The bivariate normal distribution is the statistical distribution with probability density function
(1)
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where
(2)
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and
(3)
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is the correlation of and (Kenney and Keeping 1951, pp. 92 and 202-205; Whittaker and Robinson 1967, p. 329) and is the covariance.
The probability density function of the bivariate normal distribution is implemented as MultinormalDistribution[mu1, mu2, sigma11, sigma12, sigma12, sigma22] in the Wolfram Language package MultivariateStatistics` .
The marginal probabilities are then
(4)
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(5)
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and
(6)
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(7)
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(Kenney and Keeping 1951, p. 202).
Let and be two independent normal variates with means and for , 2. Then the variables and defined below are normal bivariates with unit variance and correlation coefficient :
(8)
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(9)
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To derive the bivariate normal probability function, let and be normally and independently distributed variates with mean 0 and variance 1, then define
(10)
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(11)
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(Kenney and Keeping 1951, p. 92). The variates and are then themselves normally distributed with means and , variances
(12)
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(13)
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and covariance
(14)
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The covariance matrix is defined by
(15)
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where
(16)
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Now, the joint probability density function for and is
(17)
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but from (◇) and (◇), we have
(18)
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As long as
(19)
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this can be inverted to give
(20)
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(21)
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Therefore,
(22)
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and expanding the numerator of (22) gives
(23)
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so
(24)
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Now, the denominator of (◇) is
(25)
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so
(26)
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(27)
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(28)
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can be written simply as
(29)
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and
(30)
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Solving for and and defining
(31)
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gives
(32)
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(33)
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But the Jacobian is
(34)
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(35)
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(36)
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so
(37)
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and
(38)
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where
(39)
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Q.E.D.
The characteristic function of the bivariate normal distribution is given by
(40)
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(41)
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where
(42)
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and
(43)
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Now let
(44)
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(45)
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Then
(46)
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where
(47)
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(48)
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Complete the square in the inner integral
(49)
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Rearranging to bring the exponential depending on outside the inner integral, letting
(50)
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and writing
(51)
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gives
(52)
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Expanding the term in braces gives
(53)
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But is odd, so the integral over the sine term vanishes, and we are left with
(54)
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Now evaluate the Gaussian integral
(55)
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(56)
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to obtain the explicit form of the characteristic function,
(57)
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In the singular case that
(58)
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(Kenney and Keeping 1951, p. 94), it follows that
(59)
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(60)
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(61)
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(62)
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(63)
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so
(64)
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(65)
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where
(66)
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(67)
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The standardized bivariate normal distribution takes and . The quadrant probability in this special case is then given analytically by
(68)
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(69)
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(70)
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(Rose and Smith 1996; Stuart and Ord 1998; Rose and Smith 2002, p. 231). Similarly,
(71)
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(72)
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(73)
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