Discrete vs Continuous Distributions

Discrete distributions assign probabilities to individual values (P(X = k) > 0), while continuous distributions assign probabilities only to intervals. For continuous variables, P(X = exact value) is always 0.

This distinction determines which function describes the distribution: a probability mass function (PMF) for discrete, or a probability density function (PDF) for continuous.

The binomial counts successes in a fixed number of trials. The Poisson counts events in an interval. The geometric counts trials until the first success. The negative binomial counts trials until a target number of successes.

Choose discrete models when your outcome is a count or takes values from a finite or countable set.

The normal models symmetric measurement data. The exponential models waiting times with constant rate. The beta models proportions on [0, 1]. The gamma and Weibull cover right-skewed positive quantities like lifetimes and claim sizes.

Choose continuous models when your outcome can take any value in an interval, even if you observe rounded or binned data.

The normal approximation to the binomial works well when np and n(1−p) are both at least 10. The Poisson approximation to the binomial applies when n is large and p is small.

These approximations simplify computation and are useful when exact discrete calculations are expensive, but always check the conditions before relying on them.

Start by asking whether your variable can take non-integer values. If yes, use a continuous model. If your variable is a count with a clear upper bound, binomial is natural. If counts have no practical upper bound, consider Poisson or negative binomial.

When in doubt, plot your data and compare it to candidate PMFs or PDFs. The visual check often resolves ambiguity faster than formal tests.