The joint probability distribution of the number X ofcars and the number Y of buses per signal cycle at aproposed left-turn lane is displayed in the accompanying jointprobability table.
| y |
p(x,y) Â Â Â Â | | 0 | 1 | 2 |
| x | 0Â Â | Â Â 0.015Â Â | Â Â 0.010Â Â | Â Â 0.025Â Â |
| 1Â Â | Â Â 0.030Â Â | Â Â 0.020Â Â | Â Â 0.050Â Â |
| 2Â Â | Â Â 0.075Â Â | Â Â 0.050Â Â | Â Â 0.125Â Â |
| 3Â Â | Â Â 0.090Â Â | Â Â 0.060Â Â | Â Â 0.150Â Â |
| 4Â Â | Â Â 0.060Â Â | Â Â 0.040Â Â | Â Â 0.100Â Â |
| 5Â Â | Â Â 0.030Â Â | Â Â 0.020Â Â | Â Â 0.050Â Â |
(a) What is the probability that there is exactly one car andexactly one bus during a cycle?
(b) What is the probability that there is at most one car and atmost one bus during a cycle?
(c) What is the probability that there is exactly one car during acycle? Exactly one bus?
| P(exactly one car) | = |
| P(exactly one bus) | = |
(d) Suppose the left-turn lane is to have a capacity of five carsand one bus is equivalent to three cars. What is the probability ofan overflow during a cycle?
(e) Are X and Y independent rv's? Explain.
Yes, because p(x, y) =pX(x) ·pY(y).Yes, becausep(x, y) ≠pX(x) ·pY(y).    No,because p(x, y) =pX(x) ·pY(y).No, becausep(x, y) ≠pX(x) ·pY(y).
