# Expected Value

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The town of Smallville is considering a new pedestrian crossing system for a particular stretch of road. A red/green light controls when pedestrians cross the road. Pedestrians arrive on the left and right sides of the road, and when their light turns green, they cross the road.

The system works as follows: the pedestrian's light stays red until the total number of

waiting pedestrians equals N, and then it turns green and lets all the pedestrians cross

(this is called a "dump"). Then it turns red and waits for the next N pedestrians.

The town council wants to know some of the operating characteristics of this system, to

see if it is worth the cost, and to see what N works best (to be the least disruptive to both pedestrians and cars). You are hired not to make these trade-offs but to provide information.

Assume that the pedestrians cross instantly, and that pedestrians arrive on the left and

right sides of the roads at constant average rates of and respectively.

Calculate the e-time that an arriving pedestrian must wait until crossing.

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##### Solution Summary

The town of Smallville is considering a new pedestrian crossing system for a particular stretch of road. A red/green light controls when pedestrians cross the road. Pedestrians arrive on the left and right sides of the road, and when their light turns green, they cross the road.

The system works as follows: the pedestrian's light stays red until the total number of

waiting pedestrians equals N, and then it turns green and lets all the pedestrians cross

(this is called a "dump"). Then it turns red and waits for the next N pedestrians.

The town council wants to know some of the operating characteristics of this system, to

see if it is worth the cost, and to see what N works best (to be the least disruptive to both pedestrians and cars). You are hired not to make these trade-offs but to provide information.

Assume that the pedestrians cross instantly, and that pedestrians arrive on the left and

right sides of the roads at constant average rates of and respectively.

Calculate the e-time that an arriving pedestrian must wait until crossing.

##### Solution Preview

Solution. Assume the time that an arriving pedestrian must wait until crossing is X, then X is a random variable. Assume that a pedestrian (of the first N pedestrians) arrives at ...

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- BSc , Wuhan Univ. China
- MA, Shandong Univ.

###### Recent Feedback

- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
- "Thank you"
- "Thank you very much for your valuable time and assistance!"

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