2. 1/6 points I Previous Answers My Notes Ask Your Te The joint probability dist

Question

2. 1/6 points I Previous Answers My Notes Ask Your Te The joint probability distribution of the number X of cars and the number Y of buses per signal cycle at a proposed left-turn lane is displayed in the accompanying joint probability table. ptx, y) 0 0.010 0.015 0.025 1 0.020 0.030 0.050 2 0.050 0.075 0.125 3 0.060 0.0900.150 40.040 0.060 .100 50.020 0.030 0.050 (a) What is the probability that there is exactly one car and exactly one bus during a cycle? (b) What is the probability that there is at most one car and at most one bus during a cycle? (c) What is the probability that there is exactly one car during a cycle? Exactly one bus? P(exactly one car) exactly one bus) (d) Suppose the left-turn lane is to have a capacity of five cars and one bus is equivalent to three cars. What is the probability of an overflow during a cycle? (e) Are X and Y independent rv's? Explain. O Yes, because p(x, y)xx) p(y) O Yes, because p(x, y)pxtx) p() 0 No, because p(x, y) = px(x) , pr(y). O No, because p(x, y)Px(x) pyly)

Explanation / Answer

(a)
P(X=1,Y=1) = 0.030

(b)
P(X<=1, Y<=1) = 0.01 + 0.015 + 0.02 + 0.03 = 0.075

(c)
P(exactly one car) = 0.02 + 0.03 + 0.05 = 0.1

P(exactly one bus) = 0.015 + 0.03 + 0.075 + 0.09 + 0.06 + 0.03 = 0.3

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