Conditional Expectation of bivariate normal random variables (fixed on a observation)

Question

Im having some trouble with the conditional expectation function.

Given are 2 random variables $X \sim N(\mu_x,\sigma_x)$ and $Y \sim N(\mu_y,\sigma_y)$

I understand that $\mathbb{E}[X|Y]=\mu_x+\rho \frac{\sigma_x}{\sigma_y}(Y-\mu_y) $

However, Im reading this document (2 Pages):

Lecture Notes

and they say that $$\mathbb{E}[X|Y=y] \sim N(\mu_x+\rho \frac{\sigma_x}{\sigma_y}(y-\mu_y),(1-\rho^2)\sigma_x^2) $$

I dont understand how $\mathbb{E}[X|Y=y]$ could have a continous distribution, let alone a normal distribution. From my understanding thats a number and $\mathbb{E}[X|Y] (\omega)$ is a function in $\omega$ and what holds is:

$ \mathbb{P}(\mathbb{E}[Y|X]\leq z) $ can be calculated by using the above given normal distribution for $\mathbb{E}[X|Y]$

Edit: Question is wrong, in the link they (correctly) say $$X|Y=y \sim N(\mu_x+\rho \frac{\sigma_x}{\sigma_y}(y-\mu_y),(1-\rho^2)\sigma_x^2) $$ I just read it wrong

Answer 1

Reading your link on page 2 they correctly say that

$$X|(Y=y) \sim N\left(\mu_x+\rho \frac{\sigma_x}{\sigma_y}(y-\mu_y),(1-\rho^2)\sigma_x^2\right) $$

and not the expectation as you stated