xbeta
Suppose the random variables X 1 , X 2 bivariate normal distribution . This distribution can be characterized by five parameters:
μ 1 X 1 μ 2 X 2 σ 1 X 1 σ 2 X 2 ρ X 1 X 2
Their joint density is
f ( x 1 , x 2 ) = 1 2 π σ 1 σ 2 1 − ρ 2 exp { − 1 2 ( 1 − ρ 2 ) [ ( x 1 − μ 1 σ 1 ) 2 − 2 ρ x 1 − μ 1 σ 1 x 2 − μ 2 σ 2 + ( x 2 − μ 2 σ 2 ) 2 ] } . 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 z 1 z 2 standard normal distribution N ( 0,1 )
Then, x 1 x 2 bivariate normal distribution with parameters ( μ 1 , μ 2 , σ 1 , σ 2 , ρ )
x 1 = μ 1 + σ 1 z 1 x 2 = μ 2 + σ 2 [ z 1 ρ + z 2 1 − ρ 2 ] \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 27, 2009 18:33:27
by
Jason Blevins
(207.192.72.34)