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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
Probability distribution
quantile function of a distribution is the inverse of the cumulative distribution function. The quantile function of the standard normal distribution is called
Normal_distribution
Statistical function that defines the quantiles of a probability distribution
probability distribution's quantile function is the inverse of its cumulative distribution function. That is, the quantile function of a distribution D {\displaystyle
Quantile_function
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
Uniform distribution on an interval
\\[2pt]0&{\text{otherwise}}.\end{cases}}} The cumulative distribution function of the continuous uniform distribution is: F ( x ) = { 0 for x < a , x − a b
Continuous uniform distribution
Continuous_uniform_distribution
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
Probability distribution
{1}{\pi \gamma }}.\!} The Cauchy distribution is the probability distribution with the following cumulative distribution function (CDF): F ( x ; x 0 , γ ) =
Cauchy_distribution
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
Fourier transform of the probability density function
density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
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
Description of continuous random distribution
refer to the cumulative distribution function (CDF), or it may be a probability mass function (PMF) rather than the density. Density function itself is also
Probability_density_function
Table of probabilities related to the normal distribution
mathematical table for the values of Φ, the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic
Standard_normal_table
Probability distribution
equivalent to the cumulative distribution functions of the beta distribution and of the F-distribution: F ( k ; n , p ) = F beta-distribution ( x = 1 − p ;
Binomial_distribution
Probability distribution
parameter and c {\displaystyle c} is the scale parameter. The cumulative distribution function is F ( x ; μ , c ) = erfc ( c 2 ( x − μ ) ) = 2 − 2 Φ ( c
Lévy_distribution
Mathematical function for the probability a given outcome occurs in an experiment
practice, probability distributions are often described by functions such as cumulative distribution functions, probability mass functions, or probability density
Probability_distribution
Probability distribution
{\displaystyle \varphi } be respectively the cumulative probability distribution function and the probability density function of the N ( 0 , 1 ) {\displaystyle
Log-normal_distribution
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
Probability distribution
continuous probability distribution that has a constant failure rate. The quantile function (inverse cumulative distribution function) for Exp(λ) is F − 1
Exponential_distribution
Sigmoid shape special function
(-iz)=\operatorname {erfcx} (-iz).} The error function is essentially identical to the standard normal cumulative distribution function, denoted Φ, also named norm(x)
Error_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
Probability distribution
that the Voigt distributions are also closed under convolution. Using the above definition for z , the cumulative distribution function (CDF) can be found
Voigt_profile
Aspect of probability and statistics
y\in [c,d]} . Finding the marginal cumulative distribution function from the joint cumulative distribution function is easy. Recall that: For discrete
Marginal_distribution
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
hypergeometric function. For information on its inverse cumulative distribution function, see quantile function § Student's t-distribution. Certain values
Student's_t-distribution
Generalization of the one-dimensional normal distribution to higher dimensions
vector, its probability density function, cumulative distribution function, and inverse cumulative distribution function can be computed with the numerical
Multivariate normal distribution
Multivariate_normal_distribution
Continuous probability distribution
parameter m (the minimum) and a scale parameter s > 0 with the cumulative distribution function Pr ( X ≤ x ) = exp [ − ( x − m s ) − α ]
Fréchet_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
Class of statistical models
logit models). Alternatively, the inverse of any continuous cumulative distribution function (CDF) can be used for the link since the CDF's range is [ 0
Generalized_linear_model
Statistical distribution for dependence between random variables
statistics, a copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform on the
Copula_(statistics)
Discrete probability distribution
Excel: function POISSON( x, mean, cumulative), with a flag to specify the cumulative distribution; Mathematica: univariate Poisson distribution as PoissonDistribution[
Poisson_distribution
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
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
Variable representing a random phenomenon
and variance of a random variable, its cumulative distribution function, and the moments of its distribution. However, the definition above is valid
Random_variable
Probability distribution
where Γ ( z ) {\displaystyle \Gamma (z)} is the gamma function. The cumulative distribution function is given by: F ( x ; k ) = P ( k / 2 , x 2 / 2 ) {\displaystyle
Chi_distribution
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
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
Number of occurrences in an experiment or study
They may be used as estimators of empirical probabilities or cumulative distribution functions, for instance. The relative frequency of an event is the absolute
