In
probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
and
statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data
Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

, a probability distribution is the mathematical
function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
that gives the probabilities of occurrence of different possible outcomes for an
experiment
An experiment is a procedure carried out to support or refute a , or determine the or of something previously untried. Experiments provide insight into by demonstrating what outcome occurs when a particular factor is manipulated. Experime ...
.
It is a mathematical description of a
random
In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wikt:order, order and does not follow an intelligible pattern or combination. I ...
phenomenon in terms of its
sample space
In probability theory
Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these axiom ...

and the
probabilities
Probability is the branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathe ...

of
events
Event may refer to:
Gatherings of people
* Ceremony, an event of ritual significance, performed on a special occasion
* Convention (meeting), a gathering of individuals engaged in some common interest
* Event management, the organization of event ...
(subsets of the sample space).
For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that
the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a
survey
Survey may refer to:
Statistics and human research
* Statistical survey
Survey methodology is "the study of survey methods".
As a field of applied statistics concentrating on Survey (human research), human-research surveys, survey methodology s ...
to be conducted, etc.
Introduction

A probability distribution is a mathematical description of the probabilities of events, subsets of the
sample space
In probability theory
Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these axiom ...

. The sample space, often denoted by
, is the
set of all possible
outcomes of a random phenomenon being observed; it may be any set: a set of
real numbers
Real may refer to:
* Reality, the state of things as they exist, rather than as they may appear or may be thought to be
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish col ...

, a set of
vectors
Vector may refer to:
Biology
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector
*Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
, a set of arbitrary non-numerical values, etc. For example, the sample space of a coin flip would be .
To define probability distributions for the specific case of
random variables
In probability and statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventio ...
(so the sample space can be seen as a numeric set), it is common to distinguish between discrete and continuous
random variable
A random variable is a variable whose values depend on outcomes of a random
In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...
s. In the discrete case, it is sufficient to specify a
probability mass function
In probability
Probability is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quant ...
assigning a probability to each possible outcome: for example, when throwing a fair
die
Die, as a verb, refers to death, the cessation of life.
Die may also refer to:
Games
* Die, singular of dice, small throwable objects used for producing random numbers
Manufacturing
* Die (integrated circuit), a rectangular piece of a semiconduct ...

, each of the six values 1 to 6 has the probability 1/6. The probability of an
event
Event may refer to:
Gatherings of people
* Ceremony
A ceremony (, ) is a unified ritual
A ritual is a sequence of activities involving gestures, words, actions, or objects, performed in a sequestered place and according to a set sequence. Rit ...
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 die rolls an even value" is
:
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, consider measuring the weight of a piece of ham in the supermarket, and assume the scale has many digits of precision. The probability that it weighs ''exactly'' 500 g is zero, as it will most likely have some non-zero decimal digits. Nevertheless, one might demand, in quality control, that a package of "500 g" of ham must weigh between 490 g and 510 g with at least 98% probability, 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
and probability density function of a normal distribution .
Image:visualisation_mode_median_mean.svg, 150px, Geometric visualisation of the mode (statistics), mode, median (statistics), median and mean (statistics), mean of an arbitrary probabilit ...
describes the
infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...
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.
An alternative description of the distribution is by means of the
cumulative distribution function
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 exp ...
, which describes the probability that the random variable is no larger than a given value (i.e., for some ''x''). The cumulative distribution function is the area under the
probability density function
and probability density function of a normal distribution .
Image:visualisation_mode_median_mean.svg, 150px, Geometric visualisation of the mode (statistics), mode, median (statistics), median and mean (statistics), mean of an arbitrary probabilit ...
from
to ''x'', as described by the picture to the right.
General definition
A probability distribution can be described in various forms, such as by a probability mass function or a cumulative distribution function. One of the most general descriptions, which applies for continuous and discrete variables, is by means of a probability function
whose input space
is related to the
sample space
In probability theory
Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these axiom ...

, and gives a
real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
probability as its output.
[Chapters 1 and 2 of ]
The probability function ''P'' can take as argument subsets of the sample space itself, as in the coin toss example, where the function ''P'' was defined so that ''P'' and ''P''. However, because of the widespread use of
random variables
In probability and statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventio ...
, which transform the sample space into a set of numbers (e.g.,
,
), it is more common to study probability distributions whose argument are subsets of these particular kinds of sets (number sets), and all probability distributions discussed in this article are of this type. It is common to denote as the probability that a certain variable ''X'' belongs to a certain event ''E''.
The above probability function only characterizes a probability distribution if it satisfies all the
Kolmogorov axioms
The Kolmogorov axioms are the foundations of probability theory introduced by Andrey Kolmogorov in 1933. These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probability cases. An alternativ ...
, that is:
#
, so the probability is non-negative
#
, so no probability exceeds
#
for any disjoint family of sets
The concept of probability function is made more rigorous by defining it as the element of a
probability space
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 expres ...
, where
is the set of possible outcomes,
is the set of all subsets
whose probability can be measured, and
is the probability function, or probability measure, that assigns a probability to each of these measurable subsets
.
Probability distributions are generally divided into two classes. A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is
discrete
Discrete in science is the opposite of :wikt:continuous, continuous: something that is separate; distinct; individual.
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic c ...

