Poisson Function Calculator

Poisson Probability Formula:

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

P(k) (decimal):

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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 if 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 probability formula:

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

Where:

\( P(k) \) — Probability of k events occurring
\( \lambda \) — Average rate of occurrence (mean)
\( k \) — Number of occurrences
\( e \) — 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. Applications of Poisson Distribution

Details: The Poisson distribution is commonly used in various fields including telecommunications, traffic flow analysis, quality control, insurance, and epidemiology to model rare events occurring over time or space.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is the range of possible values for Poisson probability?
A: Poisson probability values range from 0 to 1, where 0 means impossible and 1 means certain.

Q2: When is the Poisson distribution appropriate?
A: It's appropriate when events are independent, the average rate is constant, and two events cannot occur at exactly the same instant.

Q3: What is the relationship between Poisson and binomial distributions?
A: The Poisson distribution can be derived as a limiting case of the binomial distribution when the number of trials is large and the probability of success is small.

Q4: What are the mean and variance of a Poisson distribution?
A: Both the mean and variance of a Poisson distribution are equal to λ.

Q5: Can λ be a decimal value?
A: Yes, λ can be any non-negative real number, including decimal values.