Calculate probabilities related to events occurring randomly over time or space.
The average number of events per unit of time/space.
The duration or extent of the period being considered.
The exact number of events you want to calculate the probability for.
Probability P(X=k) =
Understanding the Poisson Process and Calculator
A Poisson process is a mathematical model used to describe events that occur randomly and independently over time or space. The key characteristic is that the average rate of occurrence is constant. Examples include:
The number of customers arriving at a store per hour.
The number of phone calls received by a call center per minute.
The number of defects found in a certain length of fabric.
The number of radioactive decay events detected by a counter in a given second.
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 average rate and independently of the time since the last event.
The formula for the Poisson probability mass function is:
P(X=k) = (λt)^k * e^(-λt) / k!
Where:
P(X=k) is the probability of exactly k events occurring.
λ (lambda) is the average rate of events per unit of time or space.
t is the length of the time or space interval being considered.
λt is the expected number of events in the interval (often denoted by μ).
e is the base of the natural logarithm (approximately 2.71828).
k! is the factorial of k (k! = k * (k-1) * … * 2 * 1), with 0! defined as 1.
How this calculator works:
You input the Average Rate (λ), which is the expected number of events per unit.
You input the Time/Space Period (t), the duration or extent of your observation.
You input the specific Number of Events (k) for which you want to find the probability.
The calculator computes λt, the expected number of events in the given period.
It then applies the Poisson probability formula to calculate P(X=k).
This calculator helps in understanding the likelihood of observing a specific number of random events within a defined context, which is invaluable for risk assessment, resource allocation, and performance analysis in various fields.
// Function to calculate factorial
function factorial(n) {
if (n < 0) {
return NaN; // Factorial is not defined for negative numbers
}
if (n === 0 || n === 1) {
return 1;
}
var result = 1;
for (var i = 2; i <= n; i++) {
result *= i;
}
return result;
}
function calculatePoissonProbability() {
var lambda = parseFloat(document.getElementById("lambda").value);
var timePeriod = parseFloat(document.getElementById("timePeriod").value);
var k = parseInt(document.getElementById("k").value); // k should be an integer
var resultDisplay = document.querySelector("#result span");
// Clear previous results and styles
resultDisplay.textContent = "";
document.getElementById("result").style.borderColor = "#004a99";
// Input validation
if (isNaN(lambda) || lambda < 0) {
alert("Please enter a valid non-negative average rate (λ).");
return;
}
if (isNaN(timePeriod) || timePeriod < 0) {
alert("Please enter a valid non-negative time/space period (t).");
return;
}
if (isNaN(k) || k < 0 || !Number.isInteger(k)) {
alert("Please enter a valid non-negative integer for the number of events (k).");
return;
}
var lambdaT = lambda * timePeriod; // Expected number of events in the interval
// Calculate factorial of k
var kFactorial = factorial(k);
if (isNaN(kFactorial)) {
alert("Could not calculate factorial for k. Please ensure k is a non-negative integer.");
return;
}
// Calculate the probability using the Poisson formula
// P(X=k) = (λt)^k * e^(-λt) / k!
var probability = (Math.pow(lambdaT, k) * Math.exp(-lambdaT)) / kFactorial;
if (isNaN(probability) || !isFinite(probability)) {
alert("Calculation resulted in an invalid number. Please check your inputs.");
return;
}
// Display the result
resultDisplay.textContent = probability.toFixed(6); // Display with 6 decimal places
document.getElementById("result").style.borderColor = "#28a745"; // Success color
}