91Ó°ÊÓ

Consider the experiment of tossing two tetrahedra (regular four sided polyhedron) each with sides labeled 1 to 4 . Let \(X\) denote the number on the downturned face of the first tetrahedron and \(\mathrm{Y}\) the larger of the two downturned numbers. Find the joint density of \(\mathrm{X}\) and \(\mathrm{Y}\).

Two individuals agree to meet at a certain spot sometime between \(5: 00\) and 6:00 P.M. They will each wait 10 minutes starting from when they arrive. If the other person does not show up, they will leave. Assume the arrival times of the two individuals are independent and uniformly distributed over the hour-long interval, find the probability that the two will actually meet.

Let \((X, Y)\) have the distribution defined by the joint density function, \(f(x, y)=e^{-x-y}\) $$ x>0 $$ $$ \mathrm{y}>0 $$ \(=0 \quad\) otherwise. Find the marginal and conditional densities of \(\mathrm{Y}\) and \(\mathrm{X}\). Are \(\mathrm{X}\) and \(\mathrm{Y}\) independent?