# Probability distribution

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In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. [1] [2] In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events. [3] For instance, if the random variable 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 for X = heads, and 0.5 for X = tails (assuming the coin is fair). Examples of random phenomena can include the results of an experiment or survey.

## Contents

A probability distribution is specified in terms of an underlying sample space, which is the set of all possible outcomes of the random phenomenon being observed. The sample space may be the set of real numbers or a set of vectors, or it may be a list of non-numerical values; for example, the sample space of a coin flip would be {heads, tails} .

Probability distributions are generally divided into two classes. A discrete probability distribution (applicable to the scenarios where the set of possible outcomes is discrete, such as a coin toss or a roll of dice) can be encoded by a discrete list of the probabilities of the outcomes, known as a probability mass function. On the other hand, a continuous probability distribution (applicable to the 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) is typically described by probability density functions (with the probability of any individual outcome actually being 0). The normal distribution is a commonly encountered 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 alternative 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. The multivariate normal distribution is a commonly encountered multivariate distribution.

## Introduction

To define probability distributions for the simplest cases, it is necessary to distinguish between discrete and continuous random variables. In the discrete case, it is sufficient to specify a probability mass function ${\displaystyle p}$ 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 dice rolls an even value" is

${\displaystyle p(2)+p(4)+p(6)=1/6+1/6+1/6=1/2.}$

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, the probability that a given object weighs exactly 500 g is zero, because the probability of measuring exactly 500 g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a "500 g" package containing between 490 g and 510 g should be no less than 98%, and this demand is less sensitive to the accuracy of measurement instruments.

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. The probability that the possible values lie in some fixed interval can be related to the way sums converge to an integral; therefore, continuous probability is based on the definition of an integral.

The cumulative distribution function describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the antiderivative of the probability density function provided that the latter function exists. The cumulative distribution function is the area under the probability density function from minus infinity ${\displaystyle \infty }$ to ${\displaystyle x}$ as described by the picture to the right. [4]

## Terminology [1]

### Functions for discrete variables

• Probability function: describes the probability distribution of a discrete random variables
• 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 in a sample.
• 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.
• Discrete probability distribution function: general term to indicate the way the total probability of 1 is distributed over all various possible outcomes (i.e. over entire population) for discrete random variable
• Cumulative distribution function : function evaluating the probability that ${\displaystyle X}$ will take a value less than or equal to ${\displaystyle x}$ for a discrete random variable
• Categorical distribution : for discrete random variables with a finite set of values.

### Functions for continuous variables

• Probability density function (PDF): 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
• Continuous probability distribution function: most often reserved for continuous random variables
• Cumulative distribution function : function evaluating the probability that ${\displaystyle X}$ will take a value less than or equal to ${\displaystyle x}$ for continuous variable

### Basic terms

• Mode: for a discrete random variable, the value with highest probability (the location at which the probability mass function has its peak); for a continuous random variable, a location at which the probability density function has a local peak.
• Support: the smallest closed set whose complement has probability zero.
• Head : the range of values where the pmf or pdf is relatively high.
• Tail: the complement of the head within the support; the large set of values where the pmf or pdf is relatively low.
• 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.
• Variance : the second moment of the pmf or pdf about the mean; an important measure of the dispersion of the distribution.
• Standard deviation : the square root of the variance, and hence another measure of dispersion.
• Symmetry: a property of some distributions in which the portion of the distribution to the left of a specific value is a mirror image of the portion to its right.
• Skewness : a measure of the extent to which a pmf or pdf "leans" to one side of its mean. The third standardized moment of the distribution.
• Kurtosis : a measure of the "fatness" of the tails of a pmf or pdf. The fourth standardized moment of the distribution.

## Cumulative distribution function

Because a probability distribution P on the real line is determined by the probability of a scalar random variable X being in a half-open interval (−∞, x], the probability distribution is completely characterized by its cumulative distribution function:

${\displaystyle F(x)=\operatorname {P} [X\leq x]\qquad {\text{ for all }}x\in \mathbb {R} .}$

## Discrete probability distribution

A discrete probability distribution is a probability distribution that can take on a countable number of values. [5] For the probabilities to add up to 1, they have to decline to zero fast enough. For example, if ${\displaystyle \operatorname {P} (X=n)={\tfrac {1}{2^{n}}}}$ for n = 1, 2, ..., the sum of probabilities would be 1/2 + 1/4 + 1/8 + ... = 1.

Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution. [3] Additionally, the discrete uniform distribution is commonly used in computer programs that make equal-probability random selections between a number of choices.

When a sample (a set of observations) is drawn from a larger population, the sample points have an empirical distribution that is discrete and that provides information about the population distribution.

### Measure theoretic formulation

A measurable function ${\displaystyle X\colon A\to B}$ between a probability space ${\displaystyle (A,{\mathcal {A}},P)}$ and a measurable space ${\displaystyle (B,{\mathcal {B}})}$ is called a discrete random variable provided that its image is a countable set. In this case measurability of ${\displaystyle X}$ means that the pre-images of singleton sets are measurable, i.e., ${\displaystyle X^{-1}(\{b\})\in {\mathcal {A}}}$ for all ${\displaystyle b\in B}$. The latter requirement induces a probability mass function ${\displaystyle f_{X}\colon X(A)\to \mathbb {R} }$ via ${\displaystyle f_{X}(b):=P(X^{-1}(\{b\}))}$. Since the pre-images of disjoint sets are disjoint,

${\displaystyle \sum _{b\in X(A)}f_{X}(b)=\sum _{b\in X(A)}P(X^{-1}(\{b\}))=P\left(\bigcup _{b\in X(A)}X^{-1}(\{b\})\right)=P(A)=1.}$

This recovers the definition given above.

### Cumulative distribution function

Equivalently to the above, a discrete random variable can be defined as a random variable whose cumulative distribution function (cdf) increases only by jump discontinuities—that is, its cdf increases only where it "jumps" to a higher value, and is constant between those jumps. Note however that the points where the cdf jumps may form a dense set of the real numbers. The points where jumps occur are precisely the values which the random variable may take.

### Delta-function representation

Consequently, a discrete probability distribution is often represented as a generalized probability density function involving Dirac delta functions, which substantially unifies the treatment of continuous and discrete distributions. This is especially useful when dealing with probability distributions involving both a continuous and a discrete part. [6]

### Indicator-function representation

For a discrete random variable X, let u0, u1, ... be the values it can take with non-zero probability. Denote

${\displaystyle \Omega _{i}=X^{-1}(u_{i})=\{\omega :X(\omega )=u_{i}\},\,i=0,1,2,\dots }$

These are disjoint sets, and for such sets

${\displaystyle P\left(\bigcup _{i}\Omega _{i}\right)=\sum _{i}P(\Omega _{i})=\sum _{i}P(X=u_{i})=1.}$

It follows that the probability that X takes any value except for u0, u1, ... is zero, and thus one can write X as

${\displaystyle X(\omega )=\sum _{i}u_{i}1_{\Omega _{i}}(\omega )}$

except on a set of probability zero, where ${\displaystyle 1_{A}}$ is the indicator function of A. This may serve as an alternative definition of discrete random variables.

## Continuous probability distribution

A continuous probability distribution is a probability distribution with a cumulative distribution function that is absolutely continuous. Equivalently, it is a probability distribution on the real numbers that is absolutely continuous with respect to Lebesgue measure. Such distributions can be represented by their probability density functions. If the distribution of X is continuous, then X is called a continuous random variable. There are many examples of continuous probability distributions: normal, uniform, chi-squared, and others.

Formally, if X is a continuous random variable, then it has a probability density function ƒ(x), and therefore its probability of falling into a given interval, say [a, b], is given by the integral

${\displaystyle \operatorname {P} [a\leq X\leq b]=\int _{a}^{b}f(x)\,dx}$

In particular, the probability for X to take any single value a (that is aXa) is zero, because an integral with coinciding upper and lower limits is always equal to zero.

