Let $X$ and $Y$ be two non-constant random variables with finite variances. The correlation coefficient $\rho(X, Y)$ is defined by

$$\rho(X, Y)=\frac{\mathbb{E}[(X-\mathbb{E} X)(Y-\mathbb{E} Y)]}{(\operatorname{Var} X)^{1 / 2}(\operatorname{Var} Y)^{1 / 2}}$$

(a) Using the Cauchy-Schwarz inequality or otherwise, prove that

$$-1 \leqslant \rho(X, Y) \leqslant 1$$

(b) What can be said about the relationship between $X$ and $Y$ when either (i) $\rho(X, Y)=0$ or (ii) $|\rho(X, Y)|=1$. [Proofs are not required.]

(c) Take $0 \leqslant r \leqslant 1$ and let $X, X^{\prime}$ be independent random variables taking values $\pm 1$ with probabilities $1 / 2$. Set

$$Y= \begin{cases}X, & \text { with probability } r \\ X^{\prime}, & \text { with probability } 1-r\end{cases}$$

Find $\rho(X, Y)$.