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Bivariate Normal Density

Here is a simple algorithm for sampling from a bivariate normal distribution.  

The joint probability of observing both x1 and x2 together is given by the bivariate normal probability density:

bivariate_normal.gif (2859 bytes)

To sample from this density,

1)   Generate two, uncorrelated, standard normal variates, z1 and z2 .

standard_norma_z.gif (1160 bytes)

2)  Compute the correlated X1 and X2

bivariate_normal_X.gif (1597 bytes)

3)  X1 and X2 will have means m1 and m2, standard deviations s1 and s2, and correlation r.



1) While it is almost always possible to calculate means and standard deviations, that doesn't mean the data have a normal distribution.

2)  Using a bivariate normal density because it is convenient without checking its verisimilitude with the data is dangerous.

3)  Using estimates of parameters xbar.gif (855 bytes) and s uncritically, as though they were the populations parameters, m and s themselves, is also dangerous, especially with either small samples, or when estimating small probabilities, Pfail < 0.0001.


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Copyright � 1998-2008 Charles Annis, P.E.
Last modified: June 08, 2014