Probability distribution

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

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.

On the left is the probability density function. On the right is the cumulative distribution function, which is the area under the probability density curve.

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]

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

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:

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

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, ...

... of a continuous probability distribution, ...

... of a distribution which has both a continuous part and a discrete part.

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.

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 a ≤ X ≤ a) 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.

• 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).

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]

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 ${\displaystyle 0<p<1}$, we define

${\displaystyle {\displaystyle X={\begin{cases}1,&{\mbox{if }}U<p\\0,&{\mbox{if }}U\geq p\end{cases}}}}$

so that

${\displaystyle {\textrm {P}}(X=1)={\textrm {P}}(U<p)=p,{\textrm {P}}(X=0)={\textrm {P}}(U\geq p)=1-p.}$

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.

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.

• List of probability distributions
• Copula (statistics)
• Empirical probability
• Histogram
• Joint probability distribution
• Likelihood function
• List of statistical topics
• Kirkwood approximation
• Moment-generating function
• Quasiprobability distribution
• Riemann–Stieltjes integral application to probability theory

Wikimedia Commons has media related to Probability distribution.

• Hazewinkel, Michiel, ed. (2001) [1994], "Probability distribution", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
• Field Guide to Continuous Probability Distributions, Gavin E. Crooks.