Question

What application does the Poisson distribution have other than to estimate certain binomial probabilities?

Short answer

The Poisson distribution can be applied in a variety of real-world scenarios. One example is modeling the number of incoming calls at a call center during a certain time period. To do this, we can use the Poisson probability mass function with the average number of calls per hour (λ) as a parameter. By calculating the probability of receiving a fixed number or a range of calls in an hour using the Poisson distribution, call center management can effectively allocate resources, anticipate peak hours, and improve overall customer experience. This example highlights a practical application of the Poisson distribution that extends beyond estimating binomial probabilities.

Step-by-step solution

Example Application of Poisson Distribution: Call Center

In a call center, the number of incoming calls during a certain time period is often modeled by the Poisson distribution. This occurs because calls may come in randomly and independently, and are not limited by a fixed number of trials (unlike a binomial distribution).

Identify Parameters

For our call center example, we can use the average number of calls per hour received (λ) as a parameter for the Poisson distribution. To better understand how λ is used in the Poisson distribution, suppose a call center receives an average of 30 calls per hour. In this case, λ would be equal to 30.

Use Poisson Formula

The probability mass function of Poisson distribution is given by: P(X = k) = \frac{e^{-λ} λ^k}{k!} Where P(X = k) is the probability of receiving k calls in an hour, λ is the average number of calls per hour, and e is the base of the natural logarithm (approximately equal to 2.71828).

Calculate Probability of a Fixed Number of Calls

Suppose we want to find the probability of receiving exactly 35 calls in an hour at the call center. Plug in the values λ = 30, k = 35 in the Poisson probability mass function formula: P(X = 35) = \frac{e^{-30} * 30^{35}}{35!} Calculate the result to find the probability of receiving exactly 35 calls in an hour.

Calculate Probability of a Range of Calls

In some scenarios, it might be useful to calculate the probability of receiving a range of calls in an hour. For instance, suppose we want to know the probability of receiving between 25 and 40 calls in an hour. To do this, we will calculate the probabilities for each number of calls between 25 and 40 and sum them up: P(25 ≤ X ≤ 40) = Σ[P(X = k)] for k = 25 to 40 Compute the probabilities for each value of k and sum them to find the overall probability of receiving between 25 and 40 calls in an hour. Using the Poisson distribution, the call center management can better allocate resources, anticipate peak hours, and improve the overall customer experience. This example demonstrates just one of many real-world applications of the Poisson distribution outside of estimating binomial probabilities.

Key concepts

Probability Mass Function

The probability mass function (PMF) is a foundational concept in probability theory, particularly when dealing with discrete random variables. It describes the probability that a discrete random variable is exactly equal to some value.

For the Poisson distribution, the PMF is particularly crucial as it provides a formula to calculate the probability of observing a fixed number of events within a specific interval. Here, events occur with a constant mean rate and independently of the time since the last event. The PMF for Poisson distribution is mathematically denoted as:
\[P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}\]
where:

• \(e\) is the base of natural logarithms,
• \(\lambda\) is the average rate of success per interval,
• \(k\) is the number of occurrences of an event,
• \(k!\) stands for \(k\) factorial.

Understanding and applying the PMF allows students to solve real-world problems where the Poisson distribution is applicable, such as calculating the probability of a specific number of calls in a call center during an hour.

Average Rate of Success (λ)

The average rate of success, denoted by the Greek letter \(\lambda\) (lambda), is integral to the Poisson distribution. It represents the expected number of times an event occurs in a fixed interval of time or space.

For instance, if a call center on average receives 30 calls per hour, we say \(\lambda = 30\). This rate must be constant over the observed period. It's important to realize that the actual number of events that occur in different intervals can vary, but when averaged over time, they should converge towards \(\lambda\).

Choosing the correct value for \(\lambda\) is crucial, as it heavily influences the outcome of Poisson probability calculations. It allows businesses or researchers to predict the occurrence of events and plan accordingly, ensuring operational efficiency and preparedness for varying demand levels.

Calculating Poisson Probabilities

Calculating Poisson probabilities involves determining the chance of a given number of events happening in a set period. This is practically useful in a wide range of scenarios where managers need to make informed decisions based on expected occurrences.

With \(\lambda\) known, calculating the probability for an exact number of events (\(k\)) utilizes the PMF:
\[P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}\]
For example, to find the probability of exactly 35 calls in a call center when the average rate \(\lambda = 30\), we plug in the respective values into the PMF formula.

Beyond single events, calculating the likelihood of a range (e.g., between 25 and 40 calls) involves summing individual probabilities for all integers within that range. This requires the application of the PMF multiple times, once for each integer value, and compiling the results to obtain an overall probability. This computation allows for better resource allocation, anticipating high-traffic periods, and can drive improvements for customer service and satisfaction.