# Ask for more understanding about conditional variance

I would like to understand more about the **conditional variance** of jointly normal distribution

$$
\mathbf{var} [X|Y=y] = \sigma_X^2(1-\rho_{XY})
$$

What can we learn form this elegant formula?

Some of my observations are:
- The conditional variance is constant with respect to $Y=y$ given
- When $X$ and $Y$ are uncorrelated, $X$ behaves as it is alone. So the conditional variance is the same as the marginal variance

- When $X$ and $Y$ are fully correlated the conditional variance is **zero**.
- Being fully correlated means $X$ and $Y$ are in perfect linearity. So once $Y$ is determined, $X$ is determined; no *broadening* for $X$.

The physical meaning of edge cases above is clear.
How about the intermediate regime?