2. Cars which are passing an automatic toll are modeled by a Poisson process with rate...

2. Cars which are passing an automatic toll are modeled by a Poisson process with rate of 10 cars per hour. Some cars may violate with the probability of 0.5.

a. Calculate the probability that exactly 10 cars pass within an hour and all 10 have no violations?

b. For any fixed x ≥ 10, find the probability that x cars pass during the hour, of which 10 have no violations?

Expert Solution

orchestra answered 2 years ago

Cars get parked in a lot (with infinite capacity) according to a λ-rate Poisson process, and...

Cars get parked in a lot (with infinite capacity) according to a λ-rate Poisson process, and indepen- dently stay parked for a random duration. The parking time duration of a car follows a common distribution X, with cdf F(x) = P(X ≤ x). Let N(t) be the number of cars parked at time t. 1a. What is the distribution of N(t)? 1b. Assuming a car arrival rate of 1 per minute, and X (in min) ∼ Gamma(3, 1) (as defined...

Q1. Arrival of customers to a local store may be modeled by a Poisson process with...

Q1. Arrival of customers to a local store may be modeled by a Poisson process with an average of 1 arrival every 10 min period. Each customer on average stays for a time exponentially distributed with mean 15 minutes. This is modeled as a birth-death process. (i) What assumption is necessary for this process to be modeled as a birth-death process? (ii) Compute the probability that the number of customers in the store reaches 30, if we have 3 customers...

5. The occurrence of traffic accidents at an intersection is modeled as a Poisson process. Based...

5. The occurrence of traffic accidents at an intersection is modeled as a Poisson process. Based on accident records at that intersection, a traffic engineer has determined that the average rate of accidents occurring at that intersection is once every three years. (a) What is the probability that no accident occurs at that intersection for a period of 5 years? (b) What is the probability that there is a traffic fatality at that intersection over a period of 3 years...

Traffic on Rosedale Road in Princeton, NJ, follows a Poisson process with rate 6 cars per...

Traffic on Rosedale Road in Princeton, NJ, follows a Poisson process with rate 6 cars per minute. A deer runs out of the woods and tries to cross the road. It takes the deer a random time (independent of everything else), uniformly distributed between 2 and 5 seconds, to cross the road. If there is a car passing while the deer is on the road, it will hit the deer with portability 2/3 (independent of everything else). Find the probability...

The rate at which customers are served at an airport check-in counter is a Poisson process...

The rate at which customers are served at an airport check-in counter is a Poisson process with a rate of 10.4 per hour. The probability that more than 50 customers are served at the counter in the next 5 hours is P(Xp>50). If this is solved as a Poisson variable, the calculations will be tedious. So we use the normal approximation. Now, P(Xp > 50)=P(Z > a), where Z is the standard normal variable. What is the value of a?...

(20 pts) Inquiries arrive at an information center according to a Poisson process of rate 2...

(20 pts) Inquiries arrive at an information center according to a Poisson process of rate 2 inquiries per second. It takes a server an exponential amount of time with mean ½ second to answer each query. (a) (5 pts) If there are two servers. What is the probability that an arriving customer must wait to be served? (b) (5 pts) Find the mean number of customers in the system and the mean time spent in the system. (c) (10 pts)...

the visit of a customer at a restaurant follows a Poisson process with a rate of...

the visit of a customer at a restaurant follows a Poisson process with a rate of 3 arrivals per week. A day with no visits to the restaurant is called a risky day. a. Find the expected number of risky days in a week b. Given that a risky day was observed on a Sunday, what is the probability that the next risky day will appear on the following Wednesday? c. Find the probability that the 4th day, which is Thursday,...

The number of fires in a certain forest follows a Poisson process at a rate of...

The number of fires in a certain forest follows a Poisson process at a rate of 0.40 per week. A) What is the probability that there are no more than 3 fires in a given week? B) What is the the probability that the time between fires is at least 3 weeks?

A traffic engineer believes that the number of cars passing through a certain intersection between 2...

A traffic engineer believes that the number of cars passing through a certain intersection between 2 and 6 pm on weekdays follows a normal distribution with mean 750 and standard deviation 100. A new highway is opened, and it is hypothesized that the number or cars passing through the intersection should decrease as a result. A sample of 15 weekdays is taken, and the mean number of cars passing through the intersection is 710. Decide whether this reduction in traffic...

The traffic incidents in Melbourne and Sydney follow a Poisson process with the rate of 5...

The traffic incidents in Melbourne and Sydney follow a Poisson process with the rate of 5 and 6 incidents per hour, respectively. (a) Find the probability that no traffic accidents will occur in Melbourne in the next 30 minutes. (b) Find the expected time (in minutes) until 10 new incidents occur in Sydney. (c) Find the expected time (in minutes) until 10 new incidents in total occur in Melbourne and Sydney. Hint: For this and the next question, you can...

Question