1. What is the Poisson Formula?

The Poisson formula calculates the probability of a given number of events occurring in a fixed interval of time or space, assuming these events occur with a known constant mean rate and independently of the time since the last event.

2. How Does the Calculator Work?

The calculator uses the Poisson formula:

Where:

• \( P \) — Probability (decimal)
• \( \lambda \) — Average rate of occurrence (unitless)
• \( k \) — Number of occurrences (unitless)
• \( e \) — Euler's number (approximately 2.71828)
• \( k! \) — Factorial of k

Explanation: The formula calculates the probability of exactly k events occurring in a fixed interval when the events occur independently at a constant average rate λ.

3. Importance of Poisson Distribution

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

4. Using the Calculator

Tips: Enter λ (average rate) as a positive number, and k (number of occurrences) as a non-negative integer. Both values must be valid (λ ≥ 0, k ≥ 0).

5. Frequently Asked Questions (FAQ)

Q1: What types of events follow a Poisson distribution?
A: Events that are rare, independent, and occur at a constant average rate, such as phone calls received per hour, radioactive decay events, or website visits per minute.

Q2: What are the assumptions of the Poisson distribution?
A: Events are independent, the average rate is constant, and two events cannot occur at exactly the same instant.

Q3: When should I use Poisson instead of binomial distribution?
A: Use Poisson when the number of trials is large and the probability of success is small (typically n ≥ 100 and p ≤ 0.01).

Q4: Can λ be a decimal value?
A: Yes, λ can be any non-negative real number representing the average rate of occurrence.

Q5: What does k! mean in the formula?
A: k! (k factorial) is the product of all positive integers up to k. For example, 4! = 4 × 3 × 2 × 1 = 24. By definition, 0! = 1.