I am looking at a graph that depicts a conditional Gaussian:

enter image description here

I understand what the titled red spheres mean - that the variables are somewhat are correlated with each other.

I don't understand the significance of the blue line. I know it's related to conditional gaussian but I can't quite make sense of it.

Could someone explain what the blue line means?


1 Answer 1


Basically the bivariate normal distribution looks like this one: enter image description here

Then the conditional distribution $f_{Y|X=x}(x,y)$ is here marked with the red line. We take the joint pdf and plug in $x=1.6$

$f_{X,Y}(x,y) = \frac{1}{2 \pi \sigma_X \sigma_Y \sqrt{1-\rho^2}} \exp\left( -\frac{1}{2(1-\rho^2)}\left[ \frac{(x-\mu_X)^2}{\sigma_X^2} + \frac{(y-\mu_Y)^2}{\sigma_Y^2} - \frac{2\rho(x-\mu_X)(y-\mu_Y)}{\sigma_X \sigma_Y} \right] \right)$

$f_{Y|X=1.6}(1.6,y) = \frac{1}{2 \pi \sigma_X \sigma_Y \sqrt{1-\rho^2}} \exp\left( -\frac{1}{2(1-\rho^2)}\left[ \frac{(1.6-\mu_X)^2}{\sigma_X^2} + \frac{(y-\mu_Y)^2}{\sigma_Y^2} - \frac{2\rho(1.6-\mu_X)(y-\mu_Y)}{\sigma_X \sigma_Y} \right] \right)$

This function does depend on the variable y only-the remaining unknowns are parameters. We can just draw the area at $x=1.6$ and omit all the other from the graph above. Then basically we get an univariate distribution of a normal distributed variable. At the graph below we see one example of $f_{Y|X=x}(x,y)$ and $f_{Y|Y=y}(x,y)$ enter image description here


The red circles (ellipses) at your graph show the different values of the joint distribution.


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