In
probability theory
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 o ...
and
statistics, the Weibull distribution is a continuous
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
. It is named after Swedish mathematician
Waloddi Weibull
Ernst Hjalmar Waloddi Weibull (18 June 1887 – 12 October 1979) was a Swedish civil engineer, materials scientist, and applied mathematician. The Weibull distribution is named after him.
Education and career
Weibull joined the Swedish Coast G ...
, who described it in detail in 1951, although it was first identified by
Maurice René Fréchet and first applied by to describe a
particle size distribution
The particle-size distribution (PSD) of a powder, or granular material, or particles dispersed in fluid, is a list of values or a mathematical function that defines the relative amount, typically by mass, of particles present according to size. Si ...
.
Definition
Standard parameterization
The
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
of a Weibull
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
is
:
where ''k'' > 0 is the ''
shape parameter'' and λ > 0 is the ''
scale parameter'' of the distribution. Its
complementary cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ever ...
is a
stretched exponential function
The stretched exponential function f_\beta (t) = e^ is obtained by inserting a fractional power law into the exponential function.
In most applications, it is meaningful only for arguments between 0 and +∞. With , the usual exponential functio ...
. The Weibull distribution is related to a number of other probability distributions; in particular, it
interpolates
In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.
In engineering and science, one often has a n ...
between the
exponential distribution
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant averag ...
(''k'' = 1) and the
Rayleigh distribution (''k'' = 2 and
).
If the quantity ''X'' is a "time-to-failure", the Weibull distribution gives a distribution for which the
failure rate
Failure rate is the frequency with which an engineered system or component fails, expressed in failures per unit of time. It is usually denoted by the Greek letter λ (lambda) and is often used in reliability engineering.
The failure rate of a ...
is proportional to a power of time. The ''shape'' parameter, ''k'', is that power plus one, and so this parameter can be interpreted directly as follows:
* A value of
indicates that the
failure rate
Failure rate is the frequency with which an engineered system or component fails, expressed in failures per unit of time. It is usually denoted by the Greek letter λ (lambda) and is often used in reliability engineering.
The failure rate of a ...
decreases over time (like in case of the
Lindy effect, which however corresponds to
Pareto distributions rather than Weibull distributions). This happens if there is significant "infant mortality", or defective items failing early and the failure rate decreasing over time as the defective items are weeded out of the population. In the context of the
diffusion of innovations
Diffusion of innovations is a theory that seeks to explain how, why, and at what rate new ideas and technology spread. Everett Rogers, a professor of communication studies, popularized the theory in his book ''Diffusion of Innovations''; the boo ...
, this means negative word of mouth: the
hazard function is a monotonically decreasing function of the proportion of adopters;
* A value of
indicates that the failure rate is constant over time. This might suggest random external events are causing mortality, or failure. The Weibull distribution reduces to an exponential distribution;
* A value of
indicates that the failure rate increases with time. This happens if there is an "aging" process, or parts that are more likely to fail as time goes on. In the context of the
diffusion of innovations
Diffusion of innovations is a theory that seeks to explain how, why, and at what rate new ideas and technology spread. Everett Rogers, a professor of communication studies, popularized the theory in his book ''Diffusion of Innovations''; the boo ...
, this means positive word of mouth: the hazard function is a monotonically increasing function of the proportion of adopters. The function is first convex, then concave with an inflexion point at
.
In the field of
materials science, the shape parameter ''k'' of a distribution of strengths is known as the
Weibull modulus. In the context of
diffusion of innovations
Diffusion of innovations is a theory that seeks to explain how, why, and at what rate new ideas and technology spread. Everett Rogers, a professor of communication studies, popularized the theory in his book ''Diffusion of Innovations''; the boo ...
, the Weibull distribution is a "pure" imitation/rejection model.
