Conditional Normal Distribution

If [math]X \sim N(\mu_x,\sigma_x^2)\,[/math] and [math]Y \sim N(\mu_y,\sigma_y^2)\,[/math], then we can use

Bayes' Rule:

[math]f(X|Y) = \frac{f(XY)}{f(Y)} = \frac{f(Y|X)f(X)}{f(Y)}\,[/math]

where [math]f(XY)\,[/math] will be bivariate normal:

[math] f(x,y) = \frac{1}{2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp \left( -\frac{1}{2 (1-\rho^2)} \left[ (\frac{x-\mu_x}{\sigma_x})^2 + (\frac{y\mu_y}{\sigma_y})^2 - 2 \rho (\frac{x-\mu_x}{\sigma_x})(\frac{y\mu_y}{\sigma_y}) \right] \right) [/math]

where [math]\rho = \frac{\mathbb{E}XY}{\sigma_x \sigma_y}\,[/math], and

[math] \Sigma = \begin{bmatrix} \sigma_x^2 & \rho \sigma_x \sigma_y \\ \rho \sigma_x \sigma_y & \sigma_y^2 \end{bmatrix}. [/math]

To give that:

[math]f(X|Y) \sim N (\mu_x + \rho \sigma_x \frac{y-\mu_y}{\sigma_y},\sigma_x \sqrt{1-\rho^2})\,[/math]