Frequency_(statistics)
Continuous probability distribution, named after Benjamin Gompertz
differently (Gompertz–Makeham law of mortality). The cumulative distribution function of the Gompertz distribution is: F ( x ; η , b ) = 1 − exp ( − η ( e b
Gompertz_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
Linear combination of indicator functions of real intervals
a random variable whose cumulative distribution function is piecewise constant. In this case, it is locally a step function (globally, it may have an
Step_function
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
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
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
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
Probability distribution
Finally, the probability density function for X {\displaystyle X} is the derivative of its cumulative distribution function, which by the fundamental theorem
Rayleigh_distribution
Discrete-variable probability distribution
The discontinuity of probability mass functions is related to the fact that the cumulative distribution function of a discrete random variable is also
Probability_mass_function
Probability distribution
Laplace distribution is easy to integrate (if one distinguishes two symmetric cases) due to the use of the absolute value function. Its cumulative distribution
Laplace_distribution
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
Two-parameter family of continuous probability distributions
)={\frac {f(x/\beta ;\alpha ,1)}{\beta }}} The cumulative distribution function is the regularized gamma function F ( x ; α , β ) = Γ ( α , β x ) Γ ( α ) =
Inverse-gamma_distribution
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
Probability distribution on equally likely outcomes
distribution and n = b − a + 1. {\textstyle n=b-a+1.} In these cases the cumulative distribution function (CDF) of the discrete uniform distribution can
Discrete_uniform_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
Probability distribution
cumulative distribution function of a beta distribution can be expressed in terms of the cumulative distribution function of a binomial distribution with
Beta_distribution
Concept in probability theory and statistics
plot shows an example of the distribution of such a variable. The generalization of the cumulative distribution function from real to complex random variables
Complex_random_variable
Probability distribution
\choose k}\!\!\right)} . The cumulative distribution function can be expressed in terms of the regularized incomplete beta function: F ( k ; r , p ) ≡ Pr (
Negative binomial distribution
Negative_binomial_distribution
Set of quantities in probability theory
;\end{aligned}}} where F {\textstyle F} is the cumulative distribution function. The cumulant generating function will have vertical asymptote(s) at the negative
Cumulant
Family of probability distributions
whole real line. Since the cumulative distribution function is invertible, the quantile function for the GEV distribution has an explicit expression,
Generalized extreme value distribution
Generalized_extreme_value_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
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
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
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 used to model household income
scales the underlying variate and is a positive real. The cumulative distribution function is: F ( x ; c , k ) = 1 − ( 1 + x c ) − k {\displaystyle F(x;c
Burr_distribution
Probability distribution on the circle
)]\right)} where Ij(x) is the modified Bessel function of order j. The cumulative distribution function is not analytic and is best found by integrating
Von_Mises_distribution
Graphical representation of the distribution of income or of wealth
For a continuous distribution with the probability density function f and the cumulative distribution function F, the Lorenz curve L is given
Lorenz_curve
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
Chi-squared_distribution
Property of having a unique mode or maximum value
behavior of the cumulative distribution function (cdf). If the cdf is convex for x < m and concave for x > m, then the distribution is unimodal, m being
Unimodality
Continuous probability distribution for a non-negative random variable
in shape to the log-normal distribution but has heavier tails . Unlike the log-normal, its cumulative distribution function can be written in closed form
Log-logistic_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
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
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
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
Functional relationship between two quantities
convenient way to do this is via the (complementary) cumulative distribution (ccdf) that is, the survival function, P ( x ) = Pr ( X > x ) {\displaystyle P(x)=\Pr(X>x)}
Power_law
Statistical considerations on how many observations to make
formulas, by simulation, by Mead's resource equation, or by the cumulative distribution function: The table shown on the right can be used in a two-sample t-test
Sample_size_determination
Right continuous function with left limits
consequence of their definition, all cumulative distribution functions are càdlàg functions. For instance the cumulative at point r {\displaystyle r} correspond
Càdlàg
{n\,}}}}} Differentiating the cumulative distribution function with respect to q gives the probability density function. f R ( q ; k , ν ) = 2 π k ( k
Studentized range distribution
Studentized_range_distribution
Statistical distribution
