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.
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, Pfail < 0.0001.