The bivariate normal distribution is the statistical distribution with probability density function
(1) |
where
(2) |
and
(3) |
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) | |||
(5) |
and
(6) | |||
(7) |
(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) | |||
(9) |
To derive the bivariate normal probability function, let and
be normally and independently distributed variates with mean 0 and variance 1, then define
(10) | |||
(11) |
(Kenney and Keeping 1951, p. 92). The variates and
are then themselves normally distributed with means
and
, variances
(12) | |||
(13) |
and covariance
(14) |
The covariance matrix is defined by
(15) |
where
(16) |
Now, the joint probability density function for and
is
(17) |
but from (◇) and (◇), we have
(18) |
As long as
(19) |
this can be inverted to give
(20) | |||
(21) |
Therefore,
(22) |
and expanding the numerator of (22) gives
(23) |
so
(24) |
Now, the denominator of (◇) is
(25) |
so
(26) | |||
(27) | |||
(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 outside the inner integral, letting
(50) |
and writing
(51) |
gives
(52) |
Expanding the term in braces gives
(53) |
But is odd, so the integral over the sine term vanishes, and we are left with
(54) |
Now evaluate the Gaussian integral
(55) | |||
(56) |
to obtain the explicit form of the characteristic function,
(57) |
In the singular case that
(58) |
(Kenney and Keeping 1951, p. 94), it follows that
(59) |
(60) | |||
(61) | |||
(62) | |||
(63) |
so
(64) | |||
(65) |
where
(66) | |||
(67) |
The standardized bivariate normal distribution takes and
. The quadrant probability in this special case is then given analytically by
(68) | |||
(69) | |||
(70) |
(Rose and Smith 1996; Stuart and Ord 1998; Rose and Smith 2002, p. 231). Similarly,
(71) | |||
(72) | |||
(73) |