2-variate ND (6.8)
\begin{align*}
\pmb{X} :\thicksim N_2(\pmb{\mu}, \pmb{\Sigma})
\text{ where }
\pmb{X} = \begin{pmatrix} X_1 \\ X_2 \end{pmatrix} \text{,} \quad
\pmb{\mu} = \begin{pmatrix} \mu_1 \\ \mu_2 \end{pmatrix} \text{,} \quad
\pmb{\Sigma} = \begin{pmatrix} \sigma_{11} & \rho\sigma_{1}\sigma_{2} \\ \rho\sigma_{1}\sigma_{2} & \sigma_{22} \end{pmatrix}
\end{align*}
NOTE: \pmb{X} is 2-D Prob. vec，\pmb{\mu} is Mean vec，\pmb{\Sigma} is variance-covariance matrix.
Properties
\begin{align*}
E(\pmb{X}) &= \pmb{\mu} \text{, }\text{Var}(\pmb{X}) = \pmb{\Sigma} \\
f(x,y) &= \frac{1}{2\pi\sigma_{1}\sigma_{2}\sqrt{1 - \rho^2}} \exp\left\{-\frac{(z_1 - z_2)^2}{2 (1 - \rho^2)}\right\} \text{ where } z_i = \frac{x_i - \mu_i}{\sigma_i} \\
f(x,y) &= f_X(x) f_Y(y) \text{ if } \rho = 0 \\
E(X | Y = y ) &= \mu_x + \rho \frac{\sigma_x}{\sigma_y} (y - \mu_y) = \mu_x + \frac{\sigma_{xy}}{\sigma_y^2} (y - \mu_y)\\
Var(X | Y = y ) &= \sigma_x^2 (1 - \rho^2)
\end{align*}
Appendix
E(X | Y = y) = \mu_x + \rho \frac{\sigma_x}{\sigma_y} (y - \mu_y) = \mu_x + \frac{\sigma_{xy}}{\sigma_y^2} (y - \mu_y)
is derived from below: (3.2)
\begin{align*}
E(X | Y = y) &= \displaystyle\int_{-\infty}^\infty x f_{X|Y}(x) dx \\
f_{X|Y}(x) &= \frac{f(x,y)}{f_Y(y)} \\
f_Y(y) &= \displaystyle\int_{-\infty}^\infty f(x,y) dx
\end{align*}
Var(X | Y = y) = \sigma_x^2 (1 - \rho^2)
is derived from below: (3.2)
Var(X | Y = y) = E(X^2 | Y = y) - \left\{E(X | Y = y)\right\}^2
Discussion