f(x, y):
1) f(x, y) = k exp{-[x^2/ox^2 - 2 rho xy/(ox oy) + y^2/oy^2]/k1}
where k = 1/[2pi ox ot sqrt(1 - rho^2)], k1 = 2(1 - rho^2), rho is the
population correlation between X and Y, ox^2 is the variance of X, oy^2
is the variance of Y.
The right hand side of (1) factors into:
2) k exp(-x^2/(k1 ox^2)) exp(-y^2/(k1 oy^2)) exp(-2 rho xy/(k1 ox oy))
and since the univariate (marginal) pdf fX(x) of X is exp(-x^2/ox^2)
and similarly for fY(y) and we know that fX(x) = dFX(x)/dx where FX is
the univariate cumulative distribution function (cdf) of X, the right
hand side of (2) can be written:
3) k [(dFX(x)/dx)(dFY(y)/dy) exp(-2 rho xy/(ox oy)]^(1/k1)
which up to an exponent 1/k1 and the factor exp(-2 rho xy/(ox oy)) is
again a product of derivatives dFX(x)/dx times dFY(y)/dy.
Osher Doctorow
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