o`4E#F17$Fa\\lF(7&Fe\\lF`\\l7$$\"1n;9.)zEa\"Fao$\"1+-j8P2 Q@F17$Fh\\lF(7&F\\]lFg\\l7$$\"1n;*>kJ*)f\"Fao$\"1(o(=5L7@?F17$F_]lF(7& Fc]lF^]l7$$\"1L3nDh\"\\l\"Fao$\"1>i&4%336>F17$Ff]lF(7&Fj]lFe]l7$$\"1nT b34#oq\"Fao$\"1ar^kpT9=F17$F]^lF(7&Fa^lF\\^l7$$\"1+vy/`cghO/6li\"F17$F[_lF(7&F__ lFj^l7$$\"1+v$Q6_:(=Fao$\"1CI(*)HY)Q:F17$Fb_lF(7&Ff_lFa_l7$$\"1LLQBAaG >Fao$\"1DVOk%*f`9F17$Fi_lF(7&F]`lFh_l7$$\"1n;*G9R(y>Fao$\"1)e[oKMCQ\"F 17$F``lF(7&Fd`lF_`l7$$\"1++:&)*[_.#Fao$\"1x#[0\\zkI\"F17$Fg`lF(7&F[alF f`l7$$\"1++Di3*>4#Fao$\"18NCn8TM7F17$F^alF(7&FbalF]al7$$\"1++&[#Fao$\"1(=TXq2@6\"F17$ F\\blF(7&F`blF[bl7$$\"1LL)znt`D#Fao$\"1v1B@OM[5F17$FcblF(7&FgblFbbl7$$ \"1LL$R2$R0BFao$\"1ut^x&f>(**F.7$FjblF(7&F^clFibl7$$\"1+v[AvdjBFao$\"1 `Y6BOI3%*F.7$FaclF(7&FeclF`cl7$$\"1MLV=13:CFao$\"1F/tbM,O*)F.7$FhclF(7 &F\\dlFgcl7$$\"1+v[xwerCFao$\"187h2s1X%)F.7$F_dlF(7&FcdlF^dl7$$\"1+DYg dRDDFao$\"1\\=19Nm-!)F.7$FfdlF(7&FjdlFedl7$$\"1L$3-&)Q:e#Fao$\"1zdO&)) [dc(F.7$F]elF(7&FaelF\\el7$$\"1L3dKd4LEFao$\"1>nNirc&=(F.7$FdelF(7&Fhe lFcel7$$\"1n;H&z1()o#Fao$\"1+[z5!yoz'F.7$F[flF(7&F_flFjel7$$\"1nTDg5ZY FFao$\"1)*edIdQ:kF.7$FbflF(7&FfflFafl7$$\"1+DrA]v'z#Fao$\"1uK^S5x+hF.7 $FiflF(7&F]glFhfl7$$\"1n;uzI1^GFao$\"1]9OUgGydF.7$F`glF(7&FdglF_gl7$$ \"1++q;'or!HFao$\"1Hve5!>IY&F.7$FgglF(7&F[hlFfgl7$$\"1+]xQp0iHFao$\"1k N/QQCr^F.7$F^hlF(7&FbhlF]hl7$$\"1+vo7U;:IFao$\"1z/)*>#yP!\\F.7$FehlF(7 &FihlFdhl7$$\"1+]#eyIT2$Fao$\"1#\\O:XzHi%F.7$F\\ilF(7&F`ilF[il7$$\"1LL Vu[6FJFao$\"1p.[r5T%Q%F.7$FcilF(7&FgilFbil7$$\"1++D)p'o$=$Fao$\"1V@IiB EVTF.7$FjilF(7&F^jlFiil7$$\"1mT&)y\"\\\\B$Fao$\"1'=l@`?i$RF.7$FajlF(7& FejlF`jl7$$\"1++N5E*4H$Fao$\"1(eI'=*)o@PF.7$FhjlF(7&F\\[mFgjl7$$\"1M3# =6DPM$Fao$\"1J6il'>0`$F.7$F_[mF(7&Fc[mF^[m7$$\"1+v.qd%))R$Fao$\"1hKxi7 =TLF.7$Ff[mF(7&Fj[mFe[m7$$\"1LLe)GOFX$Fao$\"1$4^\"*)*))e;$F.7$F]\\mF(7 &Fa\\mF\\\\m7$$\"1+v=nY:4NFao$\"1c%okz>A*HF.7$Fd\\mF(7&Fh\\mFc\\m7$$\" 1nmErA\\jNFao$\"16U9%=oR$GF.7$F[]mF(7&F_]mFj\\m7$$\"1n;f>.1>OFao$\"1cn sZby!o#F.7$Fb]mF(7&Ff]mFa]m7$$\"1L3UO#oTn$Fao$\"1\"=<1!*\\q`#F.7$Fi]mF (7&F]^mFh]m7$$\"1++q)o1[s$Fao$\"1x!=7AwV(yvAF.7$Fg^mF(7&F[_mFf^m7$$\"1nm'[R`Z$QFao$\"1L;!G hm1;#F.7$F^_mF(7&Fb_mF]_m7$$\"1+v[i&*4!*QFao$\"17f(\\0JV/#F.7$Fe_mF(7& Fi_mFd_m7$$\"1+D\"p!H2VRFao$\"1we-#Ra)Q>F.7$F\\`mF(7&F``mF[`mFcz7$FdzF (7\"-%&STYLEG6#%,PATCHNOGRIDG-Fiz6&F[[l$\"*++++\"!\")Fj`mF(-%%TEXTG6%7 $$\"$N\"!\"#$!\"%Fcam%\"xG-Fiz6&F[[lF(F(Fj`m-F^am6%7$$\"\"#F($\"\")Fca m%$.25G-Fiz6&F[[lF(F(F(-%+AXESLABELSG6$Q\"~6\"Ffbm-%%VIEWG6$;F(Fdz%(DE FAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Find the \+ value of " }{TEXT 286 3 "x, " }{TEXT -1 60 "which has .75 area to the \+ left of it. We want to solve for " }{TEXT 291 1 "x" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fsolve(F(x)=.75,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+hVH'Q \"!