Question

write down the joint probability density function of the bivariate normal distribution for the random variables x1 and x2, defining all parameters

Answer 1

$f_{X_1X_2}(x_1.x_2)=\frac{1}{2\pi \sigma_{X_1}\sigma_{X_2}\sqrt{1-\rho^2}}\cdot exp\{-\frac{1}{2(1-\rho^2)}[(\frac{x_1-\mu_{X_1}}{\sigma_{X_1}})^2+(\frac{x_2-\mu_{X_2}}{\sigma_{X_2}})^2-$

$-2\rho \frac{(x-\mu_{X_1})(x_2-\mu_{X_2})}{\sigma_{X_1}\sigma_{X_2}}]\}$

where $\mu_{X_1},\ \mu_{X_2}$ are means,

$\sigma_{X_1},\ \sigma_{X_1}$ are standard deviations,

$\rho$ is correlation coefficient.