Transportation Data Analytics (TDA)

What are the characteristics of Binomial and Poisson Distributions

A traffic analyst observes that the number of accidents occurring daily at a specific intersection follows a Poisson distribution with a mean of 3 accidents per day. Find (i) probability that exactly 2 accidents occur in a day, (ii) probability that no accidents occur in a day, and (iii) probability that at least 4 accidents occur in a day.

A bus arrives at a particular stop every 10 minutes on average. The number of buses arriving in a 30-minute period follows a Poisson distribution. Find (i) probability that exactly 3 buses arrive in 30 minutes, (ii) probability that fewer than 2 buses arrive in 30 minutes, and (iii) probability that 5 or more buses arrive in 30 minutes.

A train station sells tickets to passengers, with the number of tickets sold per minute following a binomial distribution. On average, there is a 20% chance that a passenger will purchase a ticket during any given minute. If the station operates for 5 minutes, what is the probability that exactly 2 tickets are sold? (i) What is the probability that no tickets are sold in 5 minutes?, and (ii) What is the probability that at least 1 ticket is sold in 5 minutes?

A traffic study shows that 70% of drivers stop at a red light. Suppose 10 drivers are randomly selected at a red light. Find probability that (i) exactly 8 out of 10 drivers will stop at the red light, (ii) at least 9 drivers will stop at the red light, and (iii) fewer than 5 drivers will stop at the red light. Also, find the expected number of drivers who will stop at the red light?

In a large city, vehicle breakdowns occur at an average rate of 3 per hour on a busy highway.

1. What is the probability that no breakdowns will occur in the next 2 hours?

2. What is the probability that exactly 7 breakdowns will occur in a 3-hour period?

3. What is the probability that more than 10 breakdowns will occur in a 4-hour period?

4. If a new policy is implemented to reduce breakdowns, and the average rate drops to 2 per hour, what is the probability that in the next 5-hour period, no more than 5 breakdowns will occur?

A city's emergency response system is being tested for efficiency. There is an 85% chance that any given emergency call is responded to within the target time. Suppose 200 emergency calls are placed in a week.

1. What is the probability that exactly 170 calls are responded to within the target time?

2. What is the probability that fewer than 160 calls are responded to within the target time?

3. What is the probability that between 180 and 190 calls (inclusive) are responded to within the target time?

4. If the system is tested weekly for 10 weeks, what is the probability that in at least 8 weeks, 95% or more of the calls are responded to within the target time?

Traffic accidents at a particular intersection occur at an average rate of 2 per week.

1. What is the probability that there will be no accidents at this intersection in the next week?

2. What is the probability that exactly 5 accidents will occur over the next 2 weeks?

3. What is the probability that at least 3 accidents will occur in the next week?

4. If the city implements new safety measures that reduce the rate to 1.5 accidents per week, what is the probability that no more than 2 accidents will occur in the next month (4 weeks)?

A fleet of 100 autonomous vehicles is being tested in a city. Each vehicle has a probability of 0.95 of safely completing a daily route without incident.

1. What is the probability that exactly 90 vehicles will complete their route safely in one day?

2. What is the probability that at least 97 vehicles will complete their route safely?

3. Assuming each vehicle operates independently, what is the probability that fewer than 85 vehicles complete their route safely on a particular day?

4. If the testing lasts for 30 days, what is the probability that on exactly 25 days, at least 95 vehicles will complete their route safely?

The travel time for a particular route during peak hours is normally distributed with a mean of 45 minutes and a standard deviation of 8 minutes.

1. What is the probability that a randomly selected trip during peak hours will take more than 55 minutes?

2. What is the probability that a trip will take between 40 and 50 minutes?

3. If a transportation agency considers trips taking more than 60 minutes as excessively long, what percentage of trips fall into this category?

4. If the agency wants to ensure that 95% of trips take less than a certain amount of time, what should this maximum travel time be?

The fuel consumption of a fleet of trucks is normally distributed with a mean of 8 miles per gallon (mpg) and a standard deviation of 0.5 mpg.

1. What is the probability that a randomly selected truck from the fleet has a fuel consumption of less than 7.5 mpg?

2. What is the probability that a truck consumes between 7.8 and 8.2 mpg?

3. What is the fuel consumption value below which the least efficient 10% of the trucks fall?

4. If the company wants to ensure that the top 15% of its trucks have a fuel consumption of at least a certain value, what should this minimum fuel consumption be?

Vehicle emissions for a certain type of car are normally distributed with a mean of 180 grams of CO₂ per kilometer and a standard deviation of 15 grams per kilometer.

1. What is the probability that a randomly selected car emits more than 200 grams of CO₂ per kilometer?

2. What percentage of cars emit between 160 and 190 grams of CO₂ per kilometer?

3. What is the emission value below which the least polluting 5% of cars fall?

4. If the government wants to set an emissions standard such that only 10% of cars exceed it, what should this standard be?

The speed of vehicles on a highway is normally distributed with a mean of 65 mph and a standard deviation of 10 mph.

1. What is the probability that a randomly selected vehicle is traveling at more than 75 mph?

2. What is the probability that a vehicle is traveling between 60 and 70 mph?

3. What speed separates the fastest 20% of vehicles from the rest?

4. If a speed limit is set to ensure that no more than 5% of vehicles exceed it, what should the speed limit be?

The daily demand for public transportation passes in a city follows a normal distribution with a mean of 1,500 passes and a standard deviation of 300 passes.

1. What is the probability that on a given day, demand will exceed 1,800 passes?

2. What is the probability that the demand will be between 1,200 and 1,700 passes?

3. What is the daily demand level below which the lowest 15% of days fall?

4. If the city wants to stock enough passes to meet demand on 99% of days, how many passes should be available each day?