Probability Distribution Calculator

Poisson Distribution Formula:

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

Probability P(x):

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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 distribution formula:

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

Where:

\( P(x) \) — Probability of x events occurring (dimensionless)
\( \lambda \) — Average rate of occurrence (dimensionless)
\( x \) — Number of events (dimensionless)
\( e \) — Euler's number (~2.71828)
\( x! \) — Factorial of x

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

3. Importance of Probability Distributions

Details: Probability distributions like the Poisson are fundamental in statistics, used in fields from physics to finance to model random events like radioactive decay, call center arrivals, or rare disease cases.

4. Using the Calculator

Tips: Enter λ (must be positive) and x (non-negative integer). The calculator will compute the probability of exactly x events occurring.

5. Frequently Asked Questions (FAQ)

Q1: When should I use the Poisson distribution?
A: Use it when events occur independently, at a constant average rate, and you want the probability of a certain number of events in a fixed interval.

Q2: What are typical applications of Poisson distribution?
A: Modeling call center traffic, radioactive decay counts, website visits, rare disease incidence, or any low-probability events in large populations.

Q3: What's the relationship between Poisson and binomial distributions?
A: Poisson approximates binomial when number of trials is large and probability of success is small (λ = np).

Q4: What are the limitations of Poisson distribution?
A: It assumes events are independent and rate is constant. Not suitable for clustered or seasonal events.

Q5: How does λ affect the distribution shape?
A: As λ increases, the distribution becomes more symmetric and approaches normal distribution.