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Trivariate Normal Distribution

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A multivariate normal distribution in three variables. It has probability density function

P(x_1,x_2,x_3)=(e^(-w/[2(rho_(12)^2+rho_(13)^2+rho_(23)^2-2rho_(12)rho_(13)rho_(23)-1)]))/(2sqrt(2)pi^(3/2)sqrt(1-(rho_(12)^2+rho_(13)^2+rho_(23)^2)+2rho_(12)rho_(13)rho_(23))),
(1)

where

w=x_1^2(rho_(23)^2-1)+x_2^2(rho_(13)^2-1)+x_3^2(rho_(12)^2-1)+2[x_1x_2(rho_(12)-rho_(13)rho_(23))+x_1x_3(rho_(13)-rho_(12)rho_(23))+x_2x_3(rho_(23)-rho_(12)rho_(13))].
(2)

The standardized trivariate normal distribution takes unit variances and mu_1=mu_2=mu_3=0. The quadrant probability in this special case is then given analytically by

P(x_1<=0,x_2<=0,x_3<=0) =int_(-infty)^0int_(-infty)^0int_(-infty)^0P(x_1,x_2,x_3)dx_1dx_2dx_3 =1/8+1/(4pi)(sin^(-1)rho_(12)+sin^(-1)rho_(13)+sin^(-1)rho_(23))
(3)

(Rose and Smith 1996; Stuart and Ord 1998; Rose and Smith 2002, p. 231).

See also

Bivariate Normal Distribution, Multivariate Normal Distribution, Normal Distribution

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References

Rose, C. and Smith, M. D. "The Multivariate Normal Distribution." Mathematica J. 6, 32-37, 1996.Rose, C. and Smith, M. D. "The Trivariate Normal." §6.B A in Mathematical Statistics with Mathematica. New York: Springer-Verlag, pp. 226-228, 2002.Stuart, A.; and Ord, J. K. Kendall's Advanced Theory of Statistics, Vol. 1: Distribution Theory, 6th ed. New York: Oxford University Press, 1998.

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Trivariate Normal Distribution

Cite this as:

Weisstein, Eric W. "Trivariate Normal Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TrivariateNormalDistribution.html

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