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### 20.4 The Bivariate Gaussian Distribution

Function: void gsl_ran_bivariate_gaussian (const gsl_rng * r, double sigma_x, double sigma_y, double rho, double * x, double * y)

This function generates a pair of correlated Gaussian variates, with mean zero, correlation coefficient rho and standard deviations sigma_x and sigma_y in the x and y directions. The probability distribution for bivariate Gaussian random variates is,

p(x,y) dx dy = {1 \over 2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp (-(x^2/\sigma_x^2 + y^2/\sigma_y^2 - 2 \rho x y/(\sigma_x\sigma_y))/2(1-\rho^2)) dx dy


for x,y in the range -\infty to +\infty. The correlation coefficient rho should lie between 1 and -1.

Function: double gsl_ran_bivariate_gaussian_pdf (double x, double y, double sigma_x, double sigma_y, double rho)

This function computes the probability density p(x,y) at (x,y) for a bivariate Gaussian distribution with standard deviations sigma_x, sigma_y and correlation coefficient rho, using the formula given above.