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, provided these events occur with a known constant mean rate and are independent of the time since the last event. It is often used for counting occurrences of rare events in a large population, making it particularly useful in fields like telecommunications, traffic flow, and epidemiology.

5 Must Know Facts For Your Next Test

1. The Poisson distribution is defined by its parameter extlambda, which represents the average number of events in the given interval.
2. It can be applied to various scenarios like the number of phone calls received by a call center in an hour or the number of emails received in a day.
3. For large values of extlambda, the Poisson distribution closely approximates a normal distribution, allowing for simpler calculations.
4. The formula for calculating the probability of observing exactly k events is given by $$P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$$ where e is Euler's number (approximately 2.71828).
5. As extlambda increases, the Poisson distribution becomes more symmetric and resembles a bell curve.

Review Questions

• How does the Poisson distribution relate to real-world situations, and what types of events are best modeled using this distribution?
  • The Poisson distribution is particularly effective for modeling rare events that happen independently within a specified interval. Real-world situations like the number of accidents at a traffic intersection over a month or customer arrivals at a store in an hour fit this model well. It captures instances where events occur randomly but with a predictable average rate, allowing researchers and analysts to assess probabilities effectively.
• In what way does the Poisson distribution differ from the binomial distribution, and when would you choose one over the other?
  • The Poisson distribution is used when counting the number of events occurring over a continuous interval while the binomial distribution applies to scenarios with a fixed number of trials and two possible outcomes (success or failure). You would choose the Poisson distribution when dealing with rare events where you are interested in counts over time or space without a defined number of trials, while the binomial would be appropriate for fixed trials such as flipping a coin multiple times.
• Evaluate how changes in the parameter extlambda affect the shape and characteristics of the Poisson distribution.
  • Changes in extlambda significantly impact the Poisson distribution's shape. As extlambda increases, both the mean and variance increase, resulting in a distribution that becomes more symmetric and approaches normality. A small extlambda leads to a skewed right distribution where low counts are more probable, indicating that fewer events occur. This understanding helps analysts determine how likely various event counts are based on different average rates.

Related terms

Exponential distribution: A continuous probability distribution that describes the time between events in a Poisson process, often used to model waiting times.

Mean:

The average value of a set of data points, representing the central tendency, which is also the parameter ( extlambda) of the Poisson distribution.

Binomial distribution: A discrete probability distribution that represents the number of successes in a fixed number of independent Bernoulli trials, contrasting with the Poisson distribution that deals with count data over intervals.