$X$ and $Y$ dependent when $X-E[X]=\alpha(Y-E[Y])$ - why?

Question

I have read in a book that $X$ and $Y$ are linearly dependent iff: $$X-E_{\theta}[X]=\alpha(\theta)(Y-E_{\theta}[Y])$$ (for some $\alpha(\theta)$)

I was trying to understand where it comes from but I could not figure it out. Does anyone have any suggestions?

Answer 1

Suppose $X,Y$ are linearly dependent. This means $Y=aX+b$ for some fixed $a,b$. Then $E[Y]=aE[X]+b$. Subtract this from the first equation to get $(Y-E[Y])=a(X-E[X])$. Now let $\alpha=1/a$.

Otherwise, suppose $X-E[X]=\alpha(Y-E[Y])$. Then $X=\alpha Y+(E[X]-\alpha E[Y])$, which is of the form $X=aY+b$, so $X,Y$ are linearly dependent.