Note on terminology: some authors use the term "continuous distribution" to denote distributions whose cumulative distribution functions are continuous, rather than absolutely continuous. These distributions are the ones ${\displaystyle \mu }$ such that ${\displaystyle \mu \{x\}\,=\,0}$ for all ${\displaystyle \,x}$. This definition includes the (absolutely) continuous distributions defined above, but it also includes 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 properties

• The probability distribution of the sum of two independent random variables is the convolution of each of their distributions.
• Probability distributions are not a vector space—they are not closed under linear combinations, as these do not preserve non-negativity or total integral 1—but they are closed under convex combination, thus forming a convex subset of the space of functions (or measures).

## Kolmogorov definition

In the measure-theoretic formalization of probability theory, a random variable is defined as a measurable function ${\displaystyle X}$ from a probability space ${\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}$ to a measurable space ${\displaystyle ({\mathcal {X}},{\mathcal {A}})}$. Given that probabilities of events of the form ${\displaystyle \{\omega \in \Omega \mid X(\omega )\in A\}}$ satisfy Kolmogorov's probability axioms, the probability distribution of X is the pushforward measure ${\displaystyle X_{*}\mathbb {P} }$ of ${\displaystyle X}$ , which is a probability measure on ${\displaystyle ({\mathcal {X}},{\mathcal {A}})}$ satisfying ${\displaystyle X_{*}\mathbb {P} =\mathbb {P} X^{-1}}$. [7] [8] [9]

## Random number generation

Most algorithms are based on a pseudorandom number generator that produces numbers X that are uniformly distributed in the half-open interval [0,1). These random variates X 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. [10]

For example, suppose ${\displaystyle U}$ has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some ${\displaystyle 0, we define

${\displaystyle {\displaystyle X={\begin{cases}1,&{\mbox{if }}U

so that

${\displaystyle {\textrm {P}}(X=1)={\textrm {P}}(U

This random variable X has a Bernoulli distribution with parameter ${\displaystyle p}$. [10] Note that this is a transformation of discrete random variable.

For a distribution function ${\displaystyle F}$ of a continuous random variable, a continuous random variable must be constructed. ${\displaystyle F^{inv}}$, an inverse function of ${\displaystyle F}$, relates to the uniform variable ${\displaystyle U}$:

${\displaystyle {U\leq F(x)}={F^{inv}(U)\leq x}.}$

For example, suppose a random variable that has an exponential distribution ${\displaystyle F(x)=1-e^{-\lambda x}}$ must be constructed.

{\displaystyle {\begin{aligned}F(x)=u&\Leftrightarrow 1-e^{-\lambda x}=u\\&\Leftrightarrow e^{-\lambda x}=1-u\\&\Leftrightarrow -\lambda x=\ln(1-u)\\&\Leftrightarrow x={\frac {-1}{\lambda }}\ln(1-u)\end{aligned}}}

so ${\displaystyle F^{inv}(u)={\frac {-1}{\lambda }}\ln(1-u)}$ and if ${\displaystyle U}$ has a ${\displaystyle U(0,1)}$ distribution, then the random variable ${\displaystyle X}$ is defined by ${\displaystyle X=F^{inv}(U)={\frac {-1}{\lambda }}\ln(1-U)}$. This has an exponential distribution of ${\displaystyle \lambda }$. [10]

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, 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)

• Normal distribution (Gaussian distribution), for a single such quantity; the most commonly used continuous distribution

### Absolute values of vectors with normally distributed components

• Rayleigh distribution, for the distribution of vector magnitudes with Gaussian distributed orthogonal components. Rayleigh distributions are found in RF signals with Gaussian real and imaginary components.
• Rice distribution, a generalization of the Rayleigh distributions for where there is a stationary background signal component. Found in Rician fading of radio signals due to multipath propagation and in MR images with noise corruption on non-zero NMR signals.

### Some specialized applications of probability distributions

• The cache language models and other statistical language models used in natural language processing to assign probabilities to the occurrence of particular words and word sequences do so by means of probability distributions.
• In quantum mechanics, the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's wavefunction at that point (see Born rule). Therefore, the probability distribution function of the position of a particle is described by ${\displaystyle P_{a\leq x\leq b}(t)=\int _{a}^{b}dx\,|\Psi (x,t)|^{2}}$, probability that the particle's position x will be in the interval axb in dimension one, and a similar triple integral in dimension three. This is a key principle of quantum mechanics. [12]
• Probabilistic load flow in power-flow study explains the uncertainties of input variables as probability distribution and provide the power flow calculation also in term of probability distribution. [13]
• Prediction of natural phenomena occurrences based on previous frequency distributions such as tropical cyclones, hail, time in between events, etc. [14]

## Related Research Articles

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable , or just distribution function of , evaluated at , is the probability that will take a value less than or equal to .

Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event.

In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is described informally as a variable whose values depend on outcomes of a random phenomenon. The formal mathematical treatment of random variables is a topic in probability theory. In that context, a random variable is understood as a measurable function defined on a probability space that maps from the sample space to the real numbers.

This is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.

In probability theory, a probability space or a probability triple is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a dice.

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample in the sample space can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. In other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0, the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would equal one sample compared to the other sample.

In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied. The different possible notions of convergence relate to how such a behavior can be characterized: two readily understood behaviors are that the sequence eventually takes a constant value, and that values in the sequence continue to change but can be described by an unchanging probability distribution.

In mathematics, a degenerate distribution is a probability distribution in a space with support only on a space of lower dimension. If the degenerate distribution is univariate it is a deterministic distribution and takes only a single value. Examples include a two-headed coin and rolling a die whose sides all show the same number. This distribution satisfies the definition of "random variable" even though it does not appear random in the everyday sense of the word; hence it is considered degenerate.

In probability and statistics, a Bernoulli process is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variablesXi are identically distributed and independent. Prosaically, a Bernoulli process is a repeated coin flipping, possibly with an unfair coin. Every variable Xi in the sequence is associated with a Bernoulli trial or experiment. They all have the same Bernoulli distribution. Much of what can be said about the Bernoulli process can also be generalized to more than two outcomes ; this generalization is known as the Bernoulli scheme.

In probability and statistics, a probability mass function (PMF) is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete.

In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given that a certain set of "conditions" is known to occur. If the random variable can take on only a finite number of values, the “conditions” are that the variable can only take on a subset of those values. More formally, in the case when the random variable is defined over a discrete probability space, the "conditions" are a partition of this probability space.

In probability theory and statistics, given two jointly distributed random variables and , the conditional probability distribution of Y given X is the probability distribution of when is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value of as a parameter. When both and are categorical variables, a conditional probability table is typically used to represent the conditional probability. The conditional distribution contrasts with the marginal distribution of a random variable, which is its distribution without reference to the value of the other variable.

Given random variables , that are defined on a probability space, the joint probability distribution for is a probability distribution that gives the probability that each of falls in any particular range or discrete set of values specified for that variable. In the case of only two random variables, this is called a bivariate distribution, but the concept generalizes to any number of random variables, giving a multivariate distribution.

In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for each side of a k-sided die rolled n times. For n independent trials each of which leads to a success for exactly one of k categories, with each category having a given fixed success probability, the multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories.

Probability theory and statistics have some commonly used conventions, in addition to standard mathematical notation and mathematical symbols.

A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set satisfying a set of axioms weaker than those of σ-algebra. Dynkin systems are sometimes referred to as λ-systems or d-system. These set families have applications in measure theory and probability.

In statistics, binomial regression is a regression analysis technique in which the response has a binomial distribution: it is the number of successes in a series of independent Bernoulli trials, where each trial has probability of success . In binomial regression, the probability of a success is related to explanatory variables: the corresponding concept in ordinary regression is to relate the mean value of the unobserved response to explanatory variables.

In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by Maurice Fréchet (1948) who commented that the “development of probability theory and expansion of area of its applications have led to necessity to pass from schemes where (random) outcomes of experiments can be described by number or a finite set of numbers, to schemes where outcomes of experiments represent, for example, vectors, functions, processes, fields, series, transformations, and also sets or collections of sets.”

In probability theory, particularly information theory, the conditional mutual information is, in its most basic form, the expected value of the mutual information of two random variables given the value of a third.

Regular conditional probability is a concept that has developed to overcome certain difficulties in formally defining conditional probabilities for continuous probability distributions. It is defined as an alternative probability measure conditioned on a particular value of a random variable.

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