In probability theory and statistics, a probability distribution describes how probabilities are assigned to the possible results of a random phenomenon—more precisely, to events, which are sets of possible outcomes of a probabilistic experiment. Informally, a probability distribution tells us how likely different results are. Formally, it is a probability measure: a function that assigns probabilities to events in a way that satisfies the axioms of probability.
Probability distributions are closely linked to random variables. A random variable is a function that assigns a value to each outcome of a probabilistic experiment; it induces a probability distribution on the set of values it can take. For example, the result of a coin toss can be represented by a random variable that equals for heads and for tails. If the coin is fair, this distribution assigns probability to and probability to . Viewed as a probability measure, the distribution of assigns to each set }; for a fair coin, , , and .
In practice, probability distributions are often described by functions such as cumulative distribution functions, probability mass functions, or probability density functions. Which description is used depends on the nature of the distribution: probability mass functions are used for discrete distributions, while probability density functions are used for many continuous distributions.
Probability distributions that occur frequently or have special theoretical importance are often given specific names; examples are collected in the list of probability distributions.
Introduction A probability distribution is a mathematical description of the probabilities of events, i.e. subsets of the sample space. The sample space, often represented in notation by is the set of all possible outcomes of a random phenomenon being observed. The sample space may be any set of numbers, vectors, labels, or whatever else. For example, the sample space of a coin flip could be , whereas for a die roll, it could be .
To define probability distributions for the specific case of random variables (so that the sample space can mapped to a measurable space, for example the real numbers), it is common to distinguish between discrete and continuous random variables. In the discrete case, it is sufficient to specify a probability mass function assigning a probability to each possible outcome (e.g. when throwing a fair die, each of the six digits to , corresponding to the number of dots on the die, has probability of being on top when it lands). The probability of an event is then defined to be the sum of the probabilities of all outcomes that satisfy the event; for example, the probability of the event "the die rolls an even value" is
In contrast, when a random variable takes values from a continuum, then unless the probability density function has any infinitely-dense peaks, any individual outcome has probability zero. For such continuous random variables, only events that include infinitely many outcomes, such as intervals, have probability greater than 0.
For example, consider measuring the weight of a piece of ham in the supermarket, and assume the scale can provide arbitrarily many digits of precision. Then, the probability that it weighs exactly 500 g must be zero because no matter how high the level of precision chosen, it cannot be assumed that there are no non-zero digits after those output by the scale. However, for the same use case, it is possible to meet quality control requirements such as that a package of "500 g" of ham must weigh between 490 g and 510 g. This is possible because this measurement does not require infinite precision from the underlying equipment, and it provides some tolerance for variability in physical objects and processes.
Continuous probability distributions can be described by means of the cumulative distribution function, which describes the probability that the random variable is no larger than a given value (i.e., for some . The cumulative distribution function is the area under the probability density function from to , as shown in figure 1.
Most continuous probability distributions encountered in practice are not only continuous but also absolutely continuous. Such distributions can be described by their probability density function. Informally, the probability density of a random variable describes the relative value of the infinitesimal probability that takes any value — that is as becomes arbitrarily small. The probability that lies in a given interval can be computed rigorously by integrating the probability density function over that interval.
General probability definition
Let be a probability space, be a measurable space, and be a -valued random variable. Then the probability distribution of is the pushforward measure of the probability measure onto induced by . Explicitly, this pushforward measure on is given by for
Any probability distribution is a probability measure on (in general different from , unless happens to be the identity map).
A probability distribution can be described in various forms, such as by a probability mass function or a cumulative distribution function. One of the most general descriptions, which applies for absolutely continuous and discrete variables, is by means of a probability function whose input space is a σ-algebra, and gives a real number probability as its output, particularly, a number in .
The probability function can take as argument subsets of the sample space itself, as in the coin toss example, where the function was defined so that and . However, because of the widespread use of random variables, which transform the sample space into a set of numbers (e.g., , ), it is more common to study probability distributions whose argument are subsets of these particular kinds of sets (number sets), and all probability distributions discussed in this article are of this type. It is common to denote as the probability that a certain value of the variable belongs to a certain event .
The above probability function only characterizes a probability distribution if it satisfies all the Kolmogorov axioms, that is: , so the probability is non-negative , so no probability exceeds for any countable disjoint family of sets
The concept of probability function is made more rigorous by defining it as the element of a probability space , where is the set of possible outcomes, is the set of all subsets whose probability can be measured, and is the probability function, or probability measure, that assigns a probability to each of these measurable subsets .
Probability distributions usually belong to one of two classes.
A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is discrete (e.g. a coin toss, a roll of a die) and the probabilities are encoded by a discrete list of the probabilities of the outcomes; in this case probabilities are described by a probability mass function, and the probability distribution is given by a sum of the probability mass function.
An absolutely continuous probability distribution is applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In the absolutely continuous case, probabilities are described by a probability density function, and the probability distribution is by definition the integral of the probability density function. The normal distribution is a commonly encountered absolutely continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.
A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate, while a distribution whose sample space is a vector space of dimension 2 or more is called multivariate. A univariate distribution gives the probabilities of a single random variable taking on various different values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. A commonly encountered multivariate distribution is the multivariate normal distribution.
Besides the probability function, the cumulative distribution function, the probability mass function and the probability density function, the moment generating function and the characteristic function also serve to identify a probability distribution, as they uniquely determine an underlying cumulative distribution function.
Terminology
Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below.
Basic terms Random variable: takes values from a sample space; probabilities describe which values and set of values are more likely taken. Event: set of possible values (outcomes) of a random variable that occurs with a certain probability. Probability function or probability measure: describes the probability that the event occurs.Chapters 1 and 2 of Cumulative distribution function: function evaluating the probability that will take a value less than or equal to for a random variable (only for real-valued random variables). Quantile function: the inverse of the cumulative distribution function. Gives such that, with probability , will not exceed .
Discrete probability distributions Discrete probability distribution: for many random variables with finitely or countably infinitely many values. Probability mass function (pmf): function that gives the probability that a discrete random variable is equal to some value. Frequency distribution: a table that displays the frequency of various outcomes . Relative frequency distribution: a frequency distribution where each value has been divided (normalized) by a number of outcomes in a sample (i.e. sample size). Categorical distribution: for discrete random variables with a finite set of values.
Absolutely continuous probability distributions Absolutely continuous probability distribution: for many random variables with uncountably many values. Probability density function (pdf) or probability density: function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample.
Related terms Support: the set of values such that the random variable as a positive probability of falling in every open neighborhood of . Tail:More information and examples can be found in the articles Heavy-tailed distribution, Long-tailed distribution, fat-tailed distribution the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form , or a union thereof. Expected value or mean: the weighted average of the possible values, using their probabilities as their weights; or the continuous analog thereof. Median: the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half. Mode: for a discrete random variable, the value with highest probability; for an absolutely continuous random variable, a location at which the probability density function has a local peak. Quantile: the q-quantile is the value such that . Variance: the second moment of the random variable about its mean; an important measure of the dispersion of the distribution. Standard deviation: the square root of the variance, and hence another measure of dispersion.