2. (70 pts.) The joint probability distribution of $X$ and $y$ is given by the following table: (For example, $f(4,9=0$.) \begin{tabular (ICCCCC) \hline \multicolumn{2}{10}) & \multicolumn{3}{IC||}{$y$) \cline { 3 - 5 } \multicolumn{2}|CO) & 1 & 3 & 9 \hline \multirow{3}{*}{$X$) & 2 & $1 / 8$ & $1 / 24$ & $1 / 12$ \cline { 2 5} & 4 & $1 / 4$ & $1 / 4$ & 0W \cline { 2 5} & 6 & $1 / 8$ & $1 / 24$ & $1 / 12$ \hline \end{tabular) (1) Derive the marginal probability distributions of $X$ and $Y$. (2) Determine whether $X$ and $y$ are stochastically independent or not. (3) Compute population means of $X$ and $y$. (4) Compute population variances of $X$ and $y$. (5) Compute the correlation coefficient between $X$ and $y$. (6) Check whether or not $\operatorname {Pr} (X=2 \mid Y=9)=\operatorname{Pr} (Y=9 \mid X=2)$. (7) Given $X=2$, find the population mean of $y$, i.e., $\mathrm{E} (Y \mid X=2)$ S.P.PB. 271