1. What is a Poisson Process Calculator?

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

Purpose: It helps statisticians, engineers, and researchers model rare events like customer arrivals, radioactive decay, or system failures.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( P(x) \) — Probability (dimensionless)
• \( \lambda \) — Mean rate (dimensionless)
• \( x \) — Number of events (dimensionless)

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

3. Importance of Poisson Process

Details: Poisson processes model discrete events over continuous time, crucial for queueing theory, reliability analysis, and traffic engineering.

4. Using the Calculator

Tips: Enter the mean rate (λ) and desired number of events (x). Both must be ≥ 0. The calculator handles factorial computations automatically.

5. Frequently Asked Questions (FAQ)

Q1: When should I use Poisson distribution?
A: When events are independent, occur at a constant rate, and are rare (λ typically < 20).

Q2: What's a typical mean rate value?
A: Depends on context - could be 0.1 for rare events or 10 for frequent ones. Must be positive.

Q3: Can x be a decimal?
A: No, x must be a non-negative integer (0, 1, 2,...) as it represents event counts.

Q4: How accurate are the results?
A: Very accurate for λ < 20. For larger λ, normal approximation may be better.

Q5: What does e represent?
A: Euler's number (~2.71828), the base of natural logarithms.