rank-size distribution is not a probability distribution or cumulative distribution function. Rather, it is a discrete form of a quantile function (inverse
Rank–size_distribution
Family of probability distributions
^{-1}(U)-\gamma }{\delta }}\right)+\xi } where Φ is the cumulative distribution function of the normal distribution. N. L. Johnson firstly proposes the transformation :
Johnson's_SU-distribution
Probability distribution
discrete distribution; its probability mass function equals 1 in a and 0 everywhere else. In the case of a real-valued random variable, the cumulative distribution
Degenerate_distribution
Measure of inequality of a statistical distribution
calculated directly from the cumulative distribution function of the distribution F(y). Defining μ as the mean of the distribution, then specifying that F(y)
Gini_coefficient
Type of probability distribution
probability density function of the standard normal distribution and Φ ( ⋅ ) {\displaystyle \Phi (\cdot )} is its cumulative distribution function Φ ( x ) = 1
Truncated_normal_distribution
Measure of the asymmetry of random variables
} where Q is the quantile function (i.e., the inverse of the cumulative distribution function). The numerator is difference between the
Skewness
Probability distribution
Poisson binomial distribution function" by Biscarri et al. The cumulative distribution function (CDF) can be expressed as: Pr ( K ≤ k ) = ∑ ℓ = 0 k ∑ A ∈ F
Poisson_binomial_distribution
Power series derived from a discrete probability distribution
probability generating functions, then they have identical distributions. The normalization of the probability mass function can be expressed in terms
Probability generating function
Probability_generating_function
Probability distribution
= 1 θ {\displaystyle E[Y]=\mu ={\frac {1}{\theta }}} . The cumulative distribution function (CDF) is given by F Y ( y ; σ ) = ∫ 0 y 1 σ 2 π exp ( − x
Half-normal_distribution
Statistical test
empirical distribution function). If the hypothesized distribution is F {\displaystyle F} , and empirical (sample) cumulative distribution function is F n
Anderson–Darling_test
Probability distribution in mathematics
( s ) {\displaystyle \zeta (s)} is the Riemann zeta function. The cumulative distribution function is given by P ( x ≤ k ) = H k , s ζ ( s ) , {\displaystyle
Zeta_distribution
Average value of a random variable
{\displaystyle A} , in which the integral is Lebesgue. the cumulative distribution function of A {\displaystyle A} is absolutely continuous. for any Borel
Expected_value
Statistical method of dividing data into equal-sized intervals for analysis
the cumulative distribution function of a random variable is known, the q-quantiles are the application of the quantile function (the inverse function of
Quantile
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
secant function. The cumulative distribution function (cdf) of the standard distribution is a scaled and shifted version of the Gudermannian function, F (
Hyperbolic secant distribution
Hyperbolic_secant_distribution
In mathematics, a quantitative measure of the shape of a set of points
the probability distribution. More generally, if F is a cumulative probability distribution function of any probability distribution, which may not have
Moment_(mathematics)
Theorem in probability theory
the cumulative distribution function of Y n n σ , {\displaystyle {Y_{n}{\sqrt {n}} \over {\sigma }},} and Φ the cumulative distribution function of the
Berry–Esseen_theorem
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
Type of activation function
the cumulative distribution function of the standard normal distribution and erf ( z ) {\displaystyle \operatorname {erf} (z)} is the error function. This
Rectified_linear_unit
Statistics function
relation to the cumulative distribution function of the normal distribution, the Q-function can also be expressed in terms of the error function, which is an
Q-function
at level τ {\textstyle \tau } of the probability distribution with cumulative distribution function F {\textstyle F} is uniquely characterized by any
Expectile
Survival functions or complementary cumulative distribution functions are often denoted by placing an overbar over the symbol for the cumulative: F ¯ (
Notation in probability and statistics
Notation_in_probability_and_statistics
Probability distribution in economics
Dagum distribution is Statistical Size Distributions in Economics and Actuarial Sciences. The cumulative distribution function of the Dagum distribution (Type
Dagum_distribution
Statistical measure of how far values spread from their average
probability density function f ( x ) {\displaystyle f(x)} , and F ( x ) {\displaystyle F(x)} is the corresponding cumulative distribution function, then Var
Variance
Shorthand used in statistics
numerical values "68%, 95%, 99.7%" come from the cumulative distribution function of the normal distribution. The prediction interval for any standard score
68–95–99.7_rule
CUMULATIVE DISTRIBUTION-FUNCTION
CUMULATIVE DISTRIBUTION-FUNCTION
Girl/Female
Indian
Beautiful woman, Distributor, Divider
Girl/Female
Muslim
Beautiful woman, Distributor, Divider
Boy/Male
Indian
Distributor, Divider
Girl/Female
Arabic
Distributor
Surname or Lastname
English
English : of uncertain origin. Reaney suggests that it may be habitational name from Wincheap Street in Canterbury, but this origin is not supported by the present-day distribution of the surname, which is heavily concentrated in northeastern England.
Surname or Lastname
English (Yorkshire)
English (Yorkshire) : apparently a habitational name from a lost or unidentified minor place in West Yorkshire, probably in the parish of Halifax, to judge by the distribution of early occurrences of the surname.
