Theoretical formulas for the probability that an observation has a particular value, or lies within a given range of values.

Discrete probability distributions apply to observations that can take only certain distinct values, such as the integers 0, 1, 2,… or the six named faces of a die. A probability, p(r), is assigned to each event such that the total is unity. Important discrete distributions are the binomial distribution and the Poisson distribution.

Continuous probability distributions apply to observations, such as physical measurements, where no two observations are likely to be exactly the same. Since the probability of observing exactly a given value is about zero, a mathematical function, the cumulative distribution function, F(x), is used instead. This is defined as the probability that the observation does not exceed x. F(x) increases monotonically with x from 0 to 1, and the probability of observing any value between two limits, x₁ and x₂, is

This definition leads, by differential calculus, to the frequency function, f(x), which is the limiting ratio ofas h becomes small, so that the probability of an observation between x and (x + h) is h.f(x). The most important continuous distribution is the normal (or Gaussian) distribution.

Probability distributions are defined in terms of parameters, whose values determine the numerical values of the probabilities.