4. X and Y are random variables with a joint distribution given below:
\begin{tabular}{c|ccc}
& \multicolumn{3}{c}{$X = x$} \\
\cline{2-4}
$Y = y$ & 0 & 2 & 4 \\
\hline
0 & 0.05 & 0.10 & 0.25 \\
2 & 0.20 & 0.15 & 0.25 \\
\end{tabular}
(a) Obtain the marginal distributions of X and Y.
(4 marks)
(b) Evaluate E(X), Var(X), E(Y) and Var(Y).
(4 marks)
(c) Obtain the conditional distributions of Y | X = 2 and X | Y = 2.
(4 marks)
(d) Evaluate E(XY), Cov(X, Y) and Corr(X, Y).
(5 marks)
(e) Are X and Y independent? Justify your answer.
(3 marks)