# How to prove that $\mathrm{var}(X-E(X|Y)) \leq \mathrm{var}(X)$?

I tried to solve this exercise but got stuck: Assume we have the random variables $X$ and $Y$ where $E(X) = 0$. How can we prove the following inequality $\operatorname{Var} (X-E(X|Y)) \leq\operatorname{Var}(X)$?

I tried to write out the rhs: $\operatorname{Var}(X-E(X|Y)) = \operatorname{Var}(X) + \operatorname{Var}(E(X|Y)) - 2\operatorname{Cov}(X,E(X|Y))$ Since $E(X) = 0$, we have $\operatorname{Var}(E(X|Y)) = E(E(X|Y)^2)$ and $\operatorname{Cov}(X,E(X|Y)) = E(X*E(X|Y))$.

Thus we have $\operatorname{Var}(X) + E(E(X|Y)^2) - 2E(XE(X|Y)) \leq \operatorname{Var}(X)$ giving $E(E(X|Y)^2) \leq 2E(XE(X|Y))$

But that did not get me anywhere as you can see, does anybody have any tips? OR do you think there might be a typo in the exercise?

• Hint: Start rather from $$X=(X-E(X\mid Y))+E(X\mid Y)$$ hence $$\mathrm{var}(X)=\ldots$$ – Did Aug 17 '17 at 15:10
• Tnx for the reply! However, it didnt get me anywhere. I tried expanding, but nothing cancelled, care to give another tip? =)) $Var((X-E(X|Y)) + E(X|Y)) = Var(X-E(X|Y)) + Var(E(X|Y)) -2Cov(X-E(X|Y),E(X|Y))$ – LearningMath Aug 18 '17 at 11:20
• Let us turn to the covariance then (with a factor $+2$ instead of $-2$), which involves three terms: $$E((X-E(X\mid Y))E(X\mid Y))\qquad E(X-E(X\mid Y))\qquad E(E(X\mid Y))$$ What can one say about each of these terms? – Did Aug 18 '17 at 13:17
• Opps +2 and not -2 , as you said, however, i still do not see how this track gets me anywhere: We have $E(X-E(X|Y)) = 0$ and $E(E(X|Y)) = 0$ and finally $Cov(X-E(X|Y),E(X|Y)) = E((X-E(X|Y))E(X|Y)) = E(XE(X|Y))-E(E(X|Y)^2)$ But this will only yield my original $E(E(X|Y)^2)<2E(XE(X|Y))$ :( – LearningMath Aug 18 '17 at 14:04
• What you are lacking is a characterization of $E(X\mid Y)$ (until now, everything you tried would work with another random variable instead, which is a bad sign, right?). So, what is specific about $Z=E(X\mid Y)$, which makes that $E(XW)=E(ZW)$ for every $W$ such that $____$? IOW, how do you define $Z$? – Did Aug 18 '17 at 14:41

## 1 Answer

@Did's hints are spot on and you don't need $E(X)=0$.

Let $Z=E(X|Y)$, consider $$\text{Var}(X)=\text{Var}(X-Z+Z)=\text{Var}(X-Z)+\text{Var}(Z)+2\text{Cov}(X-Z,Z).$$ Note that \begin{gather} E(X-Z|Y)=E(X-E(X|Y)|Y)=E(X|Y)-E(X|Y)=0\\ \implies E(X-Z)=0\implies \text{Cov}(X-Z,Z)=E[(X-Z)Z]. \end{gather} In going from the first line to the second line, we have used the Law of Iterated Expectations: $$E(X-Z)=E[E(X-Z|Y)]=E(0)=0.$$ For $E[(X-Z)Z]$, we can condition on $Y$ again: $$E[(X-Z)Z|Y]=ZE(X|Y)-Z^2=Z^2-Z^2=0\implies E[(X-Z)Z]=0.$$ Thus, we have $$\text{Var}(X)=\text{Var}(X-Z)+\text{Var}(Z)\geq\text{Var}(X-Z)=\text{Var}[X-E(X|Y)].$$

• I think the OP has problems understanding the implication: $E(X-Z|Y) =0 \Rightarrow E(X-Z)=0$. – Harto Saarinen Nov 17 '17 at 8:30