Alternative parameterizations
First alternative
Applications in
medical statistics
Medical statistics deals with applications of statistics to medicine and the health sciences, including epidemiology, public health, forensic medicine, and clinical research. Medical statistics has been a recognized branch of statistics in the ...
and
econometrics
Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8� ...
often adopt a different parameterization. The shape parameter ''k'' is the same as above, while the scale parameter is
. In this case, for ''x'' ≥ 0, the probability density function is
:
the cumulative distribution function is
:
the hazard function is
:
and the mean is
:
Second alternative
A second alternative parameterization can also be found. The shape parameter ''k'' is the same as in the standard case, while the scale parameter ''λ'' is replaced with a rate parameter ''β'' = 1/''λ''. Then, for ''x'' ≥ 0, the probability density function is
:
the cumulative distribution function is
:
and the hazard function is
:
In all three parameterizations, the hazard is decreasing for k < 1, increasing for k > 1 and constant for k = 1, in which case the Weibull distribution reduces to an exponential distribution.
Properties
Density function
The form of the density function of the Weibull distribution changes drastically with the value of ''k''. For 0 < ''k'' < 1, the density function tends to ∞ as ''x'' approaches zero from above and is strictly decreasing. For ''k'' = 1, the density function tends to 1/''λ'' as ''x'' approaches zero from above and is strictly decreasing. For ''k'' > 1, the density function tends to zero as ''x'' approaches zero from above, increases until its mode and decreases after it. The density function has infinite negative slope at ''x'' = 0 if 0 < ''k'' < 1, infinite positive slope at ''x'' = 0 if 1 < ''k'' < 2 and null slope at ''x'' = 0 if ''k'' > 2. For ''k'' = 1 the density has a finite negative slope at ''x'' = 0. For ''k'' = 2 the density has a finite positive slope at ''x'' = 0. As ''k'' goes to infinity, the Weibull distribution converges to a
Dirac delta distribution centered at ''x'' = λ. Moreover, the skewness and coefficient of variation depend only on the shape parameter. A generalization of the Weibull distribution is the
hyperbolastic distribution of type III.
Cumulative distribution function
The
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
for the Weibull distribution is
:
for ''x'' ≥ 0, and ''F''(''x''; ''k''; λ) = 0 for ''x'' < 0.
If ''x'' = λ then ''F''(''x''; ''k''; λ) = 1 − ''e''
−1 ≈ 0.632 for all values of ''k''. Vice versa: at ''F''(''x''; ''k''; ''λ'') = 0.632 the value of ''x'' ≈ ''λ''.
The quantile (inverse cumulative distribution) function for the Weibull distribution is
:
for 0 ≤ ''p'' < 1.
The
failure rate
Failure rate is the frequency with which an engineered system or component fails, expressed in failures per unit of time. It is usually denoted by the Greek letter λ (lambda) and is often used in reliability engineering.
The failure rate of a ...
''h'' (or hazard function) is given by
:
The
Mean time between failures
Mean time between failures (MTBF) is the predicted elapsed time between inherent failures of a mechanical or electronic system during normal system operation. MTBF can be calculated as the arithmetic mean (average) time between failures of a system ...
''MTBF'' is
:
Moments
The
moment generating function of the
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
of a Weibull distributed
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
is given by
:
where is the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...
. Similarly, the
characteristic function of log ''X'' is given by
:
In particular, the ''n''th
raw moment of ''X'' is given by
:
The
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ari ...
and
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...
of a Weibull
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
can be expressed as
:
and
:
The skewness is given by
:
where
, which may also be written as
:
where the mean is denoted by and the standard deviation is denoted by .
The excess
kurtosis
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kur ...
is given by
:
where
. The kurtosis excess may also be written as:
:
Moment generating function
A variety of expressions are available for the moment generating function of ''X'' itself. As a
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
, since the raw moments are already known, one has
:
Alternatively, one can attempt to deal directly with the integral
:
If the parameter ''k'' is assumed to be a rational number, expressed as ''k'' = ''p''/''q'' where ''p'' and ''q'' are integers, then this integral can be evaluated analytically. With ''t'' replaced by −''t'', one finds
:
where ''G'' is the
Meijer G-function.
The
characteristic function has also been obtained by . The
characteristic function and
moment generating function of 3-parameter Weibull distribution have also been derived by by a direct approach.
Reparametrization tricks
Fix some
. Let
be nonnegative, and not all zero, and let
be independent samples of
, then
*
*
.