\"*" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Properties of the Dis tribution" }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 " " {TEXT -1 0 "" }{TEXT 282 61 "Calculation of Mean, Variance, and Mome nt Generating Function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "To calculate the mean of the Exponential distributio n, we can simply integrate " }{TEXT 283 1 "x" }{TEXT -1 67 "*(Exponen tialPDF), per the definition of mathematical expectation." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "interface(showassumed=0); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "assume(theta>0);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f:=x->ExponentialPDF(theta,x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%& arrowGF(-%/ExponentialPDFG6$%&thetaG9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "EX:=int(x*f(x),x=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EXG%'theta|irG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Calculating Var(" }{TEXT 256 1 "X" }{TEXT -1 36 "), we will employ the formula: Var(" }{TEXT 257 1 " X" }{TEXT -1 7 ") = E( " }{XPPEDIT 18 0 "X^2;" "6#*$%\"XG\"\"#" } {TEXT -1 5 " ) - " }{XPPEDIT 18 0 "( E(X) )^2" "6#*$-%\"EG6#%\"XG\"\"# " }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "E_X_SQ:=int((x^2)*f(x),x=0..infinity);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'E_X_SQG,$*$)%'theta|irG\"\"#\"\"\"F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "VarX:=simplify(E_X_SQ-EX^2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%VarXG*$)%'theta|irG\"\"#\"\"\" " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "The Moment Generating Function (MGF) can be calculated via Maple. Recall the moment generating function of a random variable " }{TEXT 275 1 "X" }{TEXT -1 15 " is defined as " }{TEXT 258 1 "M" }{TEXT -1 1 "(" }{TEXT 259 3 " t " }{TEXT -1 6 ") = E(" }{XPPEDIT 18 0 "exp(tX)" " 6#-%$expG6#%#tXG" }{TEXT -1 63 "), provided this expectation exists. \+ In taking the integral of" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(tx)" "6#-%$expG6#%#txG" }{TEXT -1 37 "*ExponentialPDF(x) ove r the range (0," }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 43 "), the argument of the exponential term is " }{XPPEDIT 18 0 "x(t-1 /theta)" "6#-%\"xG6#,&%\"tG\"\"\"*&\"\"\"F(%&thetaG!\"\"F," }{TEXT -1 64 ", and for the resulting integral to be convergent, we must have " }{XPPEDIT 18 0 "x(t-1/theta)" "6#-%\"xG6#,&%\"tG\"\"\"*&\"\"\"F(%&thet aG!\"\"F," }{TEXT -1 30 "<0. Equivalently, we require " }{XPPEDIT 18 0 "t<1/theta" "6#2%\"tG*&\"\"\"\"\"\"%&thetaG!