The Bivariate Normal Distribution

Abstract

In the univariate case, a random variable X is said to have a normal distribution with mean μ and variance σ² > 0 (in symbols, N(μ, σ²)) if its density function is of the form

$$ f\left( {x;\mu ,{\sigma ^2}} \right) = \frac{1}{{\sqrt {2\pi \sigma } }}{e^{ - {Q_1}\left( {x;\mu ,{\sigma ^2}} \right)/2}},x \in \Re , $$

where

$$ {Q_1}\left( {x;\mu ,{\sigma ^2}} \right) = \frac{1}{{{\sigma ^2}}}{\left( {x - \mu } \right)^2} = \left( {x - \mu } \right){\left( {{\sigma ^2}} \right)^{ - 1}}\left( {x - \mu } \right), $$

μ ∈ ℜ, and σ² ∈ (0, ∞). The bivariate normal density function given below is a natural extension of this univeriate normal density.