Poisson Distribution With Mean Calculator

Poisson Distribution Formula:

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

Probability P(k):

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1. What is 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:

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

Where:

\( P(k) \) — Probability of k events occurring
\( \mu \) — Average number of events (mean)
\( k \) — Number of occurrences
\( e \) — Euler's number (~2.71828)
\( k! \) — Factorial of k

Explanation: The formula calculates the probability of observing exactly k events when the average rate is μ.

3. Applications of Poisson Distribution

Details: The Poisson distribution is commonly used in various fields including telecommunications, astronomy, biology, and quality control to model rare events such as call arrivals, radioactive decay, mutation counts, and defect detection.

4. Using the Calculator

Tips: Enter the mean (μ) as a positive number and k as a non-negative integer. Both values must be valid (μ ≥ 0, k ≥ 0).

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's the relationship between Poisson and binomial distributions?
A: The Poisson distribution can approximate the binomial distribution when the number of trials is large and the probability of success is small.

Q3: What are the limitations of Poisson distribution?
A: It assumes events are independent and the rate is constant, which may not hold in all real-world scenarios.

Q4: How is the variance related to the mean in Poisson distribution?
A: In Poisson distribution, the variance equals the mean (σ² = μ).

Q5: Can Poisson distribution handle decimal values for k?
A: No, k must be a non-negative integer (0, 1, 2, 3, ...) as it represents a count of events.