The Bivariate Normal and Conditional Distributions

If is a normal random variable and the conditional distribution of given is (1) normal, (2) has a mean that is a linear function of , and (3) has a variance that is constant (does not depend on ), then the pair follows a bivariate normal distribution. The left image is a graph of the bivariate density function and the right image shows the conditional distribution of when takes the value of the slider.