(e.g. a coin toss, a roll of a die) and the probabilities are encoded by a discrete list of the probabilities of the outcomes; in this case the discrete probability distribution is known as
probability mass function
In probability
Probability is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quant ...
. On the other hand, continuous probability distributions are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In the case of real numbers, the continuous probability distribution is the
cumulative distribution function
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 exp ...
. In general, in the continuous case, probabilities are described by a
probability density function
and probability density function of a normal distribution .
Image:visualisation_mode_median_mean.svg, 150px, Geometric visualisation of the mode (statistics), mode, median (statistics), median and mean (statistics), mean of an arbitrary probabilit ...
, and the probability distribution is by definition the integral of the probability density function.
The
normal distribution
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 ex ...

is a commonly encountered continuous probability distribution. More complex experiments, such as those involving
stochastic processes
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 expre ...
defined in
continuous timeIn mathematical dynamics, discrete time and continuous time are two alternative frameworks within which to model variables that evolve over time.
Discrete time
Discrete sampled signal
Discrete time views values of variables as occurring at disti ...
, may demand the use of more general
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability meas ...
s.
A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called
univariate In mathematics, a univariate object is an expression, equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geom ...
, while a distribution whose sample space is a
vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of dimension 2 or more is called
multivariate. A univariate distribution gives the probabilities of a single
random variable
A random variable is a variable whose values depend on outcomes of a random
In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...
taking on various different values; a multivariate distribution (a
joint probability distribution
Given random variables X,Y,\ldots, that are defined on the same probability space, the joint probability distribution for X,Y,\ldots is a probability distribution that gives the probability that each of X,Y,\ldots falls in any particular range o ...
) gives the probabilities of a
random vector
In probability
Probability is the branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculu ...
– a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the
binomial distribution
In probability theory
Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically thes ...

, the
hypergeometric distribution
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 expre ...
, and the
normal distribution
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 ex ...

. A commonly encountered multivariate distribution is the
multivariate normal distribution
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 expre ...
.
Besides the probability function, the cumulative distribution function, the probability mass function and the probability density function, the
moment generating function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compared ...
and the
characteristic functionIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
also serve to identify a probability distribution, as they uniquely determine an underlying cumulative distribution function.
Terminology
Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below.
Functions for discrete variables
*Probability function: describes the probability
that the event
from the sample space occurs.
*
Probability mass function (pmf): function that gives the probability that a discrete random variable is equal to some value.
*
Frequency distributionIn statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a ...

: a table that displays the frequency of various outcomes in a sample.
*Relative frequency distribution: a
frequency distributionIn statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a ...

where each value has been divided (normalized) by a number of outcomes in a
sample (i.e. sample size).
*Discrete probability distribution function: general term to indicate the way the total probability of 1 is distributed over all various possible outcomes (i.e. over entire population) for discrete random variable.
*
Cumulative distribution function
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 exp ...
: function evaluating the
probability
Probability is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

that
will take a value less than or equal to
for a discrete random variable.
*
Categorical distribution
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 expres ...
: for discrete random variables with a finite set of values.
Functions for continuous variables
*
Probability density function
and probability density function of a normal distribution .
Image:visualisation_mode_median_mean.svg, 150px, Geometric visualisation of the mode (statistics), mode, median (statistics), median and mean (statistics), mean of an arbitrary probabilit ...
(pdf): function whose value at any given sample (or point) in the
sample space
In probability theory
Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these axiom ...

(the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would equal that sample.
* Continuous probability distribution function: most often reserved for continuous random variables.
*
Cumulative distribution function
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 exp ...
: function evaluating the
probability
Probability is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

that
will take a value less than or equal to
for a continuous variable.
*
Quantile function
In probability and statistics, the quantile function, associated with a probability distribution of a random variable, specifies the value of the random variable such that the probability of the variable being less than or equal to that value equa ...
: the inverse of the cumulative distribution function. Gives
such that, with probability
,
will not exceed
.
Basic terms
*
Mode: for a discrete random variable, the value with highest probability; for a continuous random variable, a location at which the probability density function has a local peak.
*
Support: set of values that can be assumed with non-zero probability by the random variable. For a random variable
, it is sometimes denoted as
.
* Tail:
[More information and examples can be found in the articles ]Heavy-tailed distribution
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 expressi ...
, Long-tailed distribution
In statistics and business, a long tail of some probability distribution, distributions of numbers is the portion of the distribution having many occurrences far from the "head" or central part of the distribution. The distribution could involve ...
, fat-tailed distribution
A fat-tailed distribution is a probability distribution
In probability theory and statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statist ...
the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form
,
or a union thereof.
*Head:
the region where the pmf or pdf is relatively high. Usually has the form
.
*
Expected value
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 exp ...
or mean: the
weighted average
The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of the possible values, using their probabilities as their weights; or the continuous analog thereof.
*
Median
In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin wi ...

: the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half.
*
Variance
In probability theory
Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these ax ...

: the second moment of the pmf or pdf about the mean; an important measure of the
dispersion
Dispersion may refer to:
Economics and finance
*Dispersion (finance), a measure for the statistical distribution of portfolio returns
*Price dispersion, a variation in prices across sellers of the same item
*Wage dispersion, the amount of variation ...
of the distribution.
*
Standard deviation
In statistics, the standard deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected v ...

: the square root of the variance, and hence another measