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" 0 "" {MPLTEXT 1 0 16 "assume(rho^2<1):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 127 "The joint (bivariate) Normal pdf - both variables have m
ean of 0 and variance equal to 1, rho is their correlation coefficient
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "f:=exp(-(x^2+y^2-2*rho
*x*y)/2/(1-rho^2))/(2*Pi*sqrt(1-rho^2));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"fG,$*(-%$expG6#,$*&,(*$)%\"xG\"\"#\"\"\"F1*$)%\"yGF0F1F1**F0
F1%%rho|irGF1F/F1F4F1!\"\"F1,&F1F1*$)F6F0F1F7F7#F7F0F1%#PiGF7F8F;#F1F0
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Verifying that the total prob
ability is 1:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "int(int(f,x=-infinity..infinity),y=-infinity..infinit
y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 56 "The marginal distribution of  Y - clerarly Normal (0,1
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "fym:=int(f,x=-infinit
y..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fymG,$*(-%$expG6#,
$*$)%\"yG\"\"#\"\"\"#!\"\"F.F/%#PiGF0F.#F/F.F3" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 110 "The conditional distribution of  X, given that  Y =
 y.  The answer is clearly Normal ( y*rho, sqrt(1-rho^2) ):" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "fxc:=si
mplify(f/fym);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fxcG,$**\"\"##\"
\"\"F'-%$expG6#,$*(,&%\"xGF)*&%%rho|irGF)%\"yGF)!\"\"F',&F2F)F)F4F4,&F
2F)F)F)F4F(F)%#PiG#F4F',&F)F)*$)F2F'F)F4F8F(" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 98 "Start anew, not making any assumptions about  rho  (oth
erwise, Maple would not let us substitute):" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "rho:='rho':" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 62 "f:=exp(-(x^2+y^2-2*rho*x*y)/2/(1-rho^2))/(2*Pi*s
qrt(1-rho^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,$*(-%$expG6#,
$*&,(*$)%\"xG\"\"#\"\"\"F1*$)%\"yGF0F1F1**F0F1%$rhoGF1F/F1F4F1!\"\"F1,
&F1F1*$)F6F0F1F7F7#F7F0F1%#PiGF7F8F;#F1F0" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 95 "Explore the shape of the resulting distribution (rotate i
t) using  rho = 0,  0.5,   0.7,  etc.:" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot3d(subs(rho=0.9,f),x=-3..3,y=-3
..3,numpoints=5000);" }}{PARA 13 "" 1 "" {GLPLOT3D 521 372 372 
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