1. What is the Poisson Distribution?

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given a constant mean rate of occurrence (λ).

2. How Does the Calculator Work?

The calculator uses the Poisson probability formula:

Where:

• \( \lambda \) — Mean number of events (must be > 0)
• \( k \) — Number of event occurrences
• \( a \) — Lower bound of events
• \( b \) — Upper bound of events
• \( e \) — Euler's number (~2.71828)

Explanation: The formula calculates the cumulative probability of observing between a and b events (inclusive) given the mean rate λ.

3. Importance of Poisson Distribution

Details: The Poisson distribution is widely used in various fields including telecommunications, biology, finance, and quality control to model rare events and count data.

4. Using the Calculator

Tips: Enter the mean rate (λ > 0), lower bound (a ≥ 0), and upper bound (b ≥ a). All values must be valid numbers within their respective ranges.

5. Frequently Asked Questions (FAQ)

Q1: When should I use the Poisson distribution?
A: Use it when events occur independently, at a constant average rate, and you want to find the probability of a certain number of events in a fixed interval.

Q2: What are typical applications of Poisson distribution?
A: Modeling call center traffic, radioactive decay, website visitors per hour, manufacturing defects, and insurance claims.

Q3: What's the difference between Poisson and binomial distributions?
A: Binomial deals with fixed number of trials, while Poisson deals with events over continuous intervals with no fixed number of trials.

Q4: Are there limitations to the Poisson distribution?
A: It assumes events are independent and the mean rate is constant. It may not be suitable for clustered or seasonal events.

Q5: How accurate is the Poisson approximation?
A: It works well when the number of trials is large and the probability of success is small (rare events).