Skellam Distribution Calculator

Mean

μ_{1}

Mean

μ_{2}

Probability type:

Difference

k

Upper bound

k_{2}

Output type Chart Table

Decimal Places: 2 3 5 8

Clear Reset

Introduction

The Skellam distribution calculator is designed to analyse the difference between two independent Poisson-distributed variables. Researchers exploring discrete probability can determine the likelihood of specific differences $k$ occurring when comparing event frequencies. This statistical tool is essential for modelling scenarios where the net outcome is the subtraction of two random counts, each defined by its own mean rate $μ_{1}$ and $μ_{2}$ .

What this calculator does

This tool performs a probability mass function analysis for the Skellam distribution. It requires two positive mean parameters, $μ_{1}$ and $μ_{2}$ , and a difference value $k$ . The calculator outputs the exact probability for a specific difference, cumulative probabilities for ranges, the distribution mean, and the variance. Results are presented through detailed calculation steps and visualised via a frequency chart or a data table.

Formula used

The probability mass function of the Skellam distribution for an integer $k$ is calculated using modified Bessel functions of the first kind. The distribution mean is the difference of the individual means, while the variance is their sum. Variable $I_{|k|}$ represents the modified Bessel function, and $μ_{1}$ and $μ_{2}$ are the expected frequencies of the independent Poisson processes.

P (K = k) = e^{- (μ_{1} + μ_{2})} {(\frac{μ_{1}}{μ_{2}})}^{k} I_{|k|} (2 \sqrt{μ_{1} μ_{2}})

Mean = μ_{1} - μ_{2}, Variance = μ_{1} + μ_{2}

How to use this calculator

1. Enter the mean values $μ_{1}$ and $μ_{2}$ for the two independent processes.
2. Select the probability type, such as equal to, less than, or between specific bounds.
3. Input the difference value $k$ and any required upper bounds.
4. Execute the calculation to view the probability results, distribution statistics, and visual charts.

Example calculation

Scenario: A researcher in sports analysis examines the difference in points scored between two teams, where Team A averages 5 points and Team B averages 3 points per session.

Inputs: $μ_{1} = 5$ , $μ_{2} = 3$ , and $k = 1$ .

Working:

Step 1: $Mean = μ_{1} - μ_{2}$

Step 2: $5 - 3$

Step 3: $2$

Step 4: $Variance = 5 + 3 = 8$

Result: P(K = 1) is approximately 0.14.

Interpretation: There is a 14% probability that Team A will outscore Team B by exactly one point.

Summary: The calculation successfully determines the discrete probability of the specified score difference.

Understanding the result

The primary output represents the probability of observing a specific difference between two counts. A positive result indicates the first process likely exceeded the second, while a negative result suggests the opposite. The mean identifies the expected centre of the difference, and the variance indicates the spread of potential outcomes around that centre.

Assumptions and limitations

This model assumes that the two underlying processes are strictly independent and follow a Poisson distribution. It requires the means to be positive. Computation is limited to means below 500 and differences within a specific range to maintain numerical stability during the evaluation of Bessel functions.

Common mistakes to avoid

Users often mistakenly assume the variance of the difference is the difference of the variances; in a Skellam distribution, the variance is always the sum of the means. Another error is applying this model to dependent processes, which violates the independence assumption required for the underlying Poisson variables.

Sensitivity and robustness

The calculation is stable for moderate mean values. Small adjustments to $μ_{1}$ or $μ_{2}$ will shift the distribution mean linearly. However, as means increase, the distribution rapidly spreads, making the probability of any single integer $k$ smaller and the overall curve more symmetrical and Bell-shaped.

Troubleshooting

If the error message regarding numeric values appears, ensure all inputs are valid numbers. If the means exceed the stability limit of 500, the calculator will halt to prevent overflow. Ensure the lower bound is less than the upper bound when using the range probability type to avoid calculation errors.

Frequently asked questions

Can the difference k be negative?

Yes, the Skellam distribution is defined for all integers, allowing for negative differences when the second process exceeds the first.

How is the variance calculated?

The variance is the sum of the two means, $μ_{1} + μ_{2}$ , reflecting the total uncertainty from both independent sources.

What happens if the means are equal?

If the means are equal, the distribution is perfectly symmetrical around a mean of zero, representing an equal chance for either process to be higher.

Where this calculation is used

This statistical method is widely applied in academic research involving the comparison of two count-based datasets. In sports analysis, it models the difference in goals or points between two competing entities. In environmental science, it may be used to compare the frequency of biological occurrences in different zones. Within probability theory, it serves as a fundamental example of the distribution of the difference of random variables, frequently appearing in coursework regarding discrete distributions and stochastic processes.

Results are based on standard mathematical and statistical methods and may involve rounding or approximation. If precise accuracy is required, please verify results independently. See full disclaimer.