Poisson Distribution Calculator

How to Use the Poisson Distribution Calculator

Introduction

The Poisson Distribution Calculator is designed to help you calculate probabilities and statistical metrics associated with a Poisson distribution. This calculator can determine the probability of a given number of events occurring in a fixed interval, as well as cumulative probabilities, mean, variance, and standard deviation.

Step-by-Step Guide

Step 1: Enter Input Values

• Mean Rate (λ): Locate the input field labeled “Mean Rate (λ)”. Enter the average number of events in the specified interval. Ensure that the value is between 0.0001 and 1000, as per the validation constraints. The input accepts decimal numbers with up to four decimal places.
• Number of Events (k): Find the input field labeled “Number of Events (k)”. Input the specific number of events you are interested in. This value should be a whole number between 0 and 1000.

Step 2: Calculate Probability

Once the values for λ and k have been entered, the calculator will automatically compute the following results:

• Probability P(X = k): This is the probability of exactly k events occurring within the interval. It is calculated using the formula (λ^k * e^(-λ)) / k!, and the result is displayed with six decimal accuracy.

Step 3: Calculate Additional Metrics

The calculator also provides additional statistical insights:

• Cumulative Probability P(X ≤ k): This value represents the probability of k or fewer events occurring. It is computed by summing the probabilities from 0 to k.
• Mean (λ): The mean of the distribution, which is simply the λ value you input. It indicates the average rate of event occurrence.
• Variance: Same as the mean (λ), the variance indicates how much variability is expected in the number of events.
• Standard Deviation: This is the square root of the variance (λ), providing insight into the spread of event occurrences around the mean.

Conclusion

By following these simple steps, you can effectively use the Poisson Distribution Calculator to understand the statistical characteristics of events in a Poisson process. Whether you are working on academic problems or practical applications, this tool provides accurate probabilities and metrics to support your analyses.