Question 4: (5 points)
There are two alternative routes (A and B) for a ship passage. Let X denote the sailing times (in
hours) for route A and Y denote the sailing times (in hours) for route B. The joint density
function of X and Y is given by:
$$f(x, y) = \begin{cases}
\frac{1}{18}e^{-(x+y)/6}, & 0 < x < y < \infty, \\
0, & \text{otherwise}.
\end{cases}$$
a) What is the probability that the sailing times for route B exceeds the sailing times for
route A by at least two hours?
b) What is the moment generating function of the sailing times (in hours) for route A?
c) Are X and Y independent?