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,
$\displaystyle \text{Var}(Y)=E\left[\text{Var}(Y|X)\right]+\text{Var}(E\left[Y|X\right])\implies E\left[\text{Var}(Y|X)\right]=\text{Var}(Y)-\text{Var}\left(E\left[Y|X\right]\right)$.
If $\displaystyle \text{Var}\left(E\left[Y|X\right]\right)=0$, we see that $\displaystyle \text{Var}(Y)-\text{Var}\left(E\left[Y|X\right]\right)=\text{Var}(Y)$.
If $\displaystyle \text{Var}\left(E\left[Y|X\right]\right)>0$, we see that $\displaystyle \text{Var}(Y)-\text{Var}\left(E\left[Y|X\right]\right)<\text{Var}(Y)$.
Therefore, we can say that $\displaystyle E\left[\text{Var}\left(Y|X\right)\right]=\text{Var}(Y)-\text{Var}\left(E\left[Y|X\right]\right)\leq\text{Var}(Y)$.
Does this make sense?