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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.

 

Cautions:

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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Last modified: June 08, 2014