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

where

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 |

r |
is the correlation coefficient of X and Y |

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

p |
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.