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}{TEXT 264 0 "" }{TEXT 265 0 "" } {TEXT 266 5 "joint" }{TEXT 267 0 "" }{TEXT -1 32 " probability density is given by" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "f(x,y) = exp((-1/(2*(1- rho^2)))*(((x-mu[1])/sigma[1])^2-2*rho*(x-mu[1])*(y-mu[2])/(sigma[1]*s igma[2])+((y-mu[2])/sigma[2])^2))/(2*Pi*sigma[1]*sigma[2]*sqrt(1-rho^2 ));" "6#/-%\"fG6$%\"xG%\"yG*&-%$expG6#*&,$*&\"\"\"\"\"\"*&\"\"#F1,&\" \"\"F1*$%$rhoG\"\"#!\"\"F1F9F9F1,(*$*&,&F'F1&%#muG6#\"\"\"F9F1&%&sigma G6#\"\"\"F9\"\"#F1*,\"\"#F1F7F1,&F'F1&F?6#\"\"\"F9F1,&F(F1&F?6#\"\"#F9 F1*&&FC6#\"\"\"F1&FC6#\"\"#F1F9F9*$*&,&F(F1&F?6#\"\"#F9F1&FC6#\"\"#F9 \"\"#F1F1F1*,\"\"#F1%#PiGF1&FC6#\"\"\"F1&FC6#\"\"#F1-%%sqrtG6#,&\"\"\" F1*$F7\"\"#F9F1F9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }{XPPEDIT 18 0 "-infinity;" "6#,$%)infinityG!\"\"" } {TEXT -1 3 " < " }{TEXT 268 1 "x" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "in finity;" "6#%)infinityG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "-infinity ;" "6#,$%)infinityG!\"\"" }{TEXT -1 3 " < " }{TEXT 269 1 "y" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 8 ", w here " }{XPPEDIT 18 0 "sigma[1];" "6#&%&sigmaG6#\"\"\"" }{TEXT -1 6 " \+ > 0, " }{XPPEDIT 18 0 "sigma[2];" "6#&%&sigmaG6#\"\"#" }{TEXT -1 15 " \+ > 0, and -1 < " }{XPPEDIT 18 0 "rho;" "6#%$rhoG" }{TEXT -1 5 " < 1." } }{PARA 0 "" 0 "" {TEXT -1 140 "The following code will draw the densit y function for the bivariate normal distribution. Note that there are five parameters at work here, " }{XPPEDIT 18 0 "mu[1],mu[2],sigma[1], sigma[2];" "6&&%#muG6#\"\"\"&F$6#\"\"#&%&sigmaG6#\"\"\"&F+6#\"\"#" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "rho;" "6#%$rhoG" }{TEXT -1 165 ". \+ Try altering these parameters, and see what happens to the distributio n. Don't forget to adjust the plot range accordingly, or you might no t see anything at all!" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 31 "with(plots,display,textplot3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "mu[1]:=0; mu[2]:=0; sigma[1]:=1; si gma[2]:=1; rho:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#muG6#\"\"\" \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#muG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&sigmaG6#\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&sigmaG6#\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "f(x, y):=exp((-1/(2*(1-rho^2)))*(((x-mu[1])/sigma[1])^2-2*rho*(x-mu[1])*(y- mu[2])/(sigma[1]*sigma[2])+((y-mu[2])/sigma[2])^2))/(2*Pi*sigma[1]*sig ma[2]*sqrt(1-rho^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6$%\"x G%\"yG,$*&-%$expG6#,&*$)F'\"\"#\"\"\"#!\"\"F1*$)F(F1F2F3F2%#PiG!\"\"# \"\"\"F1" }}}{EXCHG {PARA 0 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CD7D941EF3F052125CC6E2FAE3EF4986A82011D6D3F052125CC6E2FAE3F145D33D6AD9 B1B3F226FF1F23526933F2F5CE5A70B91183F390F120C5C4B0D3F42CF14A97756693F4 A865A28623D163F5191E0A63954E93F55DDB2F40AD2AD3F59905F89FF79083F5C13967 C85CA4A3F5CF7BF11EC320A3F5C13967C85CA4A3F59905F89FF79083F55DDB2F40AD2A D3F5191E0A63954E93F4A865A28623D163F42CF14A97756693F390F120C5C4B0D3F2F5 CE5A70B91183F226FF1F23526933F145D33D6AD9B1B3F052125CC6E2FAE3EF4986A820 11D6D-%&TITLEG6#Q1Bivariate~Normal6\"-%+AXESLABELSG6%%\"xG%\"yG%!G-%*A XESSTYLEG6#%&FRAMEG" 1 2 0 1 0 2 1 1 3 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Did you notice anything in p articular about the joint density after playing with the " }{XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT -1 7 "'s and " }{XPPEDIT 18 0 "sigma;" "6# %&sigmaG" }{TEXT -1 29 "'s? If you guessed that the " }{XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT -1 7 "'s and " }{XPPEDIT 18 0 "sigma;" "6#%&sig maG" }{TEXT -1 45 "'s are the means and standard deviations for " } {TEXT 270 1 "X" }{TEXT -1 5 " and " }{TEXT 271 1 "Y" }{TEXT -1 72 ", y ou're right! Just to be sure, let's look at the marginal density of \+ " }{TEXT 272 1 "X" }{TEXT -1 146 ". Unfortunately, Maple doesn't