Shannon entropy
The
information entropy
In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent to the variable's possible outcomes. Given a discrete random variable X, which takes values in the alphabet \ ...
is given by
:
where
is the
Euler–Mascheroni constant. The Weibull distribution is the
maximum entropy distribution
In statistics and information theory, a maximum entropy probability distribution has entropy that is at least as great as that of all other members of a specified class of probability distributions. According to the principle of maximum entropy, ...
for a non-negative real random variate with a fixed
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
of ''x''
''k'' equal to ''λ''
''k'' and a fixed expected value of ln(''x''
''k'') equal to ln(''λ''
''k'') −
.
Parameter estimation
Maximum likelihood
The
maximum likelihood estimator
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
for the
parameter given
is
:
The maximum likelihood estimator for
is the solution for ''k'' of the following equation
[.]
:
This equation defining
only implicitly, one must generally solve for
by numerical means.
When
are the
largest observed samples from a dataset of more than
samples, then the maximum likelihood estimator for the
parameter given
is
:
Also given that condition, the maximum likelihood estimator for
is
:
Again, this being an implicit function, one must generally solve for
by numerical means.
Kullback–Leibler divergence
The
Kullback–Leibler divergence
In mathematical statistics, the Kullback–Leibler divergence (also called relative entropy and I-divergence), denoted D_\text(P \parallel Q), is a type of statistical distance: a measure of how one probability distribution ''P'' is different fro ...
between two Weibulll distributions is given by
:
Weibull plot

The fit of a Weibull distribution to data can be visually assessed using a Weibull plot. The Weibull plot is a plot of the
empirical cumulative distribution function
In statistics, an empirical distribution function (commonly also called an empirical Cumulative Distribution Function, eCDF) is the distribution function associated with the empirical measure of a sample. This cumulative distribution function ...
of data on special axes in a type of
Q–Q plot. The axes are
versus
. The reason for this change of variables is the cumulative distribution function can be linearized:
:
which can be seen to be in the standard form of a straight line. Therefore, if the data came from a Weibull distribution then a straight line is expected on a Weibull plot.
There are various approaches to obtaining the empirical distribution function from data: one method is to obtain the vertical coordinate for each point using
where
is the rank of the data point and
is the number of data points.
Linear regression can also be used to numerically assess goodness of fit and estimate the parameters of the Weibull distribution. The gradient informs one directly about the shape parameter
and the scale parameter
can also be inferred.
Applications
The Weibull distribution is used
* In
survival analysis
Survival analysis is a branch of statistics for analyzing the expected duration of time until one event occurs, such as death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysi ...
* In
reliability engineering
Reliability engineering is a sub-discipline of systems engineering that emphasizes the ability of equipment to function without failure. Reliability describes the ability of a system or component to function under stated conditions for a specifi ...
and
failure analysis Failure analysis is the process of collecting and analyzing data to determine the cause of a failure, often with the goal of determining corrective actions or liability.
According to Bloch and Geitner, ”machinery failures reveal a reaction chain o ...
* In
electrical engineering to represent overvoltage occurring in an electrical system
* In
industrial engineering
Industrial engineering is an engineering profession that is concerned with the optimization of complex processes, systems, or organizations by developing, improving and implementing integrated systems of people, money, knowledge, information an ...
to represent
manufacturing
Manufacturing is the creation or production of goods with the help of equipment, labor, machines, tools, and chemical or biological processing or formulation. It is the essence of secondary sector of the economy. The term may refer to a ...
and
delivery times
* In
extreme value theory
* In
weather forecasting
Weather forecasting is the application of science and technology to predict the conditions of the atmosphere for a given location and time. People have attempted to predict the weather informally for millennia and formally since the 19th centu ...
and the
wind power industry to describe
wind speed distributions, as the natural distribution often matches the Weibull shape
* In communications systems engineering
** In
radar
Radar is a detection system that uses radio waves to determine the distance ('' ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, w ...
systems to model the dispersion of the received signals level produced by some types of clutters
** To model
fading channels in
wireless
Wireless communication (or just wireless, when the context allows) is the transfer of information between two or more points without the use of an electrical conductor, optical fiber or other continuous guided medium for the transfer. The mos ...
communications, as the
Weibull fading Weibull fading, named after Waloddi Weibull, is a simple statistical model of fading used in wireless communications and based on the Weibull distribution
In probability theory and statistics, the Weibull distribution is a continuous probabili ...
model seems to exhibit good fit to experimental fading
channel measurements
* In
information retrieval to model dwell times on web pages.
