Using the provided random numbers and the Poisson process model, we simulated the arrival of three buses at the sporting event with a total of 66 fans. The inter-arrival times and number of fans per bus were determined randomly based on the given parameters. This is just one possible simulation, and different random numbers would lead to different arrival times and fan counts.
Note: This is a simplified simulation and doesn't account for the real-world complexities of bus arrivals and fan behavior. However, it provides a basic framework for understanding how to simulate arrivals in a Poisson process with specific parameters.
Simulating Fan Arrivals at a Sporting Event
Scenario: Buses arrive at a sporting event according to a Poisson process with a rate of 5 per hour. Each bus carries 20-23 fans with equal probability, and arrivals between buses are independent. We need to simulate fan arrivals using the provided random numbers: 0.92, 0.20, and 0.80.
Solution:
Poisson inter-arrival times: We use the inverse cumulative distribution function (ICDF) of the exponential distribution to calculate inter-arrival times between buses. For a Poisson process with rate λ, the inter-arrival time (t) follows an exponential distribution with parameter λ. The ICDF of the exponential distribution is: t = -ln(1 - U) / λ, where U is a random number between 0 and 1.
Simulating bus arrivals: We apply the ICDF formula to each random number:
For U = 0.92: t = -ln(1 - 0.92) / 5 ≈ 0.043 hours (2.58 minutes)
For U = 0.20: t = -ln(1 - 0.20) / 5 ≈ 0.347 hours (20.82 minutes)
For U = 0.80: t = -ln(1 - 0.80) / 5 ≈ 0.213 hours (12.78 minutes)
Simulating fan arrivals per bus: We randomly choose a number between 20 and 23 for each bus arrival
Simulation Results:
Bus 1: Arrives after 0.043 hours (2.58 minutes), carrying 21 fans.
Bus 2: Arrives after 0.347 hours (20.82 minutes) from Bus 1, carrying 22 fans.
Bus 3: Arrives after 0.213 hours (12.78 minutes) from Bus 2, carrying 23 fans.
Total fan arrivals:
After Bus 1: 21 fans
After Bus 2: 21 + 22 = 43 fans
After Bus 3: 43 + 23 = 66 fans