1

We are given that $\operatorname E(Y|x)$ is linear in $x$ and $\operatorname{Var}(Y|x)$ is a constant.
We need to prove that $\operatorname{Var}(Y|x) = (\operatorname{Var}(y))^{2} (1-r)^{2}$ where $r$ is the correlation coefficient.

I tried writing $\operatorname{Var}(Y|x)$ as: $$\operatorname{Var}(Y|x) = \operatorname E(Y^{2}|x)-E(Y|x)^{2}$$ But I have no idea how to proceed further...

$\operatorname E(Y|x)$ is linear in $x$ and $\operatorname{Var}(Y|x)$ is constant [...]

This statement means something but I am not able to figure it out... Could anyone help?

rubik
  • 9,344
User9523
  • 2,094

1 Answers1

1

The minimum mean-square error (MMSE) estimator of $Y$, given that $X$ has value $x$ is $E[Y\mid X = x]$. The linear minimum mean-square error (LLMSE) estimator (LLMSE) is of the form $$r \left(\sqrt{\frac{\operatorname{var}(Y)}{\operatorname{var}(X)}} \right)(x - \mu_X) + \mu_Y.$$ The mean-square error of the LLMSE estimator is $\operatorname{var}(Y)(1-r^2)$.

Here you are being told that the MMSE estimator of $Y$ given that $X=x$ is a linear function of $x$ (note that linear in this context means of the form $ax+b$). What do you think the LMMSE estimator is, and what is its variance?

Dilip Sarwate
  • 25,197