Surname or Lastname
English (Lancashire)
English (Lancashire) : habitational name from a place so called, perhaps Forshaw Heath in Solihull, Warwickshire, although the modern distribution is much further north.
Boy/Male
Afghan, Arabic, German, Gujarati, Hindu, Indian, Kannada, Muslim, Pashtun, Sindhi
Divider; One who Divides; Distributor
Boy/Male
Muslim/Islamic
Divider distributor
Boy/Male
Indian
Distributor, Divider
Girl/Female
Muslim
Beautiful woman, Distributor, Divider
Girl/Female
Indian
Beautiful woman, Distributor, Divider
Surname or Lastname
English (Cambridge)
English (Cambridge) : unexplained; perhaps a habitational name from a lost or unidentified place. There are two places in England called Warland, in Durham and West Yorkshire, but the distribution of the modern surname suggests that a different souce is most probably involved.
Girl/Female
Indian, Sikh
Distributing Happiness
Surname or Lastname
English (Devon)
English (Devon) : unexplained. Reaney and Wilson suggest that this may be from an Anglo-Scandinavian personal name Tukka, but the distribution in England makes a Scandinavian connection unlikely.
Boy/Male
Muslim
Distributor, Divider
Boy/Male
Muslim
Distributor, Divider
Boy/Male
Arabic, British, Islamic, Malaysian, Muslim, Pakistani, Tamil, Urdu
Distribution
Surname or Lastname
English (Lincolnshire)
English (Lincolnshire) : unexplained. Black identified this as a Scottish name of Pictish origin. However, the modern distribution of the surname, almost exclusively in Lincolnshire and adjoining counties, suggests a more localized eastern English origin.
Surname or Lastname
English
English : unexplained; perhaps a habitational name from a lost or unidentified place. It has been suggested that it might be an altered form of Scottish Ballantine, but the distribution and variants (including Blanding) make it more probable that it is an altered form of a French original.
CUMULATIVE DISTRIBUTION-FUNCTION
CUMULATIVE DISTRIBUTION-FUNCTION
Boy/Male
Indian
Open Personality
Female
Spanish
Spanish feminine form of Latin Angelus, ÃNGELA means "angel, messenger."
Boy/Male
American, Anglo, Australian, Basque, Mexican, Russian
Spear
Boy/Male
Arabic Indian
Gentle.
Girl/Female
Arabic
Newspaper
Girl/Female
Spanish
From Dionysus god of wine.
Boy/Male
French
Famous wolf.
Surname or Lastname
English
English : metronymic or patronymic from Hill 2.
Surname or Lastname
Dutch and North German
Dutch and North German : from a Germanic personal name composed of bald ‘bold’ + gÄr, gÄ“r ‘spear’.German : habitational name from any of several places called Belgern, near Torgau and in Saxony.English : variant of Bolger.
Surname or Lastname
English
English : variant of Gascon.
CUMULATIVE DISTRIBUTION-FUNCTION
CUMULATIVE DISTRIBUTION-FUNCTION
CUMULATIVE DISTRIBUTION-FUNCTION
CUMULATIVE DISTRIBUTION-FUNCTION
CUMULATIVE DISTRIBUTION-FUNCTION
a.
Of or pertaining to distribution.
a.
Characterized by accumulation; serving to collect or amass; cumulative; additional.
a.
Joining subject and predicate; copulative.
adv.
By distribution; singly; not collectively; in a distributive manner.
n.
The act of distributing or dispensing; the act of dividing or apportioning among several or many; apportionment; as, the distribution of an estate among heirs or children.
v. i.
To make distribution.
a.
Given by same testator to the same legatee; -- said of a legacy.
n.
A distributive adjective or pronoun; also, a distributive numeral.
a.
Serving to couple, unite, or connect; as, a copulative conjunction like "and".
imp. & p. p.
of Cumulate
n.
Disposition; distribution; management.
a.
Tending to prove the same point to which other evidence has been offered; -- said of evidence.
n.
A copulative conjunction.
a.
Augmenting, gaining, or giving force, by successive additions; as, a cumulative argument, i. e., one whose force increases as the statement proceeds.
n.
Distribution; apportionment.
p. pr. & vb. n.
of Cumulate
a.
Inclined to emulation; aspiring to competition; rivaling; as, an emulative person or effort.
adv.
In a copulative manner.
n.
Distribution; dealing; apportionment.
a.
Expressing separation; denoting a taking singly, not collectively; as, a distributive adjective or pronoun, such as each, either, every; a distributive numeral, as (Latin) bini (two by two).