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Probability distribution
its probability density function is f(x)=12πσ2exp(−(x−μ)22σ2). The parameter μ is the mean or expectation of the distribution (and also its median and
Normal_distribution
Probability that random variable X is less than or equal to x
statistics, the cumulative distribution function (CDF) of a real-valued random variable X {\displaystyle X} , or just distribution function of X {\displaystyle
Cumulative distribution function
Cumulative_distribution_function
Topics referred to by the same term
Distribution function may refer to Cumulative distribution function, a basic concept of probability theory Distribution function (physics), a function
Distribution_function
Type of probability distribution
expressed in terms of a joint cumulative distribution function and either in terms of a joint probability density function (in the case of continuous variables)
Joint probability distribution
Joint_probability_distribution
Distribution function associated with the empirical measure of a sample
statistics, an empirical distribution function (a.k.a. an empirical cumulative distribution function, eCDF) is the distribution function associated with the
Empirical distribution function
Empirical_distribution_function
Probability distribution
Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution. The Cauchy distribution f (
Cauchy_distribution
Generalized function whose value is zero everywhere except at zero
Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the real
Dirac_delta_function
Mathematical function for the probability a given outcome occurs in an experiment
Informally, a probability distribution tells us how likely different results are. Formally, it is a probability measure: a function that assigns probabilities
Probability_distribution
Uniform distribution on an interval
that it is contained in the distribution's support. The probability density function of the continuous uniform distribution is f ( x ) = { 1 b − a for
Continuous uniform distribution
Continuous_uniform_distribution
Probability distribution
hypergeometric function. For information on its inverse cumulative distribution function, see quantile function § Student's t-distribution. Certain values
Student's_t-distribution
Probability distribution
moment-generating function is MX(t)=𝔼[etX]=(1−p+pet)n. Usually the mode of a binomial B(n, p) distribution is equal to ⌊(n+1)p⌋, where ⌊⋅⌋ is the floor function. However
Binomial_distribution
Spectral density of light emitted by a black body
k_{\mathrm {B} }T,} where W is the Lambert W function and e is Euler's number. However, the distribution Bλ peaks at a different energy E = [ 5 + W (
Planck's_law
Description of particle density in statistical mechanics
In statistical mechanics, the radial distribution function, (or pair correlation function) g ( r ) {\displaystyle g(r)} in a system of particles (atoms
Radial_distribution_function
Description of continuous random distribution
100%. The terms probability distribution function and probability function can also denote the probability density function. However, this use is not standard
Probability_density_function
Discrete probability distribution
Poisson distribution.. A discrete random variable X is said to have a Poisson distribution with parameter λ>0 if it has a probability mass function given
Poisson_distribution
Statistical function that defines the quantiles of a probability distribution
quantile function of a probability distribution is the inverse of its cumulative distribution function. That is, the quantile function of a distribution D {\displaystyle
Quantile_function
Function of four real variables that defines how light is reflected at an opaque surface
reflectance distribution function (BRDF), symbol f r ( ω i , ω r ) {\displaystyle f_{\text{r}}(\omega _{\text{i}},\,\omega _{\text{r}})} , is a function of four
Bidirectional reflectance distribution function
Bidirectional_reflectance_distribution_function
Discrete-variable probability distribution
often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose
Probability_mass_function
Fourier transform of the probability density function
statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Concept in probability theory and statistics
density functions or cumulative distribution functions. There are particularly simple results for the moment generating functions of distributions defined
Moment_generating_function
Distribution of distances between pairs of particles in a given volume
The pair distribution function describes the distribution of distances between pairs of particles contained within a given volume. Mathematically, if a
Pair_distribution_function
Probability distribution
and Poisson distributions. The probability density function (pdf) of an exponential distribution is f ( x ; λ ) = { λ e − λ x x ≥ 0 , 0 x < 0. {\displaystyle
Exponential_distribution
Probability distribution
multiple variables is called a Dirichlet distribution. The probability density function (PDF) of the beta distribution, for 0 ≤ x ≤ 1 {\displaystyle 0\leq
Beta_distribution
Continuous probability distribution
parameter of the distribution. Its complementary cumulative distribution function is a stretched exponential function. The Weibull distribution is related to
Weibull_distribution
Function of seven variables
In molecular kinetic theory in physics, a system's distribution function is a function of seven variables, f ( t , x , y , z , v x , v y , v z ) {\displaystyle
Distribution function (physics)
Distribution_function_(physics)
