Question

# The joint probability distribution of the number X of cars and the number Y of buses...

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.

 y p(x, y) 0 1 2 x 0 0.010 0.015 0.025 1 0.020 0.030 0.050 2 0.050 0.075 0.125 3 0.060 0.090 0.150 4 0.040 0.060 0.100 5 0.020 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) = 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.

Yes, because p(x, y) = pX(x) · pY(y).Yes, because p(x, y) ≠ pX(x) · pY(y).    No, because p(x, y) = pX(x) · pY(y).No, because p(x, y) ≠ pX(x) · pY(y).

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