Poisson Distribution Calculator

Poisson Distribution Calculator

The average number of events in a given interval (must be > 0).

The specific number of events you want to calculate the probability for (integer ≥ 0).

Calculation Results:

P(X = x):
Probability of exactly x events

P(X ≤ x):
Cumulative probability of x or fewer events

P(X > x):
Probability of more than x events

Understanding 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. These events must occur with a known constant mean rate and independently of the time since the last event.

The Poisson Formula

P(x; λ) = (e^-λ * λ^x) / x!

• P(x; λ): The probability of exactly x occurrences.
• λ (Lambda): The average number of occurrences per interval.
• e: Euler's number (approximately 2.71828).
• x: The number of occurrences (0, 1, 2, …).
• x!: The factorial of x.

Practical Examples

This calculator is essential for various fields including finance, engineering, and logistics. Common use cases include:

• Call Centers: Estimating the number of incoming calls per hour to manage staffing.
• Network Traffic: Calculating the probability of a certain number of data packets arriving at a router.
• Retail: Predicting customer arrivals during a specific time block.
• Health: Modeling the occurrence of rare diseases in a specific population over time.

Real-World Case Study

Suppose a bakery receives an average of 4 customers per hour (λ = 4). You want to know the probability of exactly 2 customers (x = 2) arriving in the next hour.

Using the Poisson formula: P(2; 4) = (e^-4 * 4²) / 2! = (0.0183 * 16) / 2 = 0.1465. There is a 14.65% chance that exactly 2 customers will arrive.

function getFactorial(n) { if (n === 0 || n === 1) return 1; var res = 1; for (var i = 2; i <= n; i++) { res *= i; } return res; } function calculatePoisson() { var lambda = parseFloat(document.getElementById("lambdaInput").value); var x = parseInt(document.getElementById("xInput").value); if (isNaN(lambda) || isNaN(x) || lambda <= 0 || x < 0) { alert("Please enter valid numbers. Lambda must be greater than 0 and x must be 0 or greater."); return; } // P(X = x) var p_exact = (Math.pow(lambda, x) * Math.exp(-lambda)) / getFactorial(x); // P(X <= x) var p_cumulative = 0; for (var i = 0; i x) var p_greater = 1 – p_cumulative; // Update UI document.getElementById("exactResult").innerText = (p_exact * 100).toFixed(4) + "% (" + p_exact.toFixed(6) + ")"; document.getElementById("lessEqualResult").innerText = (p_cumulative * 100).toFixed(4) + "% (" + p_cumulative.toFixed(6) + ")"; document.getElementById("greaterResult").innerText = (p_greater * 100).toFixed(4) + "% (" + p_greater.toFixed(6) + ")"; document.getElementById("resultsArea").style.display = "block"; }