Poisson distribution

Definition

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, under the condition that these events happen with a known constant mean rate and are independent of the time since the last event. This distribution is particularly useful for modeling rare events, such as the number of phone calls received at a call center in an hour or the number of accidents at a traffic intersection in a day.

5 Must Know Facts For Your Next Test

1. The Poisson distribution is defined for non-negative integer values, making it applicable for counting discrete events.
2. It can be used to approximate the binomial distribution when the number of trials is large, and the probability of success is small.
3. The mean and variance of a Poisson distribution are both equal to the rate parameter λ.
4. The formula for calculating the probability of observing k events is given by $$P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$$.
5. Common applications include modeling rare events like natural disasters, customer arrivals, and system failures.

Review Questions

• How does the Poisson distribution differ from the binomial distribution in terms of event occurrence and trial structure?
  • The Poisson distribution models the probability of a given number of events occurring in a fixed interval while assuming that these events happen independently at a constant mean rate. In contrast, the binomial distribution describes the number of successes in a fixed number of independent trials, each with a set probability of success. The Poisson is more suitable for rare events over time or space, while the binomial is focused on success/failure outcomes across multiple trials.
• Explain how you would use the Poisson distribution to model real-world situations and provide an example.
  • To use the Poisson distribution for modeling real-world situations, identify an event that occurs independently over a specified interval and can be counted. For example, if you wanted to model the number of emails received per hour at a business, you would collect data on the average number of emails received during that time frame to determine λ. With this rate parameter, you could calculate probabilities for various counts of emails using the Poisson formula.
• Evaluate how understanding both Poisson and binomial distributions can improve decision-making in business operations.
  • Understanding both Poisson and binomial distributions enhances decision-making in business by allowing for better modeling of different types of events. For instance, using Poisson for rare events like equipment failures helps businesses plan for maintenance schedules effectively. Conversely, applying binomial distribution can aid in determining outcomes from marketing campaigns with fixed numbers of customers contacted. By leveraging these distributions together, businesses can make informed strategic choices based on probabilistic outcomes and risk assessment.

Related terms

Exponential Distribution: A probability distribution that describes the time between events in a Poisson process, where events occur continuously and independently at a constant average rate.

Binomial Distribution: A probability distribution that represents the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

Rate Parameter (λ): In the context of the Poisson distribution, λ (lambda) represents the average number of events in a specified interval, serving as a key parameter to determine the probabilities.