The joint probability distribution of the number X of cars and the number of bus

Question

The joint probability distribution of the number X of cars and the number of buses per signal cycle at a proposed left turn lane is displayed in the accompanying joint probability table p(x, y) 0 2 0.025 0 1 0.020 0.050 0.030 2 0.050 0.125 0.075 3 0.060 0.150 0.090 4 5 0.020 0.050 0.030 0.010 0.015 0.040 0.100 0.060 (a) What is the probability that there is exactly one car and exactly one bus during a cycle? 050 (b) What is the probability that there is at most one car and at most one bus during a cycle? 05 (c) What is the probability that there is exactly one car during a cycle? Exactly one bus? P(exactly one car) = P(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)Pxx) pYy) O Yes, because p(x, y) # PX(x) . pny) O No, because p(x, y) =px(x) . py(y) O No, because p(x, y) # PX(x) . py(y)

Explanation / Answer

b) P(At most 1 car and at most one bus)

= 0.010 + 0.025 + 0.020 + 0.050

= 0.105

c) P(Exactly one car) = 0.020 + 0.050 + 0.030 = 0.10

P(Exactly one bus) = 0.025 + 0.050 + 0.125 + 0.150 + 0.100 + 0.050 = 0.50

d) P(Overflow)

= P(More than five cars)

= P(2 buses)

= 0.015 + 0.030 + 0.075 + 0.090 + 0.060 + 0.030

= 0.30

e) Option D is correct.

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