## 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:

$f(x_1, x_2)=const \times \exp \Bigg(- \frac{1}{2(1-\rho^2)} \Big( \frac{x_1 -\mu_1}{\sigma_1} \Big)^2 -2 \rho \frac{x_1 -\mu_1}{\sigma_1} \frac{x_2 -\mu_2}{\sigma_2} + \Big( \frac{x_2 -\mu_2}{\sigma_2} \Big)^2 \Bigg)$

where $const = \frac{1}{\big(2 \pi \sigma_1 \sigma_2 \sqrt{1 – \rho^2} \big)}$

To sample from this density,

1. Generate two, uncorrelated, standard normal variates, $$z_1$$ and $$z_2$$ .

$z \sim \Phi(\mu = 0, \sigma^2 = 1)$

where “$$\sim$$” is read “… is distributed as …”

1. Compute the correlated $$x_1$$ and $$x_2$$.

$x_1 =\mu_1+\sigma_1 z_1$

1. $$x_1$$ and $$x_2$$ will have means $$\mu_1$$ and $$\mu_2$$ standard deviations $$\sigma_1$$ and $$\sigma_2$$, and correlation $$\rho$$.

#### 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 $$\bar{x}$$ and $$s$$ uncritically, as though they actually were the populations parameters, $$\mu$$ and $$\sigma$$ themselves, is also dangerous, especially with either small samples – small samples notoriously underestimate $$\sigma$$ – or when estimating small probabilities, $$P_{fail}$$ < 0.001.