* In
general insurance to model the size of
reinsurance
Reinsurance is insurance that an insurance company purchases from another insurance company to insulate itself (at least in part) from the risk of a major claims event. With reinsurance, the company passes on ("cedes") some part of its own ins ...
claims, and the cumulative development of
asbestosis
Asbestosis is long-term inflammation and scarring of the lungs due to asbestos fibers. Symptoms may include shortness of breath, cough, wheezing, and chest tightness. Complications may include lung cancer, mesothelioma, and pulmonary heart diseas ...
losses
* In forecasting technological change (also known as the Sharif-Islam model)
* In
hydrology
Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and environmental watershed sustainability. A practitioner of hydrology is calle ...
the Weibull distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges.
* In
decline curve analysis
Decline curve analysis is a means of predicting future oil well or gas well production based on past production history. Production decline curve analysis is a traditional means of identifying well production problems and predicting well performan ...
to model oil production rate curve of shale oil wells.
* In describing the size of
particles generated by grinding,
milling
Milling may refer to:
* Milling (minting), forming narrow ridges around the edge of a coin
* Milling (grinding), breaking solid materials into smaller pieces by grinding, crushing, or cutting in a mill
* Milling (machining), a process of using rota ...
and
crushing operations, the 2-Parameter Weibull distribution is used, and in these applications it is sometimes known as the Rosin–Rammler distribution. In this context it predicts fewer fine particles than the
log-normal distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
and it is generally most accurate for narrow particle size distributions. The interpretation of the cumulative distribution function is that
is the
mass fraction of particles with diameter smaller than
, where
is the mean particle size and
is a measure of the spread of particle sizes.
* In describing random point clouds (such as the positions of particles in an ideal gas): the probability to find the nearest-neighbor particle at a distance
from a given particle is given by a Weibull distribution with
and
equal to the density of the particles.
* In calculating the rate of radiation-induced
single event effects onboard spacecraft, a four-parameter Weibull distribution is used to fit experimentally measured device
cross section probability data to a particle
linear energy transfer
In dosimetry, linear energy transfer (LET) is the amount of energy that an ionizing particle transfers to the material traversed per unit distance. It describes the action of radiation into matter.
It is identical to the retarding force acting o ...
spectrum. The Weibull fit was originally used because of a belief that particle energy levels align to a statistical distribution, but this belief was later proven false and the Weibull fit continues to be used because of its many adjustable parameters, rather than a demonstrated physical basis.
Related distributions
* A Weibull distribution is a
generalized gamma distribution
The generalized gamma distribution is a continuous probability distribution with two shape parameters (and a scale parameter). It is a generalization of the gamma distribution which has one shape parameter (and a scale parameter). Since many di ...
with both shape parameters equal to ''k''.
* The translated Weibull distribution (or 3-parameter Weibull) contains an additional parameter.
It has the
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
for
and
for
, where
is the
shape parameter,
is the
scale parameter and
is the
location parameter
In geography, location or place are used to denote a region (point, line, or area) on Earth's surface or elsewhere. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ...
of the distribution.
value sets an initial failure-free time before the regular Weibull process begins. When
, this reduces to the 2-parameter distribution.
* The Weibull distribution can be characterized as the distribution of a random variable
such that the random variable
is the standard
exponential distribution
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant averag ...
with intensity 1.
* This implies that the Weibull distribution can also be characterized in terms of a
uniform distribution
Uniform distribution may refer to:
* Continuous uniform distribution
* Discrete uniform distribution
* Uniform distribution (ecology)
* Equidistributed sequence In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be ...
: if
is uniformly distributed on
, then the random variable
is Weibull distributed with parameters
and
. Note that
here is equivalent to
just above. This leads to an easily implemented numerical scheme for simulating a Weibull distribution.
* The Weibull distribution interpolates between the exponential distribution with intensity
when
and a
Rayleigh distribution of mode
when
.
* The Weibull distribution (usually sufficient in
reliability engineering
Reliability engineering is a sub-discipline of systems engineering that emphasizes the ability of equipment to function without failure. Reliability describes the ability of a system or component to function under stated conditions for a specifi ...
