Probability distribution

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In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. [1] [2] It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). [3]


For instance, if X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 (1 in 2 or 1/2) for X = heads, and 0.5 for X = tails (assuming that the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. [4]


The probability mass function (pmf)
{\displaystyle p(S)}
specifies the probability distribution for the sum
{\displaystyle S}
of counts from two dice. For example, the figure shows that
{\displaystyle p(11)=2/36=1/18}
. The pmf allows the computation of probabilities of events such as
{\displaystyle P(X>9)=1/12+1/18+1/36=1/6}
, and all other probabilities in the distribution. Dice Distribution (bar).svg
The probability mass function (pmf) specifies the probability distribution for the sum of counts from two dice. For example, the figure shows that . The pmf allows the computation of probabilities of events such as , and all other probabilities in the distribution.

A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by , is the set of all possible outcomes of a random phenomenon being observed; it may be any set: a set of real numbers, a set of vectors, a set of arbitrary non-numerical values, etc. For example, the sample space of a coin flip would be Ω = {heads, tails}.

To define probability distributions for the specific case of random variables (so the sample space can be seen as a numeric set), it is common to distinguish between discrete and absolutely continuous random variables. In the discrete case, it is sufficient to specify a probability mass function assigning a probability to each possible outcome: for example, when throwing a fair die, each of the six values 1 to 6 has the probability 1/6. The probability of an event is then defined to be the sum of the probabilities of the 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 typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, consider measuring the weight of a piece of ham in the supermarket, and assume the scale has many digits of precision. The probability that it weighs exactly 500 g is zero, as it will most likely have some non-zero decimal digits. Nevertheless, one might demand, in quality control, that a package of "500 g" of ham must weigh between 490 g and 510 g with at least 98% probability, and this demand is less sensitive to the accuracy of measurement instruments.

The left shows the probability density function. The right shows the cumulative distribution function, for which the value at a equals the area under the probability density curve to the left of a. Combined Cumulative Distribution Graphs.png
The left shows the probability density function. The right shows the cumulative distribution function, for which the value at a equals the area under the probability density curve to the left of a.

Absolutely continuous probability distributions can be described in several ways. The probability density function describes the infinitesimal probability of any given value, and the probability that the outcome lies in a given interval can be computed by integrating the probability density function over that interval. [5] An alternative description of the distribution is 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 described by the picture to the right. [6]

General probability definition

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 related[ clarification needed ] to the sample space, and gives a real number probability as its output.[ citation needed ]

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 P(heads) = 0.5 and P(tails) = 0.5. 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), [7] and all probability distributions discussed in this article are of this type. It is common to denote as the probability that a certain variable belongs to a certain event . [4] [8]

The above probability function only characterizes a probability distribution if it satisfies all the Kolmogorov axioms, that is:

  1. , so the probability is non-negative
  2. , so no probability exceeds
  3. for any 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 . [9]

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 the discrete probability distribution is known as probability mass function. On the other hand, absolutely continuous probability distributions are 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. [4] [5] [8] 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. [10]

The probability density function (pdf) of the normal distribution, also called Gaussian or "bell curve", the most important absolutely continuous random distribution. As notated on the figure, the probabilities of intervals of values correspond to the area under the curve. Standard deviation diagram.svg
The probability density function (pdf) of the normal distribution, also called Gaussian or "bell curve", the most important absolutely continuous random distribution. As notated on the figure, the probabilities of intervals of values correspond to the area under the curve.


Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below. [1]

Basic terms

Discrete probability distributions

Absolutely continuous probability distributions

Cumulative distribution function

In the special case of a real-valued random variable, the probability distribution can equivalently be represented by a cumulative distribution function instead of a probability measure. The cumulative distribution function of a random variable with regard to a probability distribution is defined as

The cumulative distribution function of any real-valued random variable has the properties:

Conversely, any function that satisfies the first four of the properties above is the cumulative distribution function of some probability distribution on the real numbers. [13]

Any probability distribution can be decomposed as the sum of a discrete, an absolutely continuous and a singular continuous distribution, [14] and thus any cumulative distribution function admits a decomposition as the sum of the three according cumulative distribution functions.

