Bivariate Normal Distribution Calculator

Compute the joint probability density f(x,y) of a bivariate normal distribution from x, y, means, standard deviations, and correlation.

Input

Enter x and y, the mean and standard deviation of each axis, and the correlation ρ to compute the bivariate normal joint density f(x,y).

x value

y value

Mean μx

Mean μy

Std deviation σx

Enter a positive value

Std deviation σy

Enter a positive value

Correlation ρ

Greater than −1 and less than 1

Joint density f(1, 1)

0.0943539

Z-score zx

Z-score zy

Correlation ρ

0.5

Covariance

0.5

Marginal fx(x)

0.24197072

Marginal fy(y)

0.24197072

Squared Mahalanobis

1.33333333

Cross-section at fixed y (along x)

Cross-section at fixed x (along y)

The joint density is f(x,y) = 1 / (2 π σx σy √(1 − ρ²)) × exp( −1 / (2(1 − ρ²)) × (zx² − 2ρ zx zy + zy²) ).
Standardized scores are zx = (x − μx) / σx and zy = (y − μy) / σy.
Covariance equals ρ σx σy, and the squared Mahalanobis distance is (zx² − 2ρ zx zy + zy²) / (1 − ρ²).
Marginal densities are fx(x) = φ(zx) / σx and fy(y) = φ(zy) / σy, where φ is the standard normal density.
Enter positive standard deviations σx, σy and a correlation ρ strictly between −1 and 1.

Tell us what you think of this calculator.