1. What is a Poisson Distribution Calculator?

Definition: This calculator computes the probability of a given number of events occurring in a fixed interval using the Poisson distribution.

Purpose: It's used in statistics to model rare events, such as call center arrivals, accident rates, or natural phenomena occurrences.

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)
• \( x! \) — Factorial of x

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

3. Importance of Poisson Distribution

Details: The Poisson distribution is crucial for modeling count data where events occur independently at a constant average rate.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: When should I use the Poisson distribution?
A: Use it when modeling rare events with a known average rate, where events occur independently of each other.

Q2: What's a typical value for λ?
A: λ can be any non-negative number. In practice, values between 0.1 and 10 are common, but it depends on your specific scenario.

Q3: What if x is a large number?
A: The calculator can handle large x values (up to 170 for factorial calculation), but probabilities become very small for large x when λ is modest.

Q4: How is this different from binomial distribution?
A: Poisson is for rare events with no fixed number of trials, while binomial is for a fixed number of trials with success probability.

Q5: Can I calculate cumulative probabilities?
A: This calculator gives exact probabilities. For cumulative probabilities (P(X≤x)), you would need to sum individual probabilities.