On correlation and covariance

Question

How to prove mathematically for the following:

$ \operatorname{Corr}(x, y)=1$ implies a perfect positive linear relationship, which means that we can write $Y=a+bx $ for some constant a and some constant $b \gt0$.

Answer 1

Suppose $X$ and $Y$ are random variables with $\text{Corr}(X,Y)=1$, i.e. $\text{Cov}(X,Y) = \sqrt{\text{Var}(X) \text{Var}(Y)}$. This is equivalent to $\mathbb E[\widetilde{X} \widetilde{Y}] = \sqrt{\mathbb E[\widetilde{X}^2]\; \mathbb E[\widetilde{Y}^2]}$ where $\widetilde{X} = X - \mathbb E[X]$ and $\widetilde{Y} = Y - \mathbb E[Y]$. Now expand out $$ \mathbb E \left[\left(\sqrt{\mathbb E[\widetilde{X^2}]}\; \widetilde{Y} - \sqrt{\mathbb E[\widetilde{Y^2}]} \widetilde{X}\right)^2\right]=0$$ to conclude $$\sqrt{\mathbb E[\widetilde{X^2}]}\; \widetilde{Y} - \sqrt{\mathbb E[\widetilde{Y^2}]} \widetilde{X} = 0\ \text{almost surely}$$