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:

To sample from this density,

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

2)  Compute the correlated X₁ and X₂

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 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, P_fail < 0.0001.