5.66: Poisson Probability Distribution | Exercise Solution of Probability & Statistics by Walpole

Introduction

This tutorial focuses on solving problems related to the Poisson probability distribution, specifically based on exercise problem 5.66 from the textbook "Probability and Statistics for Engineers and Scientists" by Walpole. Understanding the Poisson distribution is crucial for modeling events that occur randomly over a fixed period, such as aircraft arrivals. We will walk through the steps to calculate probabilities for small aircraft arrivals at an airport.

Step 1: Understanding the Poisson Distribution

The Poisson distribution is characterized by the average rate (λ) at which events occur in a fixed interval. In this exercise, the average arrival rate of small aircraft is given as 6 per hour. The Poisson parameter for arrivals over a period of t hours is defined as:

• μ = 6t

This means that for a 1-hour period, μ = 6.

Key Formula

The probability of observing exactly k events in a Poisson distribution is given by the formula:

[ P(X = k) = \frac{e^{-\mu} \mu^k}{k!} ]

Where:

• ( P(X = k) ) is the probability of k events in the interval.
• ( e ) is the base of the natural logarithm (approximately equal to 2.71828).
• ( \mu ) is the average number of events (6 for 1 hour).
• ( k ) is the number of events (e.g., 4 aircraft).

Step 2: Calculate the Probability of Exactly 4 Arrivals

To find the probability that exactly 4 small aircraft arrive in a 1-hour period:

1. Use the Poisson formula with ( k = 4 ) and ( \mu = 6 ): [ P(X = 4) = \frac{e^{-6} \cdot 6^4}{4!} ]

2. Calculate ( 4! ) (factorial of 4):

   • ( 4! = 4 \times 3 \times 2 \times 1 = 24 )

3. Substitute the values into the formula: [ P(X = 4) = \frac{e^{-6} \cdot 1296}{24} ]

4. Use a calculator to compute ( e^{-6} ) and finalize the probability.

Step 3: Calculate the Probability of At Least 4 Arrivals

To find the probability of at least 4 small aircraft arriving:

1. Calculate the complement of the event (0, 1, 2, or 3 arrivals): [ P(X \geq 4) = 1 - P(X < 4) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)) ]

2. Use the Poisson formula for each value of k (0, 1, 2, 3):

   • Calculate ( P(X = 0) ), ( P(X = 1) ), ( P(X = 2) ), and ( P(X = 3) ) using the same formula: [ P(X = k) = \frac{e^{-6} \cdot 6^k}{k!} ]

3. Sum these probabilities and subtract from 1 to find ( P(X \geq 4) ).

Step 4: Calculate the Probability of At Least 75 Arrivals in a 12-Hour Period

For a 12-hour working day:

1. Calculate the new Poisson parameter: [ \mu = 6 \times 12 = 72 ]

2. To find the probability of at least 75 arrivals: [ P(X \geq 75) = 1 - P(X < 75) ]

   • This often requires using a normal approximation if ( \mu ) is large, with the normal distribution having:

   • Mean (μ) = 72

   • Standard deviation (( \sigma )) = ( \sqrt{μ} = \sqrt{72} )

3. Use the Z-score formula to standardize: [ Z = \frac{X - \mu}{\sigma} ]

4. Look up the Z-value in the standard normal distribution table to find ( P(Z) ).

Conclusion

In this tutorial, we covered the Poisson probability distribution, calculated specific probabilities for aircraft arrivals, and discussed the use of normal approximation for larger sample sizes. Understanding these calculations is essential for effectively modeling random events in various fields, including logistics and operations management. For further practice, consider solving additional problems using different arrival rates and time periods.