variance question

**cm917** · October 4th 2009, 03:51 PM

The question askes... to show E[Var (Y|X)] <= Var (Y)
I think we should apply the formula for E[Var(Y|X)] first...but I am stuck on it...
Anyone knows wha's the next step? Thanks.

**Chris L T521** · October 4th 2009, 05:18 PM

Originally Posted by cm917

The question askes... to show E[Var (Y|X)] <= Var (Y)
I think we should apply the formula for E[Var(Y|X)] first...but I am stuck on it...
Anyone knows wha's the next step? Thanks.

Note that by the law of total variance,

$\text{Var}(Y)=E\left[\text{Var}(Y|X)\right]+\text{Var}(E\left[Y|X\right])\implies E\left[\text{Var}(Y|X)\right]=\text{Var}(Y)-\text{Var}\left(E\left[Y|X\right]\right)$ .

If $\text{Var}\left(E\left[Y|X\right]\right)=0$ , we see that $\text{Var}(Y)-\text{Var}\left(E\left[Y|X\right]\right)=\text{Var}(Y)$ .

If $\text{Var}\left(E\left[Y|X\right]\right)>0$ , we see that $\text{Var}(Y)-\text{Var}\left(E\left[Y|X\right]\right)<\text{Var}(Y)$ .

Therefore, we can say that $E\left[\text{Var}\left(Y|X\right)\right]=\text{Var}(Y)-\text{Var}\left(E\left[Y|X\right]\right)\leq\text{Var}(Y)$ .

Does this make sense?

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