When does each common distribution arise — Bernoulli, Binomial, Poisson, Normal, Exponential, Uniform?
Each distribution has a natural generative story: Bernoulli is a single coin flip; Binomial sums Bernoullis; Poisson counts rare arrivals; Normal emerges from sums of many small effects; Exponential models waiting times between Poisson events; Uniform assigns equal probability across a range. Choosing correctly comes from matching that story to the data-generating process.
How to think about it
Matching a distribution to a problem is a two-step process: identify the generative story, then check the support and moment structure against your data.
Distribution taxonomy
Bernoulli(p) — a single binary trial. E[X] = p, Var = p(1-p). Use it for click/no-click, churn/retain, any single yes-or-no outcome.
Binomial(n, p) — sum of n independent Bernoulli(p) trials. E[X] = np. Use it when you have a fixed number of independent, identically distributed trials and count successes.
Poisson(λ) — number of independent events in a fixed interval when the average rate is λ and each event is rare. E[X] = Var(X) = λ. Arrivals per second, defects per batch, goals per match.
Normal(μ, σ²) — justified by the CLT: sums or averages of many small, independent, finite-variance effects converge here. Continuous, symmetric, unbounded. Use for measurement errors, aggregated financial returns, residuals in linear regression.
Exponential(λ) — waiting time until the next Poisson event. E[X] = 1/λ, Var = 1/λ². Memoryless: P(X > s+t | X > s) = P(X > t). Time between server requests, customer inter-arrival times, component lifetime under constant failure rate.
Uniform(a, b) — equal probability density over [a, b]. E[X] = (a+b)/2. Use for maximum-entropy priors when all values in a range are equally plausible, random-number seeds, and certain order-statistic problems.
Quick decision heuristic
| Data type | Generative story | Reach for |
|---|---|---|
| Single binary outcome | One trial | Bernoulli |
| Count of successes in n trials | Fixed n, constant p | Binomial |
| Count in an interval | Rare events, constant rate | Poisson |
| Continuous, symmetric, many additive sources | CLT applies | Normal |
| Time until next event | Between Poisson arrivals | Exponential |
| Continuous, no prior knowledge of shape | All values equally likely | Uniform |