4. Cars arrive at a tollbooth at a mean rate of 5 cars every 10 minutes according to a Poisson process. Find the probability that the toll collector will have to wait longer than 26.3 minutes before collecting the eighth toll
4. Cars arrive at a tollbooth at a mean rate of 5 cars every 10 minutes according to a Poisson process. Find the probability that the toll collector will have to wait longer than 26.3 minutes before collecting the eighth toll
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Transcribed Image Text:**Problem 4:** Cars arrive at a tollbooth at a mean rate of 5 cars every 10 minutes according to a Poisson process. Find the probability that the toll collector will have to wait longer than 26.3 minutes before collecting the eighth toll.
In this problem, you'll apply properties of the Poisson process and the exponential distribution. The interarrival times for a Poisson process are exponentially distributed. To find the probability of waiting more than a certain amount of time for a fixed number of events (here, 8 toll collections), consider the Erlang distribution, which is a special case of the Gamma distribution.
### Key Concepts:
- **Poisson Process:** Describes events happening independently over time at a constant average rate.
- **Exponential Distribution:** Describes the time between events in a Poisson process.
- **Erlang Distribution:** A special case of the Gamma distribution used for the waiting time until the k-th event in a Poisson process.
### Steps to Solve:
1. **Understand the rate:** With 5 cars per 10 minutes, the rate \( \lambda \) is 0.5 cars per minute.
2. **Identify the distribution:** The waiting time for the 8th car follows an Erlang distribution with parameters \( k = 8 \) and rate \( \lambda = 0.5 \).
3. **Calculate the probability** using the cumulative distribution function (CDF) of the Erlang distribution to find the likelihood of waiting longer than 26.3 minutes.
Understanding and applying these concepts will help solve the problem effectively.
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