# Conditional Variance

#### fifosm

##### New Member
Hi,

I have difficulty understanding the definition of conditional variance as:

var(X|Y) = E[(X-E[X|Y])**2 | Y].

Given the definition of variance of a random variable X:

var(X) = E[(X-E[X])**2],

and substituting X with X|Y, you get:

var(X|Y) = E[(X|Y-E[X|Y])**2]. How can one get from this equation to the definition of the conditional variance given above? I would appreciate any help on this. Thank you!

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#### BGM

##### TS Contributor
It seems that just a notation issue.

When you write the conditional expectation, we will put the conditional part at the end of the operator. The reason why we write something like $$X|Y$$ is because we can just regard the conditional distribution as a new distribution. Maybe just one more property to emphasize here:

$$E[E[X|Y]|Y] = E[X|Y]$$

#### fifosm

##### New Member
Thanks BGM. But I don't seem to be able to get it. Assuming that this is a notation convention, why the convention doesn't apply to the second term E[X|Y]? I couldn't use the law of iterated expectation that you mentioned here either. Could you please elaborate? Sorry I am new to the subject matter and am studying Dimitri Bertsekas's book - introduction to probability. Thanks!

#### BGM

##### TS Contributor
The reason why you define the conditional variance like this is to the same as we define the ordinary variance

$$Var[Z] = E[(Z - E[Z])^2]$$

So I will say it is a notation issue as the concept is like what you have said, a "substitution" (although not really a substitution).

And, of course we know that in general $$E[X|Y]$$ and $$E[X]$$ are different. The variance is calculating the average squared difference with respect to its own mean, so that why we use $$E[X|Y]$$.

If you carefully expand out the squared term, we have

$$E[(X - E[X|Y])^2]$$

$$= E[X^2 - 2XE[X|Y] + E[X|Y]^2|Y]$$

$$= E[X^2|Y] - 2E[X|Y]E[X|Y] + E[X|Y]^2$$

$$= E[X^2|Y] - E[X|Y]^2$$

whereas

$$E[(X - E[X])^2]$$

$$= E[X^2 - 2XE[X] + E[X]^2|Y]$$

$$= E[X^2|Y] - 2E[X]E[X|Y] + E[X]^2$$

so indeed they are different.

You may need to elaborate more on which part you stuck in iterated expectation if you still have some questions here.

#### fifosm

##### New Member
Thanks again! I understood your point about E[X|Y] in the definition. If you clarify one point to me, I will get the whole thing. In your first expansion, how did you calculate the second term? Meaning, how did you get from E[2XE[X|Y]|Y] to 2E[X|Y]E[X|Y]? Thanks!

#### BGM

##### TS Contributor
It is similar to something like

$$E[cX] = cE[X]$$

where $$c$$ is a constant. We pulls out the constant. This is actually can be also be interpreted more generally as the tower property of conditional expectation.

$$E[X|Y]$$ is $$\sigma(Y)$$-measurable, and in this sense you can just treat it as like a "constant" inside the conditional expectation $$E[\cdot|Y]$$. But you should note that $$E[X|Y]$$ itself is a random variable but not a constant.

#### striker_000

##### New Member
Doing a 4th post to see if this works.