Discrete probability distribution

The probability mass function of a discrete probability distribution. The probabilities of the singletons {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero. Discrete probability distrib.svg
The probability mass function of a discrete probability distribution. The probabilities of the singletons {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.
The cdf of a discrete probability distribution, ... Discrete probability distribution.svg
The cdf of a discrete probability distribution, ...
... of a continuous probability distribution, ... Normal probability distribution.svg
... of a continuous probability distribution, ...
... of a distribution which has both a continuous part and a discrete part. Mixed probability distribution.svg
... of a distribution which has both a continuous part and a discrete part.

A discrete probability distribution is the probability distribution of a random variable that can take on only a countable number of values [15] (almost surely) [16] which means that the probability of any event can be expressed as a (finite or countably infinite) sum:

where is a countable set. Thus the discrete random variables are exactly those with a probability mass function . In the case where the range of values is countably infinite, these values have to decline to zero fast enough for the probabilities to add up to 1. For example, if for , the sum of probabilities would be .

A discrete random variable is a random variable whose probability distribution is discrete.

Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, the negative binomial distribution and categorical distribution. [3] When a sample (a set of observations) is drawn from a larger population, the sample points have an empirical distribution that is discrete, and which provides information about the population distribution. Additionally, the discrete uniform distribution is commonly used in computer programs that make equal-probability random selections between a number of choices.

Cumulative distribution function

A real-valued discrete random variable can equivalently be defined as a random variable whose cumulative distribution function increases only by jump discontinuities—that is, its cdf increases only where it "jumps" to a higher value, and is constant in intervals without jumps. The points where jumps occur are precisely the values which the random variable may take. Thus the cumulative distribution function has the form

Note that the points where the cdf jumps always form a countable set; this may be any countable set and thus may even be dense in the real numbers.

Dirac delta representation

A discrete probability distribution is often represented with Dirac measures, the probability distributions of deterministic random variables. For any outcome , let be the Dirac measure concentrated at . Given a discrete probability distribution, there is a countable set with and a probability mass function . If is any event, then

or in short,

Similarly, discrete distributions can be represented with the Dirac delta function as a generalized probability density function , where

which means

for any event [17]

Indicator-function representation

For a discrete random variable , let be the values it can take with non-zero probability. Denote

These are disjoint sets, and for such sets

It follows that the probability that takes any value except for is zero, and thus one can write as

except on a set of probability zero, where is the indicator function of . This may serve as an alternative definition of discrete random variables.

One-point distribution

A special case is the discrete distribution of a random variable that can take on only one fixed value; in other words, it is a deterministic distribution. Expressed formally, the random variable has a one-point distribution if it has a possible outcome such that [18] All other possible outcomes then have probability 0. Its cumulative distribution function jumps immediately from 0 to 1.

Absolutely continuous probability distribution

An absolutely continuous probability distribution is a probability distribution on the real numbers with uncountably many possible values, such as a whole interval in the real line, and where the probability of any event can be expressed as an integral. [19] More precisely, a real random variable has an absolutely continuous probability distribution if there is a function such that for each interval the probability of belonging to is given by the integral of over : [20] [21]

This is the definition of a probability density function, so that absolutely continuous probability distributions are exactly those with a probability density function. In particular, the probability for to take any single value (that is, ) is zero, because an integral with coinciding upper and lower limits is always equal to zero. If the interval is replaced by any measurable set , the according equality still holds:

An absolutely continuous random variable is a random variable whose probability distribution is absolutely continuous.

There are many examples of absolutely continuous probability distributions: normal, uniform, chi-squared, and others.

Cumulative distribution function

Absolutely continuous probability distributions as defined above are precisely those with an absolutely continuous cumulative distribution function. In this case, the cumulative distribution function has the form

where is a density of the random variable with regard to the distribution .

Note on terminology: Absolutely continuous distributions ought to be distinguished from continuous distributions, which are those having a continuous cumulative distribution function. Every absolutely continuous distribution is a continuous distribution but the converse is not true, there exist singular distributions, which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given by the Cantor distribution. Some authors however use the term "continuous distribution" to denote all distributions whose cumulative distribution function is absolutely continuous, i.e. refer to absolutely continuous distributions as continuous distributions. [4]

For a more general definition of density functions and the equivalent absolutely continuous measures see absolutely continuous measure.

