The Pearson correlation coefficient ρ(x, y) of two random variables x and y with var(x) = var(y)

Question

Can you help with this problem please?

Answer 1

Linear regression obtains$$0=\operatorname{cov}\left(Y-a^\ast X,\,X\right)=\operatorname{cov}\left(Y,\,X\right)-a^\ast\operatorname{Var}X,$$so in the equal-variance case$$\rho=\frac{\operatorname{cov}(X,\,Y)}{\sqrt{\operatorname{Var}X\operatorname{Var}Y}}=a^\ast.$$

Answer 2

Please show some effort. Here some hints:

• Start from the linear regression framework and rewrite, using the linearity of expectation $E[(X-aY-b)^2]$ as a polynomial in $a,b$ ;

• Find the extrema of the polynomial using $\partial_a=0$ and $\partial_b=0$. This is a linear system that can be solved ;

• Compare explicitely the expression you obtain with the Pearson correlation coefficient ;