ber Y of buses per signal cycle at a left-turn lane is in the joint (x, y) 0 0.0
ID: 3063185 • Letter: B
Question
ber Y of buses per signal cycle at a left-turn lane is in the joint (x, 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 0.090 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 cyde? (b) What is the probability that there is at most one car and at gnost one bus during a cyde? (e) What is the probability that there is exactly one car during a cyde? Exacdly one bus? P(exactly one car)- Plexactly one bus)- ) 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 cyde? Yes, because pix, y) px) piy). No, because p(x, y)- (x)-pdy). OExplanation / Answer
a) P(X=1, Y=1) = 0.030
b) P(X<=1, Y<=1) = P(X=0,Y=0) + P(X=0,Y=1) + P(X=1,Y=0) + P(X=1,Y=1)
=0.010 + 0.015 + 0.020 + 0.030 = 0.075
c) P(exactly one car) = 0.020 + 0.030 + 0.050 = 0.10
P(exactly one bus) = 0.015 + 0.030 + 0.075 +0.090 +0.060 +0.030 = 0.3
d)
Overflow = more than 5 cars or the equivalent in the left turn lane.
From the problem statement one bus is equivalent to three cars. The equation is
in terms of cars.
P(X + 3Y > 5) = 1 – P(X +3Y < 5)
So find all p(x,y) that make P(X + 3Y < 5) true.
p(0,0) = p(0 + 3(0) < 5) TRUE p(0,0) = 0.010
p(1,0) = p(1 + 3(0) < 5) TRUE p(1,0) = 0.020
p(2,0) = p(2 + 3(0) < 5) TRUE p(2,0) = 0.050
p(3,0) = p(3 + 3(0) < 5) TRUE p(3,0) = 0.060
p(4,0) = p(4 + 3(0) < 5) TRUE p(4,0) = 0.040
p(5,0) = p(5 + 3(0) < 5) TRUE p(5,0) = 0.020
p(0,1) = p(0 + 3(1) < 5) TRUE p(0,1) = 0.015
p(0,2) = p(0 + 3(2) < 5) FALSE
p(1,1) = p(1 + 3(1) < 5) TRUE p(1,1) = 0.030
p(1,2) = p(1 + 3(2) < 5) TRUE p(1,2) = 0.050
p(1,3) = p(1 + 3(3) < 5) FALSE
p(1,2) and p(2,2) and p(3,2) and p(4,2) and p(5,2) and p(4,1) and p(5,1) are also FALSE
Sum all the true statements and use the equation above:
P(X+3Y>5) = 1 – P(X + 3Y < 5) = 1 – [p(0,0) + p(1,0) + p(2,0) + p(3,0) + p(4,0) + p(5,0) + p(0,1) + p(1,1) + p(1,2)
= 1 - 0.295 = 0.705
e) Yes X and Y are independent
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