Poisson Probability Distribution Calculator

Poisson's Probability Distribution Formula:

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

Poisson Probability:

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1. What is Poisson Probability Distribution?

The Poisson Probability Distribution describes the probability of a given number of events occurring in a fixed interval of time or space, assuming events occur at a constant average rate and independently of the time since the last event.

2. How Does the Calculator Work?

The calculator uses the Poisson distribution formula:

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

Where:

\( P(X = x) \) — Probability of exactly x events occurring
\( \lambda \) — Average rate of events (events per interval)
\( x \) — Number of successes/events in the sample
\( e \) — Napier's constant (approximately 2.71828)
\( x! \) — Factorial of x

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

3. Applications of Poisson Distribution

Details: Poisson distribution is commonly used in various fields including telecommunications (call arrivals), healthcare (patient arrivals), manufacturing (defect counts), and natural phenomena (earthquake occurrences).

4. Using the Calculator

Tips: Enter the average rate of events (λ) and the number of successes (x) you want to calculate the probability for. Both values must be non-negative numbers.

5. Frequently Asked Questions (FAQ)

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

Q2: What are the assumptions of Poisson distribution?
A: Events occur independently, the average rate is constant, and two events cannot occur at exactly the same instant.

Q3: Can Poisson distribution handle decimal values for x?
A: No, the number of successes (x) must be a non-negative integer (0, 1, 2, 3, ...).

Q4: What's the relationship between Poisson and binomial distributions?
A: Poisson distribution can approximate binomial distribution when the number of trials is large and the probability of success is small.

Q5: How accurate is the Poisson approximation?
A: The approximation works well when λ is fixed, n is large, and p is small (typically n ≥ 20 and p ≤ 0.05).