Understand the distribution
What the skellam distribution calculator models
Assigning probability to separate outcomes. A discrete probability mass function assigns probability directly to each allowed count or category combination. All mutually exclusive masses across the full support sum to one.
1Define the count support and parameters2Evaluate the requested mass or cumulative range3Check the trial, replacement, and dependence assumptions
Probability mass rule
If X~Poisson(λ₁) and Y~Poisson(λ₂) independently, D=X−Y follows a Skellam distribution.
A probability mass is a probability at an allowed discrete outcome.
Interpretation and limits
The sign identifies which count is larger; the mean difference is λ₁−λ₂ and variance is λ₁+λ₂.
Model boundary: The two counts must be independent Poisson variables measured over comparable exposure; the calculator reports point mass rather than a cumulative tail.
Quick guide
How to use this calculator
- Enter the observations, probabilities, model parameters, or summary statistics requested by the visible labels.
- Keep every value on the same scale and confirm that the selected sampling relationship, distribution, and tail convention match the question you are investigating.
- Read the result together with its assumptions and interpretation. Statistical output summarizes uncertainty under a model; it does not repair biased data or establish causation.
Calculation method
How the skellam distribution calculator works
If X~Poisson(λ₁) and Y~Poisson(λ₂) independently, D=X−Y follows a Skellam distribution.
The sign identifies which count is larger; the mean difference is λ₁−λ₂ and variance is λ₁+λ₂.
Worked example
Skellam Distribution example
For equal means λ₁=λ₂=1, P(D=0)=e⁻²I₀(2)≈0.308508.
If X~Poisson(λ₁) and Y~Poisson(λ₂) independently, D=X−Y follows a Skellam distribution.
Supported inputs
Precision and limits
Model and design
The two counts must be independent Poisson variables measured over comparable exposure; the calculator reports point mass rather than a cumulative tail.
Numerical scope
Inputs use double-precision numerical methods with guarded domains. Datasets accept up to 10,000 finite plain-decimal values. Extremely large parameters or probabilities deep in a numerical tail may require specialist statistical software.
Interpretation
The sign identifies which count is larger; the mean difference is λ₁−λ₂ and variance is λ₁+λ₂.
Decision boundary
The calculator does not validate how data were collected, diagnose dependence or bias, choose a scientifically meaningful effect, or replace review by a qualified statistician for consequential research, medical, regulatory, safety, or policy decisions.
Privacy
Entered values and calculated results stay in this browser and are not sent to an analytics service.
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