Learning path / Probability models
Choose a distribution
from the mechanism.
Start with what generated the observation: a count, a waiting time, an additive measurement, a proportion, or an extreme event. The formula comes second.
A practical table of contents
A random variable maps outcomes to numbers; its distribution assigns probability to those values. Support says which values are possible, parameters control shape, and a probability mass or density function describes relative likelihood.
What exactly are you counting?
One trial is Bernoulli. Fix the number of trials and count successes: binomial. Fix exposure instead and count arrivals that can happen independently at any instant: Poisson. The sparse-event limit connects the latter two—large n, small p, stable np.
Poisson
Counts independent arrivals in a fixed exposure; a sparse-event limit of the binomial.
P(X=k) = e⁻λ λᵏ / k!Counts and waiting times are two views of one process.
Poisson counts correspond to exponential gaps. Waiting for several arrivals produces gamma. Weibull relaxes the constant-hazard assumption: shape below 1 means decreasing hazard, shape above 1 means increasing hazard.
Additive effects point toward the normal.
The normal is stable under independent addition and emerges from many small effects. Student t accounts for uncertainty in scale. Cauchy is the warning label: it looks bell-shaped, but tails are so heavy that the mean and variance do not exist.
Respect the support.
Uniform says only that a value is inside an interval. Beta expresses shape over a proportion: symmetric, skewed, U-shaped, or nearly concentrated at a point. In Bayesian modeling, its greatest convenience is conjugacy with Bernoulli and binomial observations.
Continue the path
Bayesian learning
These distributions return immediately as priors and likelihoods: watch a Beta prior absorb evidence, then drive decisions.
Follow the posterior