Poisson Distribution Calculator

P(X = x):

P(X < x):

P(X ≤ x):

P(X > x):

P(X ≥ x):

1. What is the Poisson Distribution Calculator?

Definition: The Poisson Distribution Calculator computes the probability of a specific number of events occurring in a fixed interval, along with cumulative probabilities, for a Poisson distribution with a given rate of success.

Purpose: This tool helps analyze the likelihood of events occurring at a constant rate, such as phone calls at a call center or defects in manufacturing, useful in statistics, operations research, and risk assessment.

2. How Does the Calculator Work?

The calculator uses the following formulas:

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

\( P(X < x) = \sum_{k=0}^{x-1} \frac{e^{-\lambda} \lambda^k}{k!} \)

\( P(X \leq x) = \sum_{k=0}^{x} \frac{e^{-\lambda} \lambda^k}{k!} \)

\( P(X > x) = 1 - P(X \leq x) \)

\( P(X \geq x) = 1 - P(X < x) \)

Where:

\( x \): Number of occurrences;
\( \lambda \): Rate of success (average number of events);
\( P \): Probability of exactly x occurrences;
\( e \): Mathematical constant (~2.71828).

Steps:

Enter the number of occurrences (x) and rate of success (λ).
Calculate \( P(X = x) \) using the Poisson PMF formula.
Calculate \( P(X < x) \) by summing PMF from 0 to x-1.
Calculate \( P(X \leq x) \): \( P(X < x) + P(X = x) \).
Calculate \( P(X > x) \): \( 1 - P(X \leq x) \).
Calculate \( P(X \geq x) \): \( 1 - P(X < x) \).
Display results as percentages to four decimal places, using scientific notation for values less than 0.0001.

3. Importance of the Poisson Distribution Calculation

Calculating Poisson distribution probabilities is essential for:

Event Prediction: Models rare events like equipment failures or customer arrivals in fixed intervals.
Operations Research: Used in queuing theory and network traffic modeling.
Risk Assessment: Evaluates the likelihood of high-impact, low-probability events in finance and insurance.

4. Using the Calculator

Example: Calculate probabilities for a call center receiving an average of 5 calls per hour (λ = 5) with exactly 3 calls (x = 3):

Input: Number of Occurrences: 3; Rate of Success: 5.
\( P(X = 3) \): \( \frac{e^{-5} \cdot 5^3}{3!} \approx 0.1404 \).
\( P(X < 3) \): \( \sum_{k=0}^{2} \frac{e^{-5} \cdot 5^k}{k!} \approx 0.1247 \).
\( P(X \leq 3) \): \( \sum_{k=0}^{3} \frac{e^{-5} \cdot 5^k}{k!} \approx 0.2651 \).
\( P(X > 3) \): \( 1 - P(X \leq 3) \approx 0.7349 \).
\( P(X \geq 3) \): \( 1 - P(X < 3) \approx 0.8753 \).
Result: P(X = 3): 14.0400%; P(X < 3): 12.4700%; P(X ≤ 3): 26.5100%; P(X > 3): 73.4900%; P(X ≥ 3): 87.5300%.

5. Frequently Asked Questions (FAQ)

Q: What is a Poisson distribution?
A: A Poisson distribution models the probability of a given number of independent events occurring in a fixed interval of time or space with a constant mean rate [web:6].

Q: What does λ represent?
A: The parameter λ is the average rate of occurrence of events in the specified interval, equal to the mean and variance of the distribution.

Q: When should I use this calculator?
A: Use it to predict event frequencies in scenarios like call center operations, manufacturing defects, or rare event analysis in finance and science.