Choose a Probability Distribution by Its Assumptions

Choose a probability distribution by what the variable can be and how it is generated. A familiar bell curve is a poor starting point for a nonnegative event count or a bounded proportion. Support and mechanism narrow the choices before a fitting routine enters the picture.

Write down the unit, possible values, sampling process, and any dependence between observations. A count per hour differs from a waiting time in seconds. A fraction estimated from ten trials differs from a continuous measurement constrained to an interval.

Start with support and a simple mechanism

A Bernoulli variable describes one binary outcome. A binomial count describes successes across a fixed number of independent trials sharing a probability. A Poisson model describes counts under a particular constant-rate process. Exponential waiting times add a memoryless assumption that many real service durations do not satisfy.

For positive skewed quantities, a lognormal or gamma family may be worth investigating. A normal model permits negative values even if almost none are likely under a fitted parameter set. Decide whether that tail behavior is acceptable for the intended calculation.

Write parameter conventions next to the values

Gamma(shape=2, scale=3): mean = 6
Gamma(shape=2, rate=3):  mean = 2/3

These use different parameterizations of the gamma family. Passing a scale value to a function expecting a rate can produce a plausible-looking but incorrect model. The geometric distribution also has competing conventions: trials until success starts at one, while failures before success starts at zero.

A lognormal’s usual mu and sigma describe the underlying normal variable, not directly the observed variable’s mean and standard deviation. Record the library convention and test a known quantity such as the implied mean before running a larger simulation.

Check consequences beyond the fitted center

Inspect quantiles, tail behavior, zeros, and dependence. A model can fit the center well and understate the rare events that drive the application’s risk. Compare empirical and modeled summaries at the scale of the decision, and use held-out observations when the data supports them.

Do not treat a fitted distribution as proof of the mechanism. Seasonality, mixtures, changing exposure, or correlated observations can invalidate a simple model even when a plot looks convincing. Document those limitations and use a more appropriate model when they materially change the answer.

The reference table is a selection aid, not an automatic recommendation. Its value is making support, means, and parameter boundaries explicit enough to catch an unsuitable assumption early.

Reference tables to keep nearby

• Probability Distribution Selection Reference — Common probability distributions with support, mean, and practical modeling boundaries.

Primary sources

These are the specifications and project documentation behind the examples. Check the relevant implementation and version before applying a pattern.

• NIST probability distributions handbook