Probability distribution

limx→−∞F(x)=0 and F:ℝ→ℝ that satisfies the first four of the properties above is the cumulative distribution function of some probability distribution on the real numbers.

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

Discrete probability distribution

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p(S) specifies the probability distribution for the sum P(X>9)=1/12+1/18+1/36=1/6, and all other probabilities in the distribution. Figure 4: 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. Figure 5: The cdf of a discrete probability distribution, ... Figure 6: ... of a continuous probability distribution, ... Figure 7: ... 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 (almost surely) which means that the probability of any event A is a countable set with p(n)=12n for F(x)=P(X≤x)=∑ω≤xp(ω). 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

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A discrete probability distribution is often represented with Dirac measures, also called one-point distributions (see below), the probability distributions of deterministic random variables. For any outcome ω. Given a discrete probability distribution, there is a countable set p. If PX=∑ω∈Ap(ω)δω.

Similarly, discrete distributions can be represented with the Dirac delta function as a generalized probability density function P(X∈E)=∫Ef(x)dx=∑ω∈Ap(ω)∫Eδ(x−ω)=∑ω∈A∩Ep(ω) for any event u0,u1,… be the values it can take with non-zero probability. Denote X takes any value except for X(ω)=∑iui1Ωi(ω) except on a set of probability zero, where X has a one-point distribution if it has a possible outcome x to 1 at

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. More precisely, a real random variable I=[a,b]⊂ℝ the probability of f over X to take any single value [a,b] is replaced by any measurable set F has the form X with regard to the distribution

In the measure-theoretic formalization of probability theory, a random variable is defined as a measurable function (𝒳,𝒜). Given that probabilities of events of the form X∗ℙ of X∗ℙ=ℙX−1.

Other kinds of distributions

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Figure 8: 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 γ:[a,b]→ℝn within some space O a subset of the support; if the probability measure exists for the system, one would expect the frequency of observing states inside set [t2,t3], which might not happen; for example, it could oscillate similar to a sine, F=αFdiscrete+βFac+γFsingular where coefficients F(x)=∫−∞xf(t)dt. The set of values with non-zero probability density has Lebesgue measure greater than zero.

Most standard distributions in statistical applications are either purely discrete ([0, 1). These random variates 0 < p < 1, define X has a Bernoulli distribution with parameter F be the generalized left inverse of F(p) ≤ x if and only if [0, 1], then the cumulative distribution function of λ — that is, with cumulative distribution function Finv(u)=−1λln⁡(1−u), and if X=−1λln⁡(1−U) has an exponential distribution with parameter

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.

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

• Log-normal distribution, for a single such quantity whose log is normally distributed
• Pareto distribution, for a single such quantity whose log is exponentially distributed; the prototypical power law distribution

• Discrete uniform distribution, for a finite set of values (e.g. the outcome of a fair dice)
• Continuous uniform distribution, for absolutely continuously distributed values

• Basic distributions:
  • Bernoulli distribution, for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no)
  • Binomial distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed total number of independent occurrences
  • Negative binomial distribution, for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs
  • Geometric distribution, for binomial-type observations but where the quantity of interest is the number of failures before the first success; a special case of the negative binomial distribution
• Related to sampling schemes over a finite population:
  • Hypergeometric distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, using sampling without replacement
  • Beta-binomial distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, sampling using a Pólya urn model (in some sense, the "opposite" of sampling without replacement)