Probability of survival beyond any specified time
cumulative distribution function of the lifetime. Sometimes complementary cumulative distribution functions are called survival functions in general. Let the
Survival_function
Probability distribution
exponential distribution scaled by 1/2. The probability density function of the Laplace distribution is also reminiscent of the normal distribution; however
Laplace_distribution
Statistical description for the behavior of fermions
F(\varepsilon )} is called the Fermi function and is the same function that is used for the Fermi–Dirac distribution n ¯ i {\displaystyle {\bar {n}}_{i}}
Fermi–Dirac_statistics
Part of signal processing in time-frequency analysis
The Wigner distribution function (WDF) is used in signal processing as a transform in time-frequency analysis. The WDF was first proposed in physics to
Wigner_distribution_function
Particular case of the generalized extreme value distribution
his original papers describing the distribution. The cumulative distribution function of the Gumbel distribution (maximum case) is F ( x ; μ , β ) =
Gumbel_distribution
Probability distribution
number is Rayleigh-distributed. The probability density function of the Rayleigh distribution is f ( x ; σ ) = x σ 2 e − x 2 / ( 2 σ 2 ) , x ≥ 0 , {\displaystyle
Rayleigh_distribution
Probability distribution
case of the inverse-gamma distribution and a stable distribution. The probability density function of the Lévy distribution over the domain x ≥ μ {\displaystyle
Lévy_distribution
Probability distribution
incomplete gamma function. If α is a positive integer (i.e., the distribution is an Erlang distribution), the cumulative distribution function has the following
Gamma_distribution
Probability distribution
The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. This distribution has neither a
Cantor_distribution
Topological vector spaces
test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are
Spaces of test functions and distributions
Spaces_of_test_functions_and_distributions
Probability distribution
model the distribution of wealth, then the parameter α is called the Pareto index. From the definition, the cumulative distribution function of a Pareto
Pareto_distribution
Model of hadrons
partons (nonvalence partons) in addition to valence partons. A parton distribution function (PDF) within so called collinear factorization is defined as the
Parton_(particle_physics)
Statistical test comparing two probability distributions
empirical distribution function of the sample and the cumulative distribution function of the reference distribution, or between the empirical distribution functions
Kolmogorov–Smirnov_test
Class of statistical models
that have arbitrary distributions (rather than simply normal distributions)[clarification needed], and for an arbitrary function of the response variable
Generalized_linear_model
Probability distribution modeling a coin toss which need not be fair
\\\Pr(X{=}0)&=q=1-p.\end{aligned}}} The probability mass function f {\displaystyle f} of this distribution, over possible outcomes k, is f ( k ; p ) = { p if
Bernoulli_distribution
Probability distribution
distribution: X ∼ NB ( r , p ) {\displaystyle X\sim \operatorname {NB} (r,p)} The probability mass function of the negative binomial distribution is
Negative binomial distribution
Negative_binomial_distribution
Mathematical function common in physics
the complementary cumulative Weibull distribution. The stretched exponential is also the characteristic function, basically the Fourier transform, of
Stretched exponential function
Stretched_exponential_function
Specific probability distribution function, important in physics
vector. The Maxwellian distribution function for particles moving in only one direction, if this direction is x, is a normal distribution with a standard deviation
Maxwell–Boltzmann distribution
Maxwell–Boltzmann_distribution
Continuous probability distribution
integers, but the distribution is well-defined for positive real values of these parameters. The cumulative distribution function is F ( x ; d 1 , d
F-distribution
Continuous probability distribution
statistics, the logistic distribution is a continuous probability distribution. Its cumulative distribution function is the logistic function, which appears in
Logistic_distribution
Family of continuous probability distributions
the distribution are equal, 𝔼[X]=Var(X). Also, the cumulative distribution function (cdf) of the single parameter inverse Gaussian distribution is
Inverse_Gaussian_distribution
theory, there are different notions of distribution function which are used in different context. Distribution functions (in the sense of measure theory) are
Distribution function (measure theory)
Distribution_function_(measure_theory)
Mathematical function having a characteristic S-shaped curve or sigmoid curve
tangent functions have been used as the activation function of artificial neurons. Sigmoid curves are also common in statistics as cumulative distribution functions
Sigmoid_function
Indicator function of positive numbers
scaled and shifted Sigmoid function. In general, any cumulative distribution function of a continuous probability distribution that is peaked around zero
Heaviside_step_function
Distribution of variables which satisfies a stability property under linear combinations
and approaches the Dirac delta function in the limit as α → 0 {\displaystyle \alpha \rightarrow 0} . The distributions have undefined variance for α <
Stable_distribution
Techniques and methods in signal processing
