Help with correlation question? How to solve this?

Question

Let $X$ and $Y$ be random variables and $a,b$ $\in$ $\mathbb{R}$ such that $a \neq 0$. If $Y = aX + c$, then show that corr($X, Y$) = +1 or corr($X, Y$) = -1.

Answer 1

The definition of correlation is $\mathrm{corr}(X,Y) = \mathbf{E}[(X-\mathbf{E}X)(Y-\mathbf{E}Y)]/\sqrt{\mathrm{Var}{X}\cdot\mathrm{Var}{Y}}$. To apply this you'll need to use the facts that $\mathbf{E}(aX)=a\mathbf{E}X$, $\mathbf{E}(X+a)=a+\mathbf{E}X$, $\mathrm{Var}({aX}) = a^2\mathrm{Var}{X}$, and $\mathrm{Var}(X+a) = \mathrm{Var}(X)$ for any random variable $X$ and constant $a$.