Poisson Formula Calculator

Poisson Probability Formula:

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

Probability P(x):

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1. What is the Poisson Probability Formula?

Definition: The Poisson distribution calculates the probability of a given number of events occurring in a fixed interval of time or space.

Purpose: It's used for predicting probabilities of rare events in fields like physics, telecommunications, and reliability engineering.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( P(x) \) — Probability (dimensionless)
\( \lambda \) — Mean rate (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 Poisson Distribution

Details: The Poisson distribution is crucial for modeling rare events, queueing theory, and reliability analysis where events occur independently at a constant rate.

4. Using the Calculator

Tips: Enter the mean rate (λ) and the number of events (x) you want to calculate probability for. Both values must be ≥ 0.

5. Frequently Asked Questions (FAQ)

Q1: When should I use the Poisson distribution?
A: Use it when events are rare, independent, and occur at a known average rate.

Q2: What's a typical value for λ?
A: λ can be any non-negative number, but the distribution is most useful when λ is small (typically < 20).

Q3: Can x be a decimal?
A: No, x must be a non-negative integer (0, 1, 2, ...).

Q4: What if λ is very large?
A: For large λ (typically > 20), the normal distribution approximates the Poisson distribution well.

Q5: What does e represent in the formula?
A: e is Euler's number, the base of the natural logarithm (~2.71828).