I Proof request for best linear predictor

[psie]

TL;DR

In An Intermediate Course in Probability by Gut, there's a theorem stated without proof concerning best linear predictors. I was wondering if anyone knows how to prove it/or knows other sources where it has been proved.

Maybe this is a simple exercise, but I don't see how to prove the below theorem with the tools I've been given in the section (if it is possible at all).

Theorem 5.2. Suppose that $EX^2<\infty$ and $EY^2<\infty$. Set \begin{align*}\mu_x&=EX, \\ \mu_y&=EY, \\ \sigma_x^2&=\operatorname{Var}X,\\ \sigma_y^2&=\operatorname{Var}Y, \\ \sigma_{xy}&=\operatorname{Cov}(X,Y), \\ \rho&=\sigma_{xy}/(\sigma_x\sigma_y).\end{align*} The best linear predictor of $Y$ based on $X$ is $$L(X)=\alpha+\beta X,$$where $\alpha=\mu_y-\frac{\sigma_{xy}}{\sigma_x^2}\mu_x=\mu_y-\rho\frac{\sigma_y}{\sigma_x}\mu_x$ and $\beta=\frac{\sigma_{xy}}{\sigma_x^2}=\rho\frac{\sigma_y}{\sigma_x}$.

That's the theorem that I'm looking to prove. Now I'll just state some definitions and a theorem that has been given in the section prior to the above theorem. As done in the book, we confine ourselves to conditioning on a random variable, although definitions and theorems extend to conditioning on a random vector.

Definition 5.1. The function $h(x)=E(Y\mid X=x)$ is called the regression function $Y$ on $X$.

Definition 5.2. A predictor (for $Y$) based on $X$ is a function, $d(X)$. The predictor is called linear if $d$ is linear.

Definition 5.3. The expected quadratic prediction error is $$E(Y-d(X))^2.$$ Moreover, if $d_1$ and $d_2$ are predictors, we say that $d_1$ is better than $d_2$ if $E(Y-d_1(X))^2\leq E(Y-d_2(X))^2$.

Theorem 5.1. Suppose that $EY^2<\infty$. Then $h(X)=E(Y\mid X)$ (i.e. the regression function $Y$ on $X$) is the best predictor of $Y$ based on $X$.

 

[Gavran]

(https://en.wikipedia.org/wiki/Gauss–Markov_theorem)