Common Probability Distributions in Finance › Choosing a Distribution

Distribution choice begins with the variable being modeled. A normal or Student's t model can describe signed continuous returns, a lognormal model can describe a positive level generated by exponentiated normal log changes, a binomial model counts successes in a fixed number of trials, and a Poisson model counts events over an interval. A power-law tail specification is a statement about remote-tail decay rather than a complete model for the center of a distribution.

The horizon and decision also matter. A model that is adequate for central daily-return behavior may be poor for multi-day dependence, extreme quantiles, or expected shortfall. Fit diagnostics should therefore match the intended use: inspect residual dependence and stability across regimes, compare empirical and fitted quantiles, and check the particular risk measure or payoff rather than relying on one summary statistic.