xbeta
Bivariate normal distribution
Suppose the random variables? are jointly distributed according to the bivariate normal distribution. This distribution can be characterized by five parameters:
- : the mean of ,
- : the mean of ,
- : the standard deviation of ,
- : the standard deviation of ,
- : the correlation of and .
Their joint density is
f(x_1,\,x_2) = \frac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}
\exp\left\lbrace
-\frac{1}{2(1-\rho^2)}\left[
\left( \frac{x_1 - \mu_1}{\sigma_1}\right)^2
- 2 \rho \frac{x_1 - \mu_1}{\sigma_1} \frac{x_2 - \mu_2}{\sigma_2}
+ \left( \frac{x_2 - \mu_2}{\sigma_2}\right)^2
\right]
\right\rbrace.
Sampling From the Bivariate Normal Distribution
The following algorithm can be used to sample from the bivariate normal distribution:
-
Let and be independent draws from the standard normal distribution .
-
Then, and calculated as follows will have a joint bivariate normal distribution with parameters :
\begin{aligned}x_1 &= \mu_1 + \sigma_1 z_1 \\
x_2 &= \mu_2 + \sigma_2 \left[z_1 \rho + z_2 \sqrt{1 - \rho^2} \right] \\
\end{aligned}
See Also
Revised on March 7, 2007 12:20:26
by
Jason Blevins
(152.3.149.118)