time–frequency distribution function (Wigner distribution function, or WDF), Modified Wigner distribution function, Gabor–Wigner distribution function, and so
Time–frequency_analysis
Mathematical function
The definition of the bidirectional scattering distribution function (BSDF) is not well standardized. The term was probably introduced in 1980 by Bartell
Bidirectional scattering distribution function
Bidirectional_scattering_distribution_function
Relative importance of certain frequencies in a composite signal
involving distributions (in the sense of Laurent Schwartz, not in the sense of a statistical Cumulative distribution function) instead of functions. If R
Spectral_density
Probability distribution
normal distribution. Equivalently, if Y has a normal distribution, then the exponential function of Y, X = exp(Y), has a log-normal distribution. A random
Log-normal_distribution
Smooth approximation of one-hot arg max
softmax function, also known as softargmax or normalized exponential function, converts a tuple of K real numbers into a probability distribution over K
Softmax_function
Probability distribution and special case of gamma distribution
{\displaystyle k} . Tables of the chi-squared cumulative distribution function are widely available and the function is included in many spreadsheets and all statistical
Chi-squared_distribution
Statistical distribution for dependence between random variables
a copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform on the interval [0
Copula_(statistics)
Topics referred to by the same term
Probability distribution function may refer to: Probability distribution, a function that gives the probabilities of occurrence of possible outcomes for
Probability distribution function
Probability_distribution_function
Probability distribution
first success. An alternative parameterization of the distribution gives the probability mass function P ( Y = k ) = ( P Q ) k ( 1 − P Q ) {\displaystyle
Geometric_distribution
Mathematical function
incomplete beta function is the cumulative distribution function of the beta distribution, and is related to the cumulative distribution function F ( k ; n
Beta_function
Probability distribution
for any distribution that has heavier tails than the normal distribution.) The distribution of a random variable X with distribution function F is said
Heavy-tailed_distribution
Wigner distribution function in physics as opposed to in signal processing
The Wigner quasiprobability distribution, also called the Wigner function or the Wigner–Ville distribution, after Eugene Wigner and Jean-André Ville, is
Wigner quasiprobability distribution
Wigner_quasiprobability_distribution
Basic method for pseudo-random number sampling
sample numbers at random from any probability distribution given its cumulative distribution function. Inverse transformation sampling takes uniform
Inverse_transform_sampling
Probability distribution
characteristic function for the Cauchy distribution is well defined, as is the characteristic function for the normal distribution. The characteristic function for
Voigt_profile
Family of continuous probability distributions
generalized chi-squared distribution for even numbers of degrees of freedom. The cumulative distribution function of the Erlang distribution is F ( x ; k , λ
Erlang_distribution
Statistics function
In statistics, the Q-function is the tail distribution function of the standard normal distribution. In other words, Q ( x ) {\displaystyle Q(x)} is the
Q-function
Probability distribution of energy states of a system
will be in a certain state as a function of that state's energy and the temperature of the system. The distribution is expressed in the form pi∝exp(−εikBT)
Boltzmann_distribution
Two-parameter family of continuous probability distributions
inverse gamma distribution differently, as a scaled inverse chi-squared distribution. The inverse gamma distribution's probability density function is defined
Inverse-gamma_distribution
Family of probability distributions often used to model tails or extreme values
parameterization was introduced by James Pickands III . The cumulative distribution function of X ∼ GPD ( μ , σ , ξ ) {\displaystyle X\sim {\text{GPD}}(\mu
Generalized Pareto distribution
Generalized_Pareto_distribution
Probability distribution
in an ideal gas (chi distribution with three degrees of freedom). The probability density function (pdf) of the chi-distribution is f ( x ; k ) = { x
Chi_distribution
Topics referred to by the same term
generalized function used to formulate solutions of partial differential equations Distribution (number theory), its algebraic analogue Distribution (differential
Distribution
Statistical inequality
distance of an empirically determined distribution function from its associated population distribution function. It is named after Aryeh Dvoretzky, Jack
Dvoretzky–Kiefer–Wolfowitz inequality
Dvoretzky–Kiefer–Wolfowitz_inequality
Variable representing a random phenomenon
random variable and its distribution is a discrete probability distribution, i.e. can be described by a probability mass function that assigns a probability
Random_variable
Family of continuous probability distributions
probability density function, cumulative distribution function and quantile functions can be expressed in closed form. This distribution was originally proposed
Kumaraswamy_distribution
Continuous probability distribution, named after Benjamin Gompertz
distributed according to the Gompertz distribution. The probability density function of the Gompertz distribution is: f ( x ; η , b ) = b η exp ( η +
Gompertz_distribution
