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Bivariate Normal Distribution
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The bivariate normal distribution is given by
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(Kenney and Keeping 1951, pp. 92 and 202-
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The probability function of the bivariate normal distribution is implemented as
PDF[MultinormalDistribution[{mu1, mu2}, {{sigma11, sigma12},
{sigma12, sigma22}}], {x1, x2}] in the Mathematica add-on package
Statistics`MultinormalDistribution` (which can be loaded with the
command <<Statistics`).
The marginal probabilities are then
(7)
(8)
and
(9)
(10)
(Kenney and Keeping 1951, p. 202).
Let
and
be two independent normal variates with means
for i = 1, 2. Then the variables
and
and
defined below are normal
bivariates with unit variance and cross-correlation coefficient
:
(11)
(12)
To derive the bivariate normal probability function, let
and
be normally
and independently distributed variates with mean 0 and variance 1, then
define
(13)
(14)
(Kenney and Keeping 1951, p. 92). The variates
themselves normally distributed with means
and
and
are then
, variances
(15)
(16)
and covariance
(17)
The covariance matrix is defined by
(18)
where
(19)
Now, the joint probability density function for
and
is
(20)
but from (13) and (14), we have
(21)
As long as
(22)
this can be inverted to give
(23)
Therefore,
(24)
and expanding the numerator of (24) gives
(25)
so
(26
)
Now, the denominator of (24) is
(27)
so
(28)
can be written simply as
(29)
and
(30
)
Solving for
and
and defining
(31)
gives
(32)
(33)
But the Jacobian is
(34)
(35)
(36)
so
(37)
and
(38
)
where
(39)
Q.E.D.
The characteristic function of the bivariate normal distribution is given by
(40)
(41)
where
(42)
and
(43)
Now let
(44)
(45)
Then
(46
)
where
(47)
(48)
Complete the square in the inner integral
(49
)
Rearranging to bring the exponential depending on w outside the inner
integral, letting
(50)
and writing
(51)
gives
(52)
Expanding the term in braces gives
(53)
But
are left with
is odd, so the integral over the sine term vanishes, and we
(54
)
(55
)
Now evaluate the Gaussian integral
(56)
(57)
to obtain the explicit form of the characteristic function,
(58)
(59)
(60)
In the singular case that
(61)
(Kenney and Keeping 1951, p. 94), it follows that
(62)
(63)
(64)
so
(65)
(66)
where
(67)
The standardized bivariate normal distribution takes
and
. The quadrant probability in this special case is then given
analytically by
(68)
(69)
(Rose and Smith 1996; Stuart and Ord 1998; Rose and Smith 2002, p. 231).
Similarly,
(70)
(71)
Box-Muller Transformation, Correlation Coefficient--Bivariate Normal
Distribution, Multivariate Normal Distribution, Normal Distribution, Price's
Theorem, Trivariate Normal Distribution
search
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 936-937, 1972.
Holst, E. "The Bivariate Normal Distribution."
http://www.ami.dk/research/bivariate/.
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van
Nostrand, 1951.
Kotz, S.; Balakrishnan, N.; and Johnson, N. L. "Bivariate and Trivariate Normal Distributions."
Ch. 46 in Continuous Multivariate Distributions, Vol. 1: Models and Applications, 2nd ed. New
York: Wiley, pp. 251-348, 2000.
Rose, C. and Smith, M. D. "The Multivariate Normal Distribution." Mathematica J. 6, 32-37,
1996.
Rose, C. and Smith, M. D. "The Bivariate Normal." §6.4 A in Mathematical Statistics with
Mathematica. New York: Springer-Verlag, pp. 216-226, 2002.
Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill,
p. 118, 1992.
Stuart, A.; and Ord, J. K. Kendall's Advanced Theory of Statistics, Vol. 1: Distribution Theory,
6th ed. New York: Oxford University Press, 1998.
Whittaker, E. T. and Robinson, G. "Determination of the Constants in a Normal Frequency
Distribution with Two Variables" and "The Frequencies of the Variables Taken Singly." §161162 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York:
Dover, pp. 324-328, 1967.
Eric W. Weisstein. "Bivariate Normal Distribution." From MathWorld--A Wolfram Web
Resource. http://mathworld.wolfram.com/BivariateNormalDistribution.html
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