JOINTLY GAUSSIAN/NORMAL
Important Continuous Random Variable
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- Used to represent errors (amount of error in estimate).
- Has probability density function (pdf)
Where
is the center or mean of the distribution in the x direction.is the center or mean of the distribution in the y direction.
is the standard deviation of the distribution in the x direction.
is the standard deviation of the distribution in the y direction.
is the correlation coefficient between X and Y.
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The joint Gaussian distribution
can be illustrated as: or as equal probability contours:![]()
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The shape of the bell in the X & Y directions is controlled by the standard deviations, and the angle of the major/minor axes in an equal probability contour is controlled by
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- The marginal distribution function of a jointly Gaussian random variable is also Gaussian. So the marginal in X is with and
The marginal in Y behaves in a similar fashion.
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- The conditional pdf of a jointly Gaussian random variable is also Gaussian.
so
and
Note that these means and variances change with the conditioning variable y. ![]()
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