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "assume(t<(1/theta));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "int(exp(x*t)*f(x),x=0..infinity);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%,&*&%#t|irG\"\"\"%'theta|irGF)F)!\" \"F)!\"\"F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "So the moment generatin g function for a Exponential(" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" } {TEXT -1 30 " ) random variable is given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 260 1 "M" }{TEXT -1 2 "( " }{TEXT 261 1 "t" }{TEXT -1 5 " ) = " }{XPPEDIT 18 0 "(1-thet a*t)^(-1);" "6#),&\"\"\"\"\"\"*&%&thetaG\"\"\"%\"tGF&!\"\",$\"\"\"F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "We will define " }{TEXT 262 1 "M" }{TEXT -1 2 "( " } {TEXT 263 1 "t" }{TEXT -1 21 " ) as a function of " }{TEXT 264 1 "t" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "M(t):=(1-theta*t)^(-1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"MG6#%\"tG*&\"\"\"F),&\"\"\"F+*&%& thetaGF+F'F+!\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 117 "The moment generating function provides \+ us alternative ways to calculate the mean and variance by way of the f ormula:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "M^r;" "6#)%\"MG%\"rG" }{TEXT -1 9 " (0) = E(" } {XPPEDIT 18 0 "X^r" "6#)%\"XG%\"rG" }{TEXT -1 10 ") , where " } {XPPEDIT 18 0 "M^r" "6#)%\"MG%\"rG" }{TEXT -1 1 "(" }{TEXT 265 1 "t" } {TEXT -1 14 ") denotes the " }{TEXT 266 1 "r" }{TEXT -1 18 " th deriva tive of " }{TEXT 267 1 "M" }{TEXT -1 1 "(" }{TEXT 268 1 "t" }{TEXT -1 18 ") with respect to " }{TEXT 269 1 "t" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "This formula holds a s long as " }{TEXT 270 1 "M" }{TEXT -1 1 "(" }{TEXT 271 1 "t" }{TEXT -1 209 ") exists in an open interval containing zero. See, for exampl e, Mathematical Statistics and Data Analysis by John A. Rice for more \+ on the moment generating function. Looking at the first derivative, w e find:" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "M_p:=diff(M(t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M_pG*&%&thetaG\"\"\"*$),&\"\"\"F+*&F&F+%\"tGF+!\"\"\"\"#F'!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify(M_p);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&%&thetaG\"\"\"*$),&!\"\"\"\"\"*&F$F* %\"tGF*F*\"\"#F%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "si mplify(subs(t=0,M_p));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&thetaG" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "T his agrees with our previous result, " }{TEXT -1 3 " E(" }{TEXT 273 1 "X" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 45 ". Turning to the second derivative, we find:" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "M_pp:= diff(M_p,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%M_ppG,$*&*$)%&theta G\"\"#\"\"\"F+*$),&\"\"\"F/*&F)F/%\"tGF/!