Trivariate Normal Distribution
المؤلف:
Rose, C. and Smith, M. D.
المصدر:
"The Multivariate Normal Distribution." Mathematica J
الجزء والصفحة:
...
2-3-2021
1695
Trivariate Normal Distribution
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))),](https://mathworld.wolfram.com/images/equations/TrivariateNormalDistribution/NumberedEquation1.gif) |
(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))].](https://mathworld.wolfram.com/images/equations/TrivariateNormalDistribution/NumberedEquation2.gif) |
(2)
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The standardized trivariate normal distribution takes unit variances and 
. The quadrant probability in this special case is then given analytically by
 |
(3)
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(Rose and Smith 1996; Stuart and Ord 1998; Rose and Smith 2002, p. 231).
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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