like to use subscripts when assuming conditions for its variables, so we'l l have to temporarily endure the use of " }{XPPEDIT 18 0 "mu1,mu2,sigm a1;" "6%%$mu1G%$mu2G%'sigma1G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sig ma2;" "6#%'sigma2G" }{TEXT -1 32 " for the subscripted versions. " }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 145 "f(x,y):=exp((-1/(2*(1-rho^2)))*(((x-mu1)/sigma1)^2 -2*rho*(x-mu1)*(y-mu2)/(sigma1*sigma2)+((y-mu2)/sigma2)^2))/(2*Pi*sigm a1*sigma2*sqrt(1-rho^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6$ %\"xG%\"yG,$*&-%$expG6#,$*&,(*&*$),&F'\"\"\"%$mu1G!\"\"\"\"#\"\"\"F9*$ )%'sigma1G\"\"#F9!\"\"F5*&*(%$rhoGF5F4F5,&F(F5%$mu2GF7F5F9*&F<\"\"\"%' sigma2G\"\"\"F>!\"#*&*$)FBF8F9F9*$)FF\"\"#F9F>F5F9,&F8F5*$)FAF8F9FHF>F 7F9**%#PiG\"\"\"F<\"\"\"FF\"\"\"-%%sqrtG6#,&F5F5FPF7F9F>#F5F8" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "assume(rho>-1):additionally( rho<1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "assume(sigma1>0) :assume(sigma2>0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "g(x): =int(f(x,y),y=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>-%\"gG6#%\"xG,$*&*&-%$expG6#,$*&,(*$)F'\"\"#\"\"\"\"\"\"*&F'F5%$mu1G F5!\"#*$)F7F3F4F5F4*$)%(sigma1|irG\"\"#F4!\"\"#!\"\"F3F5-%%sqrtG6#F3F4 F4*&-FC6#%#PiGF4F=\"\"\"F?#F5F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "But this can be written as " }{XPPEDIT 18 0 "g(x) = exp(1*((x-mu[1 ])/sigma[1])^2/2)/(sigma[1]*sqrt(2*Pi));" "6#/-%\"gG6#%\"xG*&-%$expG6# *(\"\"\"\"\"\"*$*&,&F'F.&%#muG6#\"\"\"!\"\"F.&%&sigmaG6#\"\"\"F6\"\"#F .\"\"#F6F.*&&F86#\"\"\"F.-%%sqrtG6#*&\"\"#F.%#PiGF.F.F6" }{TEXT -1 50 ", which is simply a normal distribution with mean " }{XPPEDIT 18 0 "m u[1];" "6#&%#muG6#\"\"\"" }{TEXT -1 24 " and standard deviation " } {XPPEDIT 18 0 "sigma[1];" "6#&%&sigmaG6#\"\"\"" }{TEXT -1 39 ". Simil arly, the marginal density for " }{TEXT 273 1 "Y" }{TEXT -1 36 " is a \+ normal distribution with mean " }{XPPEDIT 18 0 "mu[2];" "6#&%#muG6#\" \"#" }{TEXT -1 24 " and standard deviation " }{XPPEDIT 18 0 "sigma[2]; " "6#&%&sigmaG6#\"\"#" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "That leaves us with " }{XPPEDIT 18 0 "rho;" "6#%$rhoG" } {TEXT -1 47 ", whose meaning may have be harder to intuit. " } {XPPEDIT 18 0 "rho;" "6#%$rhoG" }{TEXT -1 15 " is called the " }{TEXT 274 23 "correlation coefficient" }{TEXT -1 108 ", and it measures how \+ strongly the two random variables vary together. When an expression i s found for Cov(" }{TEXT 275 1 "X" }{TEXT -1 2 ", " }{TEXT 276 1 "Y" } {TEXT -1 25 ") and is standardized by " }{XPPEDIT 18 0 "sigma[1];" "6# &%&sigmaG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sigma[2];" "6#& %&sigmaG6#\"\"#" }{TEXT -1 34 ", the expression will simplify to " } {XPPEDIT 277 0 "rho;" "6#%$rhoG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 96 "sigma[12]:=value(Doubleint((x-mu1)*(y-mu2)*f(x ,y),x=-infinity..infinity,y=-infinity..infinity));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%&sigmaG6#\"#7*(%%rho|irG\"\"\"%(sigma1|irGF*%(sigm a2|irGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "sigma[12]/(sigm a1*sigma2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%rho|irG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Voila! " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 192 "Now that we have talked about the parameters at some length, l et's develop more intuition about how the parameters affect the shape \+ of the density. The next bit of code shows what happens as " } {XPPEDIT 18 0 "mu[1];" "6#&%#muG6#\"\"\"" }{TEXT -1 10 " changes, " } {TEXT 278 15 "ceteris paribus" }{TEXT -1 3 ". " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "with(plots,display,textplot3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "mu_start:=-1.5; mu[2]:=0: sigma[1]:=1: sigma[2]:=1: r ho:=0:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)mu_startG$!#:!