The Bivariate Normal and Conditional Distributions

x

10

μ

x

= mean of X

10

μ

y

= mean of Y

12

σ

x

= the standard deviation of X

3

σ

y

= the standard deviation of Y

4

ρ = the correlation coefficient of X and Y

0.6

If

X

is a normal random variable and the conditional distribution of

Y

given

X=x

is (1) normal, (2) has a mean that is a linear function of

x

, and (3) has a variance that is constant (does not depend on

x

), then the pair

(X,Y)

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

Y

when

X

takes the value of the

x

slider.