A roadway has an average hourly volume of 300 vph. Assume that the arrival of ve

Question

A roadway has an average hourly volume of 300 vph. Assume that the arrival of vehicles is Poison discerned. Then estimate the probabilities of: p(x) = e^ lambda t( lambda t)^x/x! A. Having exactly 2 vehicles per 20 seconds B. Having at most 3 vehicles per minute C. Having at least 4 vehicles per minute Given the following measurements of traffic speed upsilon and density k, apply the method of least squares to find the best-fitting straight line upsilon = beta_0 + beta_1 k beta_1 = sigma (x_1 - x)(y_1 - y)/ sigma (x_i - x) beta_1 = y - beta, x

Explanation / Answer

Solution :-

Roadway has an average hourly volume of 360 vph.

Given, that its Poisson distribution, using the given formula :-

a) Arrival rate, = 360 vph or 0.1 vehicles pes sec

Using t = 20 seconds

Probability of exactly two vehicles = P(2) = [(0.1 x 20)2 e-0.1(20)] / 2!

= 0.271

b) At most 3 vehicles per minute

Volume of vehicles per second = 0.1

Therefore, for t = 60 seconds

   Probability of at most 3 vehicles = P(X < 3) = P(X=0) + P(X=1) + P(X=2) + P(X=3)

= 0.151

c) Probability of at least 4 vehicles per minute,

t = 60 seconds, i.e., 360 / 60 = 6 vpm (vehicles per minute)

P(X < 4) = P(X=4) + P(X=5) + P(X=6)

= 0.849

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