\"\"\"\"$F+!\"\"F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "simplify(subs(t=0,M_pp));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%&thetaG\"\"#\"\"\"F'" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Th erefore, E(" }{XPPEDIT 18 0 "X^2" "6#*$%\"XG\"\"#" }{TEXT -1 2 ")=" } {XPPEDIT 18 0 "2theta^2" "6#*&\"\"#\"\"\"*$%&thetaG\"\"#F%" }{TEXT -1 109 ", which again is in agreement with the value calculated previousl y. The variance is now quickly calculated as" }}{PARA 0 "" 0 "" {TEXT -1 4 "Var(" }{TEXT 274 1 "X" }{TEXT -1 7 ") = E( " }{XPPEDIT 18 0 "X^2 ;" "6#*$%\"XG\"\"#" }{TEXT -1 5 " ) - " }{XPPEDIT 18 0 "( E(X) )^2" "6 #*$-%\"EG6#%\"XG\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 14 " = " }{XPPEDIT 18 0 "2*theta^2-theta^2;" "6#,&*&\"\"#\"\"\"*$%&thetaG\"\"#F &F&*$F(\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 14 " = " } {XPPEDIT 18 0 "theta^2;" "6#*$%&thetaG\"\"#" }{TEXT -1 3 " ." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Simulation" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 51 "We will now simulate some data from a n Exponential(" }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 238 ") d istribution in order to compare the sampling distribution to the theor etical distribution. The next bit of code will simulate a random samp le of size 500 from an Exponential(1) distribution, but you are invite d to change the value of " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "theta:=1: n:=500:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sample:=ExponentialS(theta, n):" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "Remember that whe n you plot histograms, the number of bins that you select can make a h uge difference in the plot." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "num_bins:=28:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "a:=Histogram(sample,0..7,num_bins): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "b:=plot(ExponentialPDF( theta,x),x=0..7,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "display([a,b],title=\"Simulated vs. Theoretical\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7\\r7$\"\"!F( 7$F($\"1++++++S!*!#;7$$\"1+++++++DF,F*7$F.F(F07$F.$\"1++++++?vF,7$$\"1 +++++++]F,F27$F5F(F77$F5$\"1++++++g\\F,7$$\"1+++++++vF,F97$F7$F<$ \"1++++++gTF,7$$\"\"\"F(F@7$FCF(FE7$FC$\"1++++++gLF,7$$\"1++++++]7!#:F G7$FJF(FM7$FJ$\"1+++++++?F,7$$\"1+++++++:FLFO7$FRF(FT7$FR$\"1++++++!3# F,7$$\"1++++++]F,7$$\"\"#F(Fgn7$FjnF(F\\ o7$Fjn$\"1+++++++!)!