) is a special case of the three parameter
exponentiated Weibull distribution In statistics, the exponentiated Weibull family of probability distributions was introduced by Mudholkar and Srivastava (1993) as an extension of the Weibull family obtained by adding a second shape parameter.
The cumulative distribution function ...
where the additional exponent equals 1. The exponentiated Weibull distribution accommodates
unimodal
In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object.
Unimodal probability distribution
In statistics, a unimodal pr ...
,
bathtub shaped and
monotone failure rate
Failure rate is the frequency with which an engineered system or component fails, expressed in failures per unit of time. It is usually denoted by the Greek letter λ (lambda) and is often used in reliability engineering.
The failure rate of a ...
s.
* The Weibull distribution is a special case of the
generalized extreme value distribution. It was in this connection that the distribution was first identified by
Maurice Fréchet Maurice may refer to:
People
*Saint Maurice (died 287), Roman legionary and Christian martyr
* Maurice (emperor) or Flavius Mauricius Tiberius Augustus (539–602), Byzantine emperor
* Maurice (bishop of London) (died 1107), Lord Chancellor and ...
in 1927. The closely related
Fréchet distribution, named for this work, has the probability density function
* The distribution of a random variable that is defined as the minimum of several random variables, each having a different Weibull distribution, is a
poly-Weibull distribution In probability theory and statistics, the poly-Weibull distribution is a continuous probability distribution. The distribution is defined to be that of a random variable defined to be the smallest of a number of statistically independent random var ...
.
* The Weibull distribution was first applied by to describe particle size distributions. It is widely used in
mineral processing
In the field of extractive metallurgy, mineral processing, also known as ore dressing, is the process of separating commercially valuable minerals from their ores.
History
Before the advent of heavy machinery the raw ore was broken up using ...
to describe
particle size distribution
The particle-size distribution (PSD) of a powder, or granular material, or particles dispersed in fluid, is a list of values or a mathematical function that defines the relative amount, typically by mass, of particles present according to size. Si ...
s in
comminution
Comminution is the reduction of solid materials from one average particle size to a smaller average particle size, by crushing, grinding, cutting, vibrating, or other processes. In geology, it occurs naturally during faulting in the upper part ...
processes. In this context the cumulative distribution is given by
where
**
is the particle size
**
is the 80th percentile of the particle size distribution
**
is a parameter describing the spread of the distribution
* Because of its availability in
spreadsheet
A spreadsheet is a computer application for computation, organization, analysis and storage of data in tabular form. Spreadsheets were developed as computerized analogs of paper accounting worksheets. The program operates on data entered in ce ...
s, it is also used where the underlying behavior is actually better modeled by an
Erlang distribution.
* If
then
(
Exponential distribution
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant averag ...
)
* For the same values of k, the
Gamma distribution
In probability theory and statistics, the gamma distribution is a two- parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma dis ...
takes on similar shapes, but the Weibull distribution is more
platykurtic
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real number, real-valued random variable. Like skew ...
.
* From the viewpoint of the
Stable count distribution,
can be regarded as Lévy's stability parameter. A Weibull distribution can be decomposed to an integral of kernel density where the kernel is either a
Laplace distribution
In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two expo ...
or a
Rayleigh distribution :
where
is the
Stable count distribution and
is the
Stable vol distribution.
See also
*
Fisher–Tippett–Gnedenko theorem
*
Logistic distribution
Logistic may refer to:
Mathematics
* Logistic function, a sigmoid function used in many fields
** Logistic map, a recurrence relation that sometimes exhibits chaos
** Logistic regression, a statistical model using the logistic function
** Logit ...
*
Rosin–Rammler distribution for particle size analysis
*
Rayleigh distribution
*
Stable count distribution
References
Bibliography
*.
*
*
*
*.
*
*.
*
*
External links
*
Mathpages – Weibull analysisThe Weibull DistributionReliability Analysis with Weibull* Interactive graphic
Online Weibull Probability Plotting
{{DEFAULTSORT:Weibull Distribution
Continuous distributions
Survival analysis
Exponential family distributions
Extreme value data