Bivariate Normal Distribution

Two variables follow the bivariate normal distribution if for each value of one variable, the corresponding values of another variable are normally distributed. The bivariate normal probability distribution function for a pair of continuous random variables (X and Y) is given by:

f(x) = {1/[2ps1 s2*(1-r2)1/2]} * e^{-1/2(1-r2)*[(x-m1)/s1]2 -

2r[(x-m1 )/s1 ]*[(y-m2 )/s2] + [(y-m2)/s2]2 }

¥<x<¥, -¥<y<¥,-¥<m1<¥, -¥<m2<¥, s1>0, s2>0, and -1<r<1


m1, m2

are the respective means of the random variables X and Y

s1, s2

are the respective standard deviations of the random variables X and Y


is the correlation coefficient of X and Y


is the base of the natural logarithm, sometimes called Euler's e (2.71...)


is the constant Pi (3.14...)

See also, Normal Distribution, Elementary Concepts (Normal Distribution), Basic Statistics - Tests of Normality, Distribution Fitting Introductory Overview - Types of Distributions, Q-Q Plots - Normal Distribution, and P-P Plots - Normal Distribution.