# Easy problem

## Main Question or Discussion Point

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

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

Thank You So Much!!!