Poisson Distribution Calculator

Poisson Distribution Calculator

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

The exact number of events you want to find the probability for (k ≥ 0, integer).

Enter values and click "Calculate Probability"

function factorial(n) { if (n < 0) { return NaN; // Factorial is not defined for negative numbers } if (n === 0) { return 1; } var result = 1; for (var i = 1; i <= n; i++) { result *= i; } return result; } function calculatePoisson() { var lambdaInput = document.getElementById("averageRate").value; var kInput = document.getElementById("numEvents").value; var resultDiv = document.getElementById("poissonResult"); var lambda = parseFloat(lambdaInput); var k = parseInt(kInput); if (isNaN(lambda) || isNaN(k)) { resultDiv.innerHTML = "Please enter valid numbers for both fields."; return; } if (lambda < 0) { resultDiv.innerHTML = "Average Rate (λ) must be non-negative."; return; } if (k < 0 || !Number.isInteger(k)) { resultDiv.innerHTML = "Number of Events (k) must be a non-negative integer."; return; } // Handle the special case where lambda is 0 if (lambda === 0) { if (k === 0) { resultDiv.innerHTML = "The probability P(X=0) is: 1.0000"; } else { resultDiv.innerHTML = "The probability P(X=" + k + ") is: 0.0000"; } return; } var euler = Math.E; // Euler's number var lambda_pow_k = Math.pow(lambda, k); var e_pow_neg_lambda = Math.exp(-lambda); var k_factorial = factorial(k); if (k_factorial === 0) { // Should not happen for k >= 0, but as a safeguard resultDiv.innerHTML = "Error: Factorial calculation resulted in zero."; return; } var probability = (lambda_pow_k * e_pow_neg_lambda) / k_factorial; resultDiv.innerHTML = "The probability P(X=" + k + ") is: " + probability.toFixed(4) + ""; }

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 if these events occur with a known constant mean rate and independently of the time since the last event. It's widely used in various fields, including statistics, physics, engineering, and biology, to model rare events.

When to Use the Poisson Distribution?

You would typically use the Poisson distribution when you are counting the number of times an event occurs in a fixed interval, and:

• The events occur independently.
• The average rate of events (λ) is constant over the interval.
• Two events cannot occur at exactly the same instant.
• The probability of an event occurring in a very short interval is proportional to the length of the interval.

The Poisson Formula

The probability mass function for the Poisson distribution is given by:

P(X=k) = (λ^k * e^-λ) / k!

Where:

• P(X=k): The probability of observing exactly k events.
• λ (lambda): The average rate of events (the mean number of events in the given interval). This is a positive real number.
• k: The actual number of observed events (a non-negative integer: 0, 1, 2, …).
• e: Euler's number, approximately 2.71828.
• k!: The factorial of k (k! = k × (k-1) × … × 2 × 1). Note that 0! = 1.

Practical Examples

Here are a few scenarios where the Poisson distribution is applicable:

• Customer Service: The number of calls received by a call center per hour.
• Manufacturing: The number of defects in a product per square meter.
• Biology: The number of mutations in a given stretch of DNA after exposure to radiation.
• Traffic Management: The number of cars passing a certain point on a road in a 10-minute interval.
• Web Analytics: The number of visitors to a website per minute.

How to Use the Calculator

Our Poisson Distribution Calculator simplifies the process of finding probabilities:

1. Average Rate (λ): Enter the known average number of events that occur in your specified interval. For example, if a call center receives an average of 5 calls per hour, enter '5'.
2. Number of Events (k): Enter the specific number of events for which you want to calculate the probability. If you want to know the probability of receiving exactly 3 calls in an hour, enter '3'.
3. Calculate Probability: Click the "Calculate Probability" button. The calculator will instantly display P(X=k), the probability of observing exactly 'k' events.

This tool helps you quickly understand the likelihood of specific outcomes in situations governed by the Poisson process, without needing to manually perform complex factorial and exponential calculations.