# [R] bivariate normal cdf and rho

Jun Yan jyan at stat.wisc.edu
Tue May 14 17:11:19 CEST 2002

```A while back, I asked the following quesiton:

> Suppose F(x, y; rho) is the cdf of a bivariate normal distribution, with
> standardized marginals and correlation parameter rho. For any fixed x and
> y, I wonder if F(x, y; rho) is a monotone increasing function of rho,
> i.e., there is a 1 to 1 map from rho to F(x, y; rho).

Actually there is a beautiful result on this and Professor S. Le Cessie
from Netherlands kindly gave me an elegant proof of the following:

d  F(x, y; rho) / d rho = f(x, y; rho),

where f is the standard bivariate normal density.

It can be proved by showing that

d f(x, y; rho) / d rho = d f(x, y; rho) / dx dy.

My thanks to Prof. Thomas Lumley, Chong Gu, and John Fox for pointing me
to the references.

Jun

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```