1. What is Poisson Distribution?

Definition: 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.

Purpose: It's used to model rare events that occur independently with a known constant mean rate, such as call arrivals, radioactive decay, or traffic accidents.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( P(x) \) — Probability of x events occurring (dimensionless)
• \( \lambda \) — Mean rate of occurrence (dimensionless)
• \( x \) — Number of events (dimensionless)
• \( e \) — Euler's number (~2.71828)

Explanation: The formula calculates the probability of observing exactly x events when the average rate is λ.

3. Applications of Poisson Distribution

Details: Commonly used in queuing theory, reliability analysis, and other fields where you need to predict the probability of rare events.

4. Using the Calculator

Tips: Enter the mean rate (λ) and the number of events (x) you want to calculate the probability for. Both must be ≥ 0.

5. Frequently Asked Questions (FAQ)

Q1: What's a typical value for λ?
A: λ can be any non-negative number. Small values (λ < 1) indicate rare events, while larger values indicate more frequent events.

Q2: What's the relationship between Poisson and Binomial distributions?
A: Poisson approximates Binomial when number of trials is large and probability is small.

Q3: When is Poisson distribution appropriate?
A: When events are independent, occur at a constant rate, and the probability of more than one event in a small interval is negligible.

Q4: What does P(x) represent?
A: The probability of observing exactly x events in the given interval.

Q5: How accurate is this calculator?
A: It provides exact Poisson probabilities up to floating-point precision limits.