Probability distribution

A probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. In other words, a probability distribution is a probability space whose underlying σ-algebra is the Borel algebra on the reals.

Every random variable gives rise to a probability distribution, and this distribution contains most of the important information about the variable. If X is a random variable, the corresponding probability distribution assigns to the interval [a, b] the probability Pr[a ≤ X ≤ b], i.e. the probability that the variable X will take a value in the interval [a, b].

The probability distribution of the variable X can be uniquely described by its cumulative distribution function F(x), which is defined by

F(x) = Pr[ X ≤ x ]

for any x in R.

Many probability distributions can be expressed by a probability density function: a non-negative Lebesgue integrable function f defined on the reals such that

Pr[ a ≤ X ≤ b] = ∫_a^b f(x) dx

for all a and b. Some distributions do not admit such a density however, for example those of discrete random variables.

Several probability distributions are so important that they have been given specific names:

Discrete distributions
- The discrete uniform distribution, where all elements of a finite set are equally likely. It doesn't appear in nature, but pseudo-random number generators on computers attempt to produce a discrete uniform distribution, which can then be used to generate other distributions.
- The binomial distribution, a discrete distribution which describes the number of successes in a series of Yes/No experiments.
- The geometric distribution, a discrete distribution which describes the number of attempts to get the first success in a series of Yes/No experiments.
- The negative binomial distribution, a generalization of the geometric distribution to the n'th success.
- The Poisson distribution, which describes the number of random events that happen in a certain time interval.
Continuous distributions
- The uniform distribution, where all points in a finite interval are equally likely. It doesn't appear in nature.
- The normal distribution, also called the Gaussian or the bell curve. It is ubiquitous in nature and statistics.
- The exponential distribution, which describes the time between random events.
- The Weibull distribution, used to model the lifetime of technical devices.
- The Cauchy distribution, an example of a distribution which does not have an expected value or a variance.
- The Maxwell-Boltzmann distribution, a discrete distribution important in physics which describes the probabilities of the various energy levels of a system.
- The chi-square distribution, used in statistics to check whether a given model fits the data, and whether changes in frequencies are statistically significant.
- The Student's t distribution, useful for estimating unknown expected values.