1. What is a Poisson Function 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 probability theory and statistics to model rare events, such as call center arrivals, traffic flow, or equipment failures.

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)
• \( ! \) — Factorial function

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). Both must be non-negative values.

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 are independent.

Q2: What's a typical value for λ?
A: λ can be any non-negative number. Values between 0.1 and 10 are common in practical applications.

Q3: What if x is large (e.g., >20)?
A: The calculator handles large x values using efficient factorial computation methods.

Q4: Can λ be zero?
A: Yes, but then P(0) = 1 and P(x>0) = 0, meaning no events will occur.

Q5: How accurate are the results?
A: Results are accurate to 6 decimal places, sufficient for most statistical applications.