In a bivariate distribution against a linear regression setting, if $E[y|x]$ = $\alpha$ + $\beta$$x$ and if $Var[y|x]$ is a constant, then
$$Var[y|x] = Var[y](1-Corr^2[y,x]) =\sigma^2_y(1-\rho^2_{xy})$$
I have attempted the following proof.
I have been able to decompose the variance into regression variance and residual variance as follows:
$$Var[y] = Var_x[E[y|x]] + E_x[Var[y|x]]$$
where,
$Var_x[E[y|x]]$ is the regression variance, and,
$E_x[Var[y|x]]$ is the residual variance.
$Var[y] = {\sigma^2_y}$ is the total variance.
Next, I invoked the formula for coefficient of determination:
$$\rho^2_{xy} = \frac{\text{regression variance}}{\text{total variance}}$$
$$= \frac{Var_x[E[y|x]]}{Var[y]}$$
$$\implies [1-\rho^2_{xy}] = [1 - \frac{Var_x[E[y|x]]}{Var[y]}]$$
$$\implies [\sigma^2_y (1-\rho^2_{xy})] = Var[y]- Var_x[E[y|x]]$$
$$\implies [\sigma^2_y (1-\rho^2_{xy})]= E_x[Var[y|x]] = Var[y|x]$$
The last equality is by virtue of the initial restriction that $Var[y|x]$ is constant.
Please guide me if my proof has any errors. I would also highly appreciate any alternative proofs, especially those that involve the definition of covariance.