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "f(x,y):=exp((-1/(2*(1-rho^2 )))*(((x-mu1)/sigma[1])^2-2*rho*(x-mu1)*(y-mu[2])/(sigma[1]*sigma[2])+ ((y-mu[2])/sigma[2])^2))/(2*Pi*sigma[1]*sigma[2]*sqrt(1-rho^2)):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "step:=.1;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%stepG$\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "for k from 0 to 30" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " mu1:=mu_start+k*step:" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " num:=convert(mu1,string):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 " mu_value:=textplot3d([-3,2,0.12,` mu[1] is `.num],color=blue);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " d ensity:=plot3d(f(x,y),x=-3..3,y=-3..3);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " pic[k]:=display(\{density,mu_value\})" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "film :=[seq(pic[k],k=0..30)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "display(film,insequence=true,axes=frame,title=\"mu[1] is changing!\") ;" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6%-%(ANIMATEG 6A7%-%%TEXTG6%7%$!\"$\"\"!$\"\"#F-$\"#7!\"#%.mu[1]~is~-1.5G-%'COLOURG6 &%$RGBGF-F-$\"*++++\"!\")-%%GRIDG6%;F+$\"\"$F-F>W(\\bm\":\":3F42CF14A9 7756693F534BF3BA4729903F629900DF73D9673F70D6881EE99C0B3F7CA476223E8988 3F86E29EB9F5DB8D3F912D6A47EEC4E53F9839596EAF2D603FA00BB9FF0CE3503FA3F8 25F615A54E3FA758B4F4D6E39E3FA9A420DA7482073FAA747F02A1FC693FA9A420DA74 82073FA758B4F4D6E39E3FA3F825F615A54E3FA00BB9FF0CE3503F9839596EAF2D603F 912D6A47EEC4E53F86E29EB9F5DB8D3F7CA476223E89883F70D6881EE99C0B3F629900 DF73D9673F534BF3BA4729903F42CF14A97756693F4A865A28623D163F5B3672F22E47 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This time, however, we \+ will let the standard deviation of " }{TEXT 279 1 "Y" }{TEXT -1 16 " g et smaller as " }{XPPEDIT 18 0 "mu[1];" "6#&%#muG6#\"\"\"" }{TEXT -1 17 " is increasing. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res tart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "with(plots,display ,textplot3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "mu_start:= -1.5; mu[2]:=0: sigma[1]:=1: sigma_start:=1; rho:=0:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)mu_startG$!#:!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,sigma_startG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "f(x,y):=exp((-1/(2*(1-rho^2)))*(((x-mu1)/sigma[1])^2-2*rho*(x-mu1 )*(y-mu[2])/(sigma[1]*sigma2)+((y-mu[2])/sigma2)^2))/(2*Pi*sigma[1]*si gma2*sqrt(1-rho^2)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ste p1:=.1; step2:=.021;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&step1G$\"\" \"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&step2G$\"#@!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "for k from 0 to 30" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " \+ mu1:=mu_start+k*step1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " sigma2 :=sigma_start-k*step2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " num1:=c onvert(mu1,string):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " num2:=conv ert(sigma2,string):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 " mu_value:= textplot3d([-3,2,0.12,`mu[1] is `.num1],color=blue);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 " sigma_value:=textplot3d([2,-3,0.12,`sig[2] is \+ `.num2],color=blue);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " density:= plot3d(f(x,y),x=-3..3,y=-3..3);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 " pic[k]:=display(\{density,mu_value,sigma_value\})" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f ilm:=[seq(pic[k],k=0..30)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "display(film,insequence=true,axes=frame,title=\"sigma[2] is changi ng, too!\");" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6% -%(ANIMATEG6A7&-%%TEXTG6%7%$!\"$\"\"!$\"\"#F-$\"#7!\"#%.mu[1]~is~-1.5G -%'COLOURG6&%$RGBGF-F-$\"*++++\"!\")-F(6%7%F.F+F0%,sig[2]~is~1GF4-%%GR 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