Question
The time between arrivals of vehicles at a particular intersection follows an exponentialprobability distribution with a mean of 12 seconds.a. Sketch this exponential probability distribution.b. What is the probability that the arrival time between vehicles is 12 seconds or less?c. What is the probability that the arrival time between vehicles is 6 seconds or less?d. What is the probability of 30 or more seconds between vehicle arrivals?
Step 1
In this case, $\mu = 12$ seconds. So, the function becomes $f(x) = \frac{1}{12} e^{-x/12}$. Show more…
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The time between arrivals of vehicles at a particular intersection follows an exponential probability distribution with a mean of 12 seconds. a. Sketch this exponential probability distribution. b. What is the probability that the arrival time between vehicles is 12 seconds or less? c. What is the probability that the arrival time between vehicles is 6 seconds or less? d. What is the probability of 30 or more seconds between vehicle arrivals?
Traffic flow is traditionally modeled as a Poisson distribution. A traffic engineer monitors the traffic flowing through an intersection with an average of six cars per minute. To set the timing of a traffic signal, the following probabilities are used. (a) What is the probability that no cars pass through the intersection within 30 seconds? (b) What is the probability that three or more cars pass through the intersection within 30 seconds? (c) Calculate the minimum number of cars through the intersection so that the probability of this number or fewer cars in 30 seconds is at least $90 \%$. (d) If the variance of the number of cars through the intersection per minute is $20,$ is the Poisson distribution appropriate? Explain.
Discrete Random Variables and Probability Distributions
Poisson Distribution
Traffic flow is traditionally modeled as a Poisson distribution. A traffic engineer monitors the traffic flowing through an intersection with an average of six cars per minute. To set the timing of a traffic signal, the following probabilities are used. (a) What is the probability of no cars through the intersection within 30 seconds? (b) What is the probability of three or more cars through the intersection within 30 seconds? (c) Calculate the minimum number of cars through the intersection so that the probability of this number or fewer cars in 30 seconds is at least $90 \%$ (d) If the variance of the number of cars through the intersection per minute is 20 , is the Poisson distribution appropriate? Explain.
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