multivariate normal distribution

Random variables X₁, X₂,..., X_n (each with ranges from−∞ to ∞) have a multivariate normal distribution if their joint probability density function f is given by

where x is the n×1 vector of values, μ is the n×1 vector of means, Σ is the n×n variance–covariance matrix, and det(Σ) is the determinant of Σ.

For the special case of the bivariate normal distribution, with random variables X and Y, the joint probability density function f is given by


where the mean and variance of X are μ_x and σ²_x, the mean and variance of Y are μ_y and σ_y², and ρ is the correlation coefficient between the two variables.

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A Dictionary of Statistics. Second edition revised. Copyright © Oxford University Press, 2008. All rights reserved.

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