a spherical contact distribution function, first contact distribution function, or empty space function is a mathematical function that is defined in relation
Spherical contact distribution function
Spherical_contact_distribution_function
Probability distribution used to model household income
"generalized log-logistic distribution". The Burr (Type XII) distribution has probability density function: f ( x ; c , k ) = c k x c − 1 ( 1 + x c ) k + 1 f (
Burr_distribution
Inputs for which a function's value is non-zero
in a natural way to functions taking values in more general sets than ℝ and to other objects, such as measures or distributions. The most common situation
Support_(mathematics)
Type of probability distribution
probability density function is sometimes referred to as a mixture density. The cumulative distribution function (and the probability density function if it exists)
Mixture_distribution
Probability distribution
triangular distribution with parameters a , b {\displaystyle a,b} and c {\displaystyle c} . This can be obtained from the cumulative distribution function. The
Triangular_distribution
Continuous probability distribution
the generalized extreme value distribution. It has the cumulative distribution function Pr ( X ≤ x ) = e − x − α if x > 0 . {\displaystyle
Fréchet_distribution
Property of having a unique mode or maximum value
of the distribution, not just to the strict definition of mode which is usual in statistics. If there is a single mode, the distribution function is called
Unimodality
Graph of wealth or income distribution
For a continuous distribution with the probability density function f and the cumulative distribution function F, the Lorenz curve L is given
Lorenz_curve
Distribution within a group of stars of the ratio of iron to hydrogen in a star
The metallicity distribution function is an important concept in stellar and galactic evolution. It is a curve of what proportion of stars have a particular
Metallicity distribution function
Metallicity_distribution_function
Discrete probability distribution
0 < p < 1. Because of the identity above, the distribution is properly normalized. The cumulative distribution function is F ( k ) = 1 + B ( p ; k + 1 , 0 ) ln
Logarithmic_distribution
Kth smallest value in a statistical sample
continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution. For
Order_statistic
Objects that generalize functions
In mathematical analysis, distributions are a type of generalized function that makes it possible to differentiate functions whose derivatives do not exist
Distribution (mathematical analysis)
Distribution_(mathematical_analysis)
Fundamental theorem in probability theory and statistics
distribution to a normal 𝒩(0,σ2): n(X¯n−μ)⟶d𝒩(0,σ2). In the case σ>0, convergence in distribution means that the cumulative distribution functions of
Central_limit_theorem
Probability theory
the distribution of the reciprocal, Y = 1 / X. If the distribution of X is continuous with density function f(x) and cumulative distribution function F(x)
Inverse_distribution
Statistical distribution
density function, within the support of the distribution, being proportional to the reciprocal of the variable. The reciprocal distribution is an example
Reciprocal_distribution
Conditional probability used in Bayesian statistics
are related as follows: Given a prior belief that a probability distribution function is p ( θ ) {\displaystyle p(\theta )} and that the observations
Posterior_probability
Functional relationship between two quantities
looser sense, a power-law probability distribution is a distribution whose density function (or mass function in the discrete case) has the form, for
Power_law
Type of probability distribution
probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function involves the arcsine and the square
Arcsine_distribution
Measure of inequality of a statistical distribution
When the income (or wealth) distribution is given as a continuous probability density function p(x), the Gini coefficient is again half of
Gini_coefficient
Aspect of probability and statistics
[c,d]} . Finding the marginal cumulative distribution function from the joint cumulative distribution function is easy. Recall that: For discrete random
Marginal_distribution
Statistic which divides data into four same-sized parts for analysis
quartile. The quantile function is the inverse of the cumulative distribution function if the cumulative distribution function is monotonically increasing
Quartile
Continuous probability distribution
hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional
Hyperbolic secant distribution
Hyperbolic_secant_distribution
Type of mathematical function
distribution, the Pareto distribution, the log-normal distribution, and the F-distribution. Note that the cumulative distribution function (CDF) of all log-concave
Logarithmically concave function
Logarithmically_concave_function
Generalization of the one-dimensional normal distribution to higher dimensions
statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional
Multivariate normal distribution
Multivariate_normal_distribution
Statistic which divides a data set into 100 parts and analyzes it as a percentage
the percentile approximates the percentile function, the inverse of the cumulative distribution function. A related quantity is the percentile rank of
Percentile
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DISTRIBUTION FUNCTION
DISTRIBUTION FUNCTION
DISTRIBUTION FUNCTION
DISTRIBUTION FUNCTION
DISTRIBUTION FUNCTION
DISTRIBUTION FUNCTION
DISTRIBUTION FUNCTION
DISTRIBUTION FUNCTION
DISTRIBUTION FUNCTION
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