Poisson Distribution Calculator

Poisson Distribution Formula:

\[ P(k) = \frac{\lambda^k \times e^{-\lambda}}{k!} \]

Probability P(k):

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1. What is the Poisson Distribution?

The 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, given the average number of times the event occurs over that interval.

2. How Does the Calculator Work?

The calculator uses the Poisson distribution formula:

\[ P(k) = \frac{\lambda^k \times e^{-\lambda}}{k!} \]

Where:

\( \lambda \) — Average rate/mean number of events
\( k \) — Actual number of events
\( e \) — 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 events occur at a constant average rate λ.

3. Importance of Poisson Distribution

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

4. Using the Calculator

Tips: Enter the mean rate (λ) as a positive number and the number of events (k) as a non-negative integer. Both values must be valid.

5. Frequently Asked Questions (FAQ)

Q1: What types of events follow a Poisson distribution?
A: Events that are independent, occur at a constant average rate, and where the probability of more than one event in a small interval is negligible.

Q2: What are common applications of Poisson distribution?
A: Modeling call arrivals at a call center, number of mutations in DNA, insurance claims, website visits, and radioactive decay.

Q3: How does Poisson differ from binomial distribution?
A: Poisson models events over continuous intervals with known average rate, while binomial models successes in a fixed number of trials.

Q4: What are the limitations of Poisson distribution?
A: Assumes events are independent and occur at a constant rate. Not suitable for events that cluster or have varying rates.

Q5: When should I use Poisson distribution?
A: When you need to calculate the probability of a specific number of events occurring in a fixed interval with a known average rate.