Bivariate Normal Distribution — Definition, Formula & Examples

The bivariate normal distribution describes the joint probability distribution of two random variables that are each normally distributed and linked by a linear correlation. It is the simplest multivariate extension of the familiar bell curve.

A continuous random vector $(X, Y)$ follows a bivariate normal distribution if its joint probability density function is determined entirely by five parameters: the means $\mu_X$ and $\mu_Y$ , the standard deviations $\sigma_X$ and $\sigma_Y$ , and the correlation coefficient $\rho \in (-1, 1)$ . Every linear combination $aX + bY$ is univariate normal, and the conditional distributions $X \mid Y = y$ and $Y \mid X = x$ are also normal.

Key Formula

f(x,y) = \frac{1}{2\pi\sigma_X\sigma_Y\sqrt{1-\rho^2}} \exp\!\left(-\frac{1}{2(1-\rho^2)}\left[\frac{(x-\mu_X)^2}{\sigma_X^2} - \frac{2\rho(x-\mu_X)(y-\mu_Y)}{\sigma_X\sigma_Y} + \frac{(y-\mu_Y)^2}{\sigma_Y^2}\right]\right)

Where:

$\mu_X, \mu_Y$ = Means of X and Y
$\sigma_X, \sigma_Y$ = Standard deviations of X and Y (both > 0)
$\rho$ = Correlation coefficient between X and Y, where −1 < ρ < 1

How It Works

The distribution produces an elliptical, mound-shaped surface over the

xy

-plane. When

\rho = 0

the ellipse axes align with the coordinate axes and

X

Y

are independent. As

|\rho|

increases toward 1, the ellipse narrows and tilts, reflecting stronger linear dependence. Contours of equal probability density are ellipses centered at

(\mu_X, \mu_Y)

. You can compute marginal distributions by integrating out one variable — each marginal is simply a univariate normal. Conditional distributions are also normal, with a mean that shifts linearly with the conditioning value.

Worked Example

Problem: Suppose (X, Y) follows a bivariate normal distribution with μ_X = 50, μ_Y = 100, σ_X = 10, σ_Y = 20, and ρ = 0.6. Find the conditional distribution of Y given X = 60.

Conditional mean: The conditional mean of Y given X = x is given by the formula below.

\mu_{Y|X} = \mu_Y + \rho\frac{\sigma_Y}{\sigma_X}(x - \mu_X) = 100 + 0.6 \cdot \frac{20}{10}(60 - 50) = 100 + 12 = 112

Conditional standard deviation: The conditional standard deviation does not depend on the observed value of X.

\sigma_{Y|X} = \sigma_Y\sqrt{1-\rho^2} = 20\sqrt{1-0.36} = 20\times 0.8 = 16

Answer: Given X = 60, Y is normally distributed with mean 112 and standard deviation 16, i.e., Y | X = 60 ~ N(112, 16²).

Why It Matters

The bivariate normal distribution underpins simple linear regression — the assumption that errors are normal and linearly related leads directly to this model. It appears routinely in fields like finance (modeling correlated asset returns) and engineering (tolerance analysis of paired measurements).

Common Mistakes

Mistake: Assuming that two individually normal variables are automatically bivariate normal.

Correction: Two marginals can each be normal while their joint distribution is not bivariate normal. Bivariate normality requires that every linear combination aX + bY also be normal, which is a stronger condition.