Problem 50 - Basic Statistics for Business and Economics with Student

Question

An internal study by the Technology Services department at Lahey Electronics revealed company employees receive an average of two emails per hour. Assume the arrival of these emails is approximated by the Poisson distribution. a. What is the probability Linda Lahey, company president, received exactly 1 email between 4 p.m. and 5 p.m. yesterday? b. What is the probability she received 5 or more email during the same period? c. What is the probability she did not receive any email during the period?

Step-by-Step Solution

Answer

(a) 0.2707; (b) 0.0527; (c) 0.1353.

Step 1: Understanding the Poisson Distribution

The Poisson distribution models the number of events (such as email arrivals) occurring within a fixed interval of time or space. We use it here because we are counting the number of emails receiving per hour. The key parameter of the Poisson distribution is the average rate (\(\lambda\)), which in this case is 2 emails per hour.

Step 2: Calculating Probability of Receiving Exactly 1 Email

To find the probability of receiving exactly 1 email, we use the Poisson probability formula: \[P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!}\] Here, \(\lambda = 2\) and \(k = 1\). Therefore, the probability is: \[P(X = 1) = \frac{e^{-2} \cdot 2^1}{1!} = \frac{2e^{-2}}{1} = 2e^{-2} \approx 0.2707\].

Step 3: Calculating Probability of Receiving 5 or More Emails

To find the probability of receiving 5 or more emails, we calculate the complement of receiving 0 to 4 emails, and subtract it from 1. \[P(X \geq 5) = 1 - \sum_{k=0}^{4} \frac{e^{-2} \cdot 2^k}{k!}\]. Computing these probabilities and summing them gives: \[P(X \geq 5) = 1 - (P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4))\]. After calculations: \[P(X \geq 5) \approx 1 - 0.9473 = 0.0527\]

Step 4: Calculating Probability of Receiving 0 Emails

To find the probability of receiving 0 emails, use the formula for \(k = 0\): \[P(X = 0) = \frac{e^{-2} \cdot 2^0}{0!} = e^{-2} \approx 0.1353\].

Step 5: Conclusion

The calculations show different scenarios for the Poisson distribution at an average rate of 2 emails per hour: Linda has a probability of 0.2707 of receiving exactly 1 email, a probability of 0.0527 of receiving 5 or more emails, and a probability of 0.1353 of receiving no email.

Key Concepts

Probability Calculation

Probability calculation is a key aspect of the Poisson distribution. In this context, it helps determine the likelihood of an event occurring a specific number of times within a fixed interval. When working with the Poisson distribution, the average rate, denoted by \( \lambda \), is crucial. It represents the average number of events expected in the interval. In our email arrival example, \( \lambda \) is 2, meaning on average, 2 emails arrive per hour.
To calculate probabilities for specific events, the Poisson probability formula is used:

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

This formula gives the probability of \( k \) events occurring, where \( k \) is a non-negative integer. Calculating this requires an understanding of factorials and the mathematical constant \( e \) (approximately 2.71828). For example, the probability of receiving exactly 1 email is determined by plugging \( k = 1 \) into the formula.
This step-by-step approach not only helps in calculating probabilities but also builds an understanding of the statistical framework guiding these calculations.

Statistical Modeling

Statistical modeling with the Poisson distribution is a powerful tool to represent real-world scenarios where events happen randomly over time. In this context, it simplifies complex problems by providing a mathematical framework to forecast event frequencies.
The Poisson model assumes:

• The average rate \( \lambda \) is constant over time.
• Events occur independently of each other.

These assumptions fit well with scenarios like the arrival of emails, where emails can come at any time independently. The predictability it offers is vital in decision-making processes and resource allocation, providing insights into expected workload or system demands. For instance, understanding that the expected email arrival rate is 2 per hour helps in planning bandwidth for server capacities or managing staff email workloads more effectively.
Thus, using statistical modeling in this manner aids in forming precise business strategies, optimizing operations, and improving service delivery.

Email Arrival Rate

The email arrival rate is a specific application of the Poisson distribution concept. In practical terms, it reflects the average number of emails an employee might expect to receive within a given time frame. In our scenario, this rate is 2 emails per hour.
By leveraging this rate, businesses can forecast email volumes, which can be particularly useful for resource planning and management. For instance, if a department knows the average email volume per day, it can plan employee shifts accordingly to ensure smooth operations without overload.
Understanding such arrival rates also helps in designing systems to handle peak periods efficiently without sacrificing performance. It ensures adequate server capacity is available, or that there is enough human resource allocation to manage correspondence. Additionally, knowing the variance in email arrival rates across different times can aid in scheduling staff breaks and handling customer queries in call centers more effectively.
Therefore, the email arrival rate is more than just a number; it's a foundational metric for operational efficiency and strategic planning.