#<7$$\"1++++++]AFLF^o7$FboF(Fdo7$Fbo$\"1++++++?6F, 7$$F/FLFfo7$FioF(Fjo7$Fio$\"1+++++++cF`o7$$\"1++++++]FFLF\\p7$F_pF(Fap 7$F_p$\"1+++++++[F`o7$$\"\"$F(Fcp7$FfpF(Fhp7$Ffp$\"1+++++++KF`o7$$\"1+ +++++]KFLFjp7$F]qF(F_q7$F]q$\"1+++++++CF`o7$$\"1+++++++NFLFaq7$FdqF(Ff q7$Fdq$\"1+++++++;F`o7$$\"1++++++]PFLFhq7$F[rF(F]r7$F[rFaq7$$\"\"%F(Fa q7$F`rF(Fbr7$F`rFhq7$$\"1++++++]UFLFhq7$FerF(Fgr7$Fer$F_o!#=7$$\"1++++ +++XFLFir7$F\\sF(F^s7$F\\sFhq7$$\"1++++++]ZFLFhq7$FasF(Fcs7$FasFaq7$$ \"\"&F(Faq7$FfsF(Fhs7$FfsFhq7$$\"1++++++]_FLFhq7$F[tF(F]t7$F[tFir7$$\" 1+++++++bFLFir7$F`tF(FbtF_t7$$\"1++++++]dFLFir7$FdtF(FftFct7$$\"\"'F(F ir7$FhtF(FjtFjt7$$\"1++++++]iFLF(F[uF[uF[u7$$\"1+++++++lFLF(F^uF^uF^u7 $$\"1++++++]nFLF(FauFauFau7$$\"\"(F(F(Fdu-%'COLOURG6&%$RGBG$\"*++++\"! \")F(F(-%*LINESTYLEG6#Fgp-F$6$7W7$F(FC7$$\"1LLLeR+HwF`o$\"1z\\_@VZl#*F ,7$$\"1nmm\"z+e_\"F,$\"1xQGS9!\\e)F,7$$\"1+](oM'f*=#F,$\"1jFS^9aL!)F,7 $$\"1ML3->R`GF,$\"1%=Dq@#fQ9d&F,7$$\"1ML3Ft3XtF,$\"1fJ&>#3T(z%F,7$$\"1nmTNj&=t) F,$\"1/J*pY!>wTF,7$$\"1+](=`xn,\"FL$\"1Ar\\,#)e?F,7$$\"1++]F_m]FL$\"1l$f!38I#\\\"F,7$$\"1++]s2O[?FL$\"1K;OC5Y*G\"F,7$$ \"1n;aG\"H5=#FL$\"14!e@c_#H6F,7$$\"1ML$ej%yQBFL$\"1)*\\Bc#yWk*F`o7$$\" 1MLLVUUsCFL$\"1&z.rI0!Q%)F`o7$$\"1+](o()yyi#FL$\"1eEHz6:BsF`o7$$\"1LLL oD[lFFL$\"1zC#3krXH'F`o7$$\"1+](oibk\"HFL$\"1>5fe(=DT&F`o7$$\"1,]i!o<- 1$FL$\"1'3*zy*[xo%F`o7$$\"1ML3-$=-@$FL$\"1%4#3I/yMSF`o7$$\"1M$3xplzM$F L$\"1HV$G*)zb^$F`o7$$\"1nm\"H([a'\\$FL$\"1[F`o7$$\"1+++&\\@-3%FL$\"1M5$[6s.p\"F`o7$$\"1++v$opoA%FL$\"1> Vg]:!)f9F`o7$$\"1,](oMf(oVFL$\"1G,(4h%pm7F`o7$$\"1++DEOIEXFL$\"1Sn2A*f ?3\"F`o7$$\"1LLLoT'ym%FL$\"1d\"z$4&3BR*Fjr7$$\"1,+]i-,>[FL$\"1rC<$4vZ2 )Fjr7$$\"1m;a)3rf&\\FL$\"1%\\rmS41igFjr 7$$\"1L$3-\"QfY_FL$\"1n*o&=HUl_Fjr7$$\"1+]PWF'QR&FL$\"1!eY#*e&QWXFjr7$ $\"1LL$e/Xy`&FL$\"19aGZu*\\$RFjr7$$\"1+](=<\"e)o&FL$\"1\">EUV\"R%Q$Fjr 7$$\"1ommwzvLeFL$\"1\"ybeZcq#HFjr7$$\"1nm\"zAAA)fFL$\"1O6F.H@BDFjr7$$ \"1M$3-7d%HhFL$\"1u4L.Hwx@Fjr7$$\"1,++&p]ZE'FL$\"168A%e(=->Fjr7$$\"1ML e*R7)>kFL$\"1?x.+='*G;Fjr7$$\"1nmmO9]elFL$\"1=EP&44!=9Fjr7$$\"1+](o(GP 1nFL$\"1bn^&R#4B7Fjr7$$\"1,]78Z!z%oFL$\"1ziw$)ynh5Fjr7$Feu$\"1n^ab'>)= \"*!#>-Fhu6&FjuF(F(F(-%&TITLEG6#Q:Simulated~vs.~Theoretical6\"-%+AXESL ABELSG6$%!GFbgl-%%VIEWG6$;F(Feu%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 63 "Now let's compare the simulated mean \+ to the theoretical mean. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "theo_mean:=1/theta;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*theo_meanG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sim_mean:=Mean(sample);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)sim_meanG$\"+oGR!*)*!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "relative_error:=(sim_mean-theo_mean)/theo_mean;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%/relative_errorG$!*Krg4\"!#5" }}}}} {MARK "0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }