Probability Distributions: A Field Guide
A map of eight common probability distributions organized by the random process each one describes, with a pick-a-distribution selector table.
This is the hub for an eight-part series on the probability distributions that turn up most often in practice. Each post follows the same shape: the question the distribution answers, a compact model card, what its parameters do to the shape, one fully worked example (including “at least” and “at most” probabilities, not just exact ones), when the model fits and when it fails, and a short SciPy check.
The organizing idea is that a distribution is a story about a random process, not a curve to pattern-match against a histogram. Recognizing the process is what lets us calculate something useful, and knowing the assumptions is what keeps us honest.
Pick a Distribution
Start from the question, not the data’s size or a shape it happens to resemble:
| The question | Candidate distribution |
|---|---|
| How many successes in $n$ independent trials? | Binomial |
| How many attempts until the first success? | Geometric |
| How many attempts (or failures) until the $r$-th success? | Negative Binomial |
| How many events occur in a fixed interval? | Poisson |
| How long until the next constant-rate event? | Exponential |
| How does symmetric, additive variation spread around a center? | Normal |
| What happens when many positive factors multiply? | Log-Normal |
| Symmetric variation with a sharp center and more extremes than Normal? | Laplace |
The table names a candidate family. Whether it actually fits depends on the process assumptions and on model checking, never on the question alone.
The Conceptual Path
Chronologically the posts run Binomial through Laplace, but they group into three families that share a mechanism.
Group A, repeated binary trials. Binomial counts successes in a fixed number of trials; Geometric counts trials until the first success; Negative Binomial counts until the $r$-th success (and doubles as an overdispersed-count model).
Group B, events in continuous time. Poisson counts events in an interval; Exponential measures the waiting time to the next one. They are two views of the same constant-rate process: a count in a window versus the time between arrivals.
Group C, continuous measurements. Normal captures additive variation around a center; Log-Normal captures positive multiplicative variation; Laplace captures symmetric deviations with a sharper peak and heavier-than-Normal tails.
One Habit to Carry Through
A dataset being positive, bell-shaped, or overdispersed is evidence worth investigating, not proof that a named distribution generated it. For every distribution in the series, the useful questions are the same: what is the random variable, what are the parameters and their units, what process would produce this shape, and what observed pattern would tell us the model is wrong?