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Chapter 3: Problem 87

The number of requests for assistance received by a towing service is a Poisson process with rate \(\alpha=4\) per hour. a. Compute the probability that exactly ten requests are received during a particular 2-hour period. b. If the operators of the towing service take a 30 -min break for lunch, what is the probability that they do not miss any calls for assistance? c. How many calls would you expect during their break?

Short Answer

Expert verified
a. Use Poisson formula for 2-hour interval (位=8). b. Use Poisson with 位=2 for 30-min interval, compute P(X=0). c. Expect 2 calls (using 位=2 for 30-min).

Step by step solution

01

Identify the Poisson Process Parameter for 2-hour Period

The rate of requests per hour is given as \( \alpha = 4 \). For a 2-hour period, the equivalent rate is \( \lambda = 2 \times 4 = 8 \). This is the average number of requests expected during the 2-hour period.
02

Calculate Probability for Exactly Ten Requests

The probability of receiving exactly \( k = 10 \) requests in a 2-hour period is given by the Poisson probability formula:\[P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}\]Substituting the values, we get:\[P(X = 10) = \frac{e^{-8} \times 8^{10}}{10!}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Calculation
Probability calculation in the context of a Poisson process involves finding the likelihood of a certain number of events occurring within a defined period. In our exercise, the Poisson process is defined with a rate of 4 requests per hour. However, since we're looking at a 2-hour time frame, the average or effective rate becomes 8 requests.

To find the probability of receiving exactly 10 requests, we use the Poisson probability formula:
  • Calculate the probability of exactly k events in a given time period with the formula:
    \[P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}\]
  • Here, \(\lambda\) is the mean number of events, \(k\) is the number of events we're calculating the probability for, and \(e\) is the base of the natural logarithm.
  • In our example, when you substitute \(\lambda = 8\), and \(k = 10\), you calculate the probability of exactly 10 requests over 2 hours.
This calculation gives insight into how likely a rare occurrence, such as exactly 10 requests, is to happen given a known average rate.
Expected Value
In a Poisson process, the expected value provides the average number of events expected over a certain period. For our towing service example, this is crucial in planning resources.

The expected value for calls during a 30-minute lunch break can directly impact the service's operational planning.
  • First, consider the rate parameter \(\alpha\) as the average number of calls per hour. Here, \(\alpha = 4\).
  • For a 30-minute or 0.5-hour period, multiply the hourly rate by 0.5 to find the expected number of events (calls): \[E(X) = \alpha \times \text{time period} = 4 \times 0.5 = 2\]
Thus, during a 30-minute lunch, we expect around 2 calls. This expected value allows the towing service to anticipate the workload and adjust staffing or break times accordingly.
Rate Parameter
The rate parameter or rate of occurrence, denoted as \(\alpha\), is vital in defining a Poisson process. It represents the average number of events happening in a fixed unit of time. In our towing service example, this rate is given as 4 calls per hour.

Understanding the rate parameter is essential for calculating probabilities and expected values. Here鈥檚 how it works:
  • The rate parameter is context-specific, giving the average rate of calls in this exercise. It's essential to scale this parameter up or down based on changes in time units (e.g., from hours to half-hours as in the lunch break scenario).
  • Calculating over different periods involves multiplying the rate by the time period, illustrating how the expected frequency adapts based on varying durations.
  • This is why the rate parameter is crucial for decision-makers in a business, informing risk assessments and operational strategies.
A firm grasp of the rate parameter ensures you're equipped to solve other Poisson-related problems by accurately interpreting and applying this foundational component.

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