Kolmogorov definition

In the measure-theoretic formalization of probability theory, a random variable is defined as a measurable function from a probability space to a measurable space . Given that probabilities of events of the form satisfy Kolmogorov's probability axioms, the probability distribution of is the pushforward measure of , which is a probability measure on satisfying . [22] [23] [24]

Other kinds of distributions

One solution for the Rabinovich-Fabrikant equations. What is the probability of observing a state on a certain place of the support (i.e., the red subset)? Rabinovich Fabrikant 2314.png
One solution for the Rabinovich–Fabrikant equations. What is the probability of observing a state on a certain place of the support (i.e., the red subset)?

Absolutely continuous and discrete distributions with support on or are extremely useful to model a myriad of phenomena, [4] [6] since most practical distributions are supported on relatively simple subsets, such as hypercubes or balls. However, this is not always the case, and there exist phenomena with supports that are actually complicated curves within some space or similar. In these cases, the probability distribution is supported on the image of such curve, and is likely to be determined empirically, rather than finding a closed formula for it. [25]

One example is shown in the figure to the right, which displays the evolution of a system of differential equations (commonly known as the Rabinovich–Fabrikant equations) that can be used to model the behaviour of Langmuir waves in plasma. [26] When this phenomenon is studied, the observed states from the subset are as indicated in red. So one could ask what is the probability of observing a state in a certain position of the red subset; if such a probability exists, it is called the probability measure of the system. [27] [25]

This kind of complicated support appears quite frequently in dynamical systems. It is not simple to establish that the system has a probability measure, and the main problem is the following. Let be instants in time and a subset of the support; if the probability measure exists for the system, one would expect the frequency of observing states inside set would be equal in interval and , which might not happen; for example, it could oscillate similar to a sine, , whose limit when does not converge. Formally, the measure exists only if the limit of the relative frequency converges when the system is observed into the infinite future. [28] The branch of dynamical systems that studies the existence of a probability measure is ergodic theory.

Note that even in these cases, the probability distribution, if it exists, might still be termed "absolutely continuous" or "discrete" depending on whether the support is uncountable or countable, respectively.

Random number generation

Most algorithms are based on a pseudorandom number generator that produces numbers that are uniformly distributed in the half-open interval [0, 1). These random variates are then transformed via some algorithm to create a new random variate having the required probability distribution. With this source of uniform pseudo-randomness, realizations of any random variable can be generated. [29]

For example, suppose has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some , we define

so that

This random variable X has a Bernoulli distribution with parameter . [29] Note that this is a transformation of discrete random variable.

For a distribution function of an absolutely continuous random variable, an absolutely continuous random variable must be constructed. , an inverse function of , relates to the uniform variable :

For example, suppose a random variable that has an exponential distribution must be constructed.

so and if has a distribution, then the random variable is defined by . This has an exponential distribution of . [29]

A frequent problem in statistical simulations (the Monte Carlo method) is the generation of pseudo-random numbers that are distributed in a given way.

Common probability distributions and their applications

The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the science of statistics. There is spread or variability in almost any value that can be measured in a population (e.g. height of people, durability of a metal, sales growth, traffic flow, etc.); almost all measurements are made with some intrinsic error; in physics, many processes are described probabilistically, from the kinetic properties of gases to the quantum mechanical description of fundamental particles. For these and many other reasons, simple numbers are often inadequate for describing a quantity, while probability distributions are often more appropriate.

The following is a list of some of the most common probability distributions, grouped by the type of process that they are related to. For a more complete list, see list of probability distributions, which groups by the nature of the outcome being considered (discrete, absolutely continuous, multivariate, etc.)

All of the univariate distributions below are singly peaked; that is, it is assumed that the values cluster around a single point. In practice, actually observed quantities may cluster around multiple values. Such quantities can be modeled using a mixture distribution.

Linear growth (e.g. errors, offsets)

Exponential growth (e.g. prices, incomes, populations)

Uniformly distributed quantities

Bernoulli trials (yes/no events, with a given probability)

Categorical outcomes (events with K possible outcomes)

Poisson process (events that occur independently with a given rate)

Absolute values of vectors with normally distributed components

Normally distributed quantities operated with sum of squares

As conjugate prior distributions in Bayesian inference

Some specialized applications of probability distributions

See also


Related Research Articles

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