Bivariate Normal

If $X_1,\,X_2$ have a bivariate normal distribution with parameters $\mu_1,\,\mu_2,\,\sigma^2_1,\,\sigma^2_2, \,\rho$, then the joint pdf of $X_1$ and $X_2$ is

$$f(x_1,\,x_2) \, = \, k e^{-\frac{1}{2(1-\rho^2)}\left[ \frac{... ...sigma_1\sigma_2}+\frac{(x_2-\mu_2)^2}{\sigma^2_2}\right] } \ ,$$

where $k=1/2 \pi \sigma_1 \sigma_2 \sqrt{1-\rho^2}$.

Let ${\bf X}'=(X_1,\,X_2), \ \mbox{\boldmath$\mu$}' = (\mu_1,\, \mu_2)$, and define $\Sigma$ by

$$\mbox{\boldmath$\Sigma$}= \left[ \begin{array}{ll} \sigma^2_... ..._2\\ \rho\sigma_1\sigma_2 & \sigma^2_2 \end{array} \right]$$

and we see that the joint pdf can be written in matrix notation as

$$f_{\bf X} ({\bf x}) = \frac{1}{2\pi\vert\mbox{\boldmath$\Sigm... ...mbox{\boldmath$\Sigma$}^{-1}({\bf x} - \mbox{\boldmath$\mu$})}$$

where $\vert\mbox{\boldmath$\Sigma$}\vert$ is the determinant of $\Sigma$.

Check that $\vert\mbox{\boldmath$\Sigma$}\vert = \sigma^2_1\sigma^2_2(1-\rho^2)$ and that

$$\mbox{\boldmath$\Sigma$}^{-1} = \frac{1}{1-\rho^2}\left[ \beg... ...gma_1\sigma_2} & \frac{1}{\sigma^2_2} \end{array} \right] \ .$$

We write ${\bf X} \sim N_2(\mbox{\boldmath$\mu$},\,\mbox{\boldmath$\Sigma$})$. Read HC 3.5 to revise some of the properites of the bivariate normal distribution, which can be regarded as a special case of the multivariate normal distribution. This will be considered in the remainder of this chapter.