

Bivariate Normal Density Here is a simple algorithm for sampling from a bivariate normal distribution. The joint probability of observing both x_{1} and x_{2} together is given by the bivariate normal probability density: To sample from this density, 1) Generate two, uncorrelated, standard normal variates, z_{1} and z_{2} . 2) Compute the correlated X_{1} and X_{2} 3) X_{1} and X_{2} will have means m_{1} and m_{2}, standard deviations s_{1} and s_{2}, 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.


