Poisson Calculation

Poisson Distribution Calculator

Enter the average number of events occurring in a fixed interval of time or space.

Enter the specific number of events for which you want to calculate the probability (must be a non-negative integer).

function factorial(n) { if (n < 0) return NaN; if (n === 0) return 1; var result = 1; for (var i = 1; i <= n; i++) { result *= i; } return result; } function poissonPMF(lambda, k) { if (lambda < 0 || k =0) return numerator / denominator; } function calculatePoisson() { var lambdaInput = document.getElementById("lambda").value; var kInput = document.getElementById("k").value; var lambda = parseFloat(lambdaInput); var k = parseInt(kInput); var resultsDiv = document.getElementById("poissonResults"); resultsDiv.innerHTML = ""; // Clear previous results if (isNaN(lambda) || isNaN(k) || lambda < 0 || k < 0 || !Number.isInteger(k)) { resultsDiv.innerHTML = "Please enter a valid non-negative number for Lambda (average rate) and a non-negative integer for k (number of occurrences)."; return; } // Calculate P(X=k) var prob_X_eq_k = poissonPMF(lambda, k); // Calculate P(X<=k) var prob_X_le_k = 0; for (var i = 0; i <= k; i++) { prob_X_le_k += poissonPMF(lambda, i); } // Calculate P(X 0) { for (var i = 0; i =k) var prob_X_ge_k = 1 – prob_X_lt_k; // P(X>=k) = 1 – P(Xk) var prob_X_gt_k = 1 – prob_X_le_k; // P(X>k) = 1 – P(X<=k) resultsDiv.innerHTML += "

Calculation Results:

"; resultsDiv.innerHTML += "Probability P(X = " + k + "): " + (prob_X_eq_k * 100).toFixed(4) + "%"; resultsDiv.innerHTML += "Cumulative Probability P(X ≤ " + k + "): " + (prob_X_le_k * 100).toFixed(4) + "%"; resultsDiv.innerHTML += "Cumulative Probability P(X < " + k + "): " + (prob_X_lt_k * 100).toFixed(4) + "%"; resultsDiv.innerHTML += "Cumulative Probability P(X ≥ " + k + "): " + (prob_X_ge_k * 100).toFixed(4) + "%"; resultsDiv.innerHTML += "Cumulative Probability P(X > " + k + "): " + (prob_X_gt_k * 100).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 particularly useful for modeling rare events.

When to Use the Poisson Distribution

The Poisson distribution is applicable under specific conditions:

• Events are independent: The occurrence of one event does not affect the probability of another event occurring.
• Events occur at a constant average rate: The average rate (λ) of events is constant over the given interval.
• Events are rare: While not strictly a requirement, it's often used for events that occur infrequently.
• Fixed interval: The events are counted within a defined interval of time, space, or other dimension.

Common applications include modeling the number of customer calls to a call center per hour, the number of defects in a manufactured product per square meter, or the number of traffic accidents at an intersection per month.

Key Parameters: Lambda (λ) and k

• Lambda (λ): Average Rate of Occurrence
  This is the mean number of events that occur in the specified fixed interval. It's the most crucial parameter of the Poisson distribution. For example, if a call center receives an average of 10 calls per hour, then λ = 10.
• k: Number of Occurrences
  This is the specific number of events for which you want to calculate the probability. It must be a non-negative integer (0, 1, 2, 3, …). For instance, if you want to know the probability of receiving exactly 7 calls, then k = 7.

The Poisson Probability Mass Function (PMF)

The probability of observing exactly k events in an interval, given an average rate of λ, is calculated using the formula:

P(X=k) = (λ^k * e^-λ) / k!

Where:

• P(X=k) is the probability of exactly k occurrences.
• λ (lambda) is the average rate of occurrence.
• e is Euler's number (approximately 2.71828).
• k! is the factorial of k (k * (k-1) * … * 1).

How to Use This Calculator

1. Enter Average Rate (λ): Input the known average number of events that occur in your chosen interval. This value must be non-negative.
2. Enter Number of Occurrences (k): Input the specific number of events you are interested in. This must be a non-negative integer.
3. Click "Calculate": The calculator will then compute various probabilities based on your inputs.

Interpreting the Results

The calculator provides several key probabilities:

• P(X = k): The probability of observing exactly k events.
• P(X ≤ k): The cumulative probability of observing k or fewer events (i.e., P(X=0) + P(X=1) + … + P(X=k)).
• P(X < k): The cumulative probability of observing fewer than k events (i.e., P(X=0) + P(X=1) + … + P(X=k-1)).
• P(X ≥ k): The cumulative probability of observing k or more events (i.e., 1 – P(X < k)).
• P(X > k): The cumulative probability of observing more than k events (i.e., 1 – P(X ≤ k)).

Example Scenario

Imagine a small bakery that receives an average of 5 customer orders per hour during peak times. We want to understand the probability of receiving a certain number of orders in the next hour.

• Average Rate (λ) = 5
• Let's say we want to find the probability of receiving exactly 3 orders (k=3).

Using the calculator with λ=5 and k=3, you would get results similar to these:

• P(X = 3): Approximately 14.04% (The probability of receiving exactly 3 orders).
• P(X ≤ 3): Approximately 26.50% (The probability of receiving 3 or fewer orders).
• P(X ≥ 3): Approximately 87.53% (The probability of receiving 3 or more orders).

This information can help the bakery staff plan their resources, such as how many bakers to have on duty or how much raw material to prepare, based on the likelihood of different order volumes.