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I have to show that that if the conditional mean $E( Y\mid X)$ is linear in $X$, and $\operatorname{Var} (Y\mid X)$ is a constant, then

$\operatorname{Var}(Y\mid X) =( \operatorname{Var} Y) [1 - r^2 ] $, where $r$ is the correlation coefficient.

Regression equation is given by

$$E(Y\mid X=x) = E(Y) - [ r \operatorname{Var} Y ( x - E(X) ] / \operatorname{Var} X $$

Please suggest how to proceed.

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    You can use $\TeX$ by enclosing formulas in dollar signs -- single dollar signs for inline, double dollar signs for displayed.2012-04-19
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    @joriki: Thanks for your help.2012-04-19
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    Are you sure that your regression equation is correctly stated? or did you make a mistake in typing it in from your homework?2012-04-19
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    @DilipSarwate: Thanks, I have edited the equation now.2012-04-19
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    Then I suggest you ask your instructor whether you are to assume that $X$ and $Y$ have equal (unconditional) variance since without this assumption the regression equation is incorrect. (Look at what your textbook says).2012-04-19
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    Was that "thanks, but no thanks"? I don't see any formatted equations in the new version.2012-04-19
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    @joriki: It is a thanks... i tried but not familiar with latex.2012-04-19
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    @DilipSarwate: It is an exercise question from Miller and Miller.2012-04-19
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    You can see how things are done by looking around the site -- you can see the $\TeX$ code for anything you see on the site by right-clicking on it and selecting "Show Math As ... TeX Commands".2012-04-19
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    @joriki: Thanks for the help. I managed to do it second time around.2012-04-19
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    Thanks for the effort. Two more hints: Displayed equations (equations on a line by themselves, as opposed to inline) look better and are easier to read when you format them in display style by enclosing them in double dollar signs. And function names such as "Var" get interpreted as a juxtaposition of variables names if you just write them out like that; that causes them to be italicized and to get the wrong spacing. To get proper formatting for those, you can use `\operatorname{Var}` to produce $\operatorname{Var}$. There are also predefined commands for frequently used functions like `\sin`.2012-04-19

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Actually you need $$ \mathbb{E}(Y \mid X=x) = \mathbb{E}(Y) + \frac{r\sqrt{\operatorname{var}(Y)}}{\sqrt{\operatorname{var}(X)}}(X - \mathbb{E}(X)). $$

So $$ \operatorname{var}(\mathbb{E}(Y\mid X)) = \left(\frac{r\sqrt{\operatorname{var}(Y)}}{\sqrt{\operatorname{var}(X)}}\right)^2 \operatorname{var}(X-\mathbb{X}) = \frac{r^2\operatorname{var}(Y)}{\operatorname{var}(X)} \operatorname{var}(X) = r^2\operatorname{var}(Y). $$

In other words, the square of the correlation is what fraction of the variability of $Y$ is "explained" by the variability of $X$.

The "law of total variance" partitions $\operatorname{var}(Y)$ into "explained" and "unexplained" parts: $$ \operatorname{var}(Y) = \operatorname{var}(\mathbb{E}(Y\mid X)) + \mathbb{E}(\operatorname{var}(Y\mid X)). $$ In the special case you're considering, the second term---the "unexplained" part---is the expected value of a constant $\operatorname{var}(Y\mid X)=(1-r^2)\operatorname{var}(Y)$.