Suppose vehicles arrive at a signalized road intersection according to a Poisson

Question

Suppose vehicles arrive at a signalized road intersection according to a Poisson distribution at an average rate of 360 per hour and the cycle of the traffic lights is set at 40 seconds. (a) What is the value of lambda? (b) In what percentage of cycles will the number of vehicles arriving be exactly 5? (c) In what percentage of cycles will the number of vehicles arriving be less than 5? (d) If, after the lights change to green, there is time to clear only 5 vehicles before the signal changes to red again, what is the probability that waiting vehicles are not cleared in one cycle?

Explanation / Answer

road average rate = 360/hr = 360/3600 = 0.1/s

cycle of traffic = 40 sec

x = no of vehicles arriving at signal during that cycle

in 40s we can expect 4 vehicles be arrived

=>

a)

lambda = 360 *40 / 60*604

b)

P(x=5) = e^-4 * (4)^5/5!

= 0.1562

c)

P(x<5) = P(x=0) + ….+ P(x=5)

= e^-4 [ 4^0/1! + 4/1! + 4^2/2! + 4^3/3! + 4^4/4!]

= 0.6283

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