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Intuitive Parameterization of the Bivariate Normal Distribution

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This Demonstration shows a parameterization of a bivariate normal distribution by specifying an angle of cross-correlation . The controls and are the and spreads of the blob.

Contributed by: Robert L. Brown (March 2011)
Open content licensed under CC BY-NC-SA

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A bivariate normal distribution can be represented by three polar variables: , , or with the parameters , , , where

,

,

.

.

Conversely, define:

,

,

.

Then this gives , , from , , :

,

,

.

See the Wikipedia article Gaussian function.

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