Answer to Question #237149 in Statistics and Probability for prob

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.