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# System prepended metadata

title: Ask for more understanding about conditional variance

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# 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 
![](https://i.imgur.com/PJlxNjt.png =100x)

- 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 $X$.
![](https://i.imgur.com/cO9AQOv.png =100x)


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