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Bivariate normal distribution

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Suppose the random variables? X 1 ,X 2 are jointly distributed according to the bivariate normal distribution. This distribution can be characterized by five parameters:

  • μ 1 : the mean of X 1 ,
  • μ 2 : the mean of X 2 ,
  • σ 1 : the standard deviation of X 1 ,
  • σ 2 : the standard deviation of X 2 ,
  • ρ: the correlation of X 1 and X 2 .

Their joint density is

f(x 1 ,x 2 )=1 2 πσ 1 σ 2 1 −ρ 2 exp{−1 2 (1 −ρ 2 )[(x 1 −μ 1 σ 1 ) 2 −2 ρx 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:

  1. Let z 1 and z 2 be independent draws from the standard normal distribution N(0,1 ).

  2. Then, x 1 and x 2 calculated as follows will have a joint 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

category: Statistics

Revised on March 7, 2007 12:20:26 by Jason Blevins (152.3.149.118)
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