Let x and y denote the number of buses and cars

Question

Consider a ferry that can carry both buses and cars across a waterway. Let X and Y denote the number of buses and cars, respectively, carried on a given trip. The joint PMF for X and Y is given by

x

y 0 1 2

0 0.01 0.01 0.03

1 0.03 0.08 0.07

2 0.03 0.06 0.06

3 0.07 0.07 0.13

4 0.12 0.04 0.03

5 0.08 0.06 0.02

1. Give the marginal PMFs of X and Y.
2. Find the expected value and the standard deviation of X.
3. Find the expected value and the standard deviation of Y.
4. Give the conditional distribution of X given Y = 2.
5. Find E[X|Y = 2].
6. If the ferry carries only 1 bus across the waterway on a given trip, what is the expected number of cars carried on the same trip?
7. Are X and Y independent? Give reason.
8. Find corr(X, Y).