The Mystery of the ARROW: Can Someone Explain?

[tuanle007]

can someone explain to me how did it derived to the ARROW...
I am not quite sure how that happened.thanks

 

[HallsofIvy]

You have
[tex]e^{-\frac{x^2+ y^2- 2\rho xy}{2\sigma(1-\rho)^2}}[/itex]<br /> They complete the square:<br /> $$x^2- 2\rho xy+ \rho^2y^2- \rho^2y^2+ y^2= (x- \rho y)^2+ (y^2- \rho^2 y^2)$$<br /> so the exponential becomes<br /> [tex]e^{-\frac{(x-\rho y)^2- (y^2- \rho^2 y^2)}{2\sigma(1-\rho)^2}}[/itex]<br /> Then separate the exponentials:<br /> $$e^{-\frac{(x- \rho y)^2}{2\sigma(1-\rho)^2}}e^{-\frac{(y^2- \rho^2 y^2}{2\sigma(1-\rho)^2}}$$<br /> and finally factor out y² in the last exponential<br /> $$e^{-\frac{(x- \rho y)^2}{2\sigma(1-\rho)^2}}e^{-\frac{y^2(1- \rho^2}{2\sigma(1-\rho)^2}}$$[/tex][/tex]

 

[tuanle007]

Thank You So Much!