Problem 2. X and Y are two random variables and Y = g(X) where
\begin{equation*}
g(x) = \begin{cases}
\frac{1}{3}x, & 0 \le x < 1\\ \frac{1}{3}(2x - 1), & 1 \le x \le 2\\ 0, & o.w.
\end{cases}
\end{equation*}
(a) Find the PDF, $f_Y(y)$, of Y in terms of $f_X(x)$. Assume that $f_X(x)$ is nonzero only for $0 \le x \le 2$.
(b) Plot $f_Y(y)$ if X is uniformly distributed in $[0, 2]$.
(c) Describe verbally the formation of $f_Y(y)$ from the specific $f_X(x)$ and $g(x)$ in this problem.
(d) Find the covariance and correlation coefficient of X and Y if X is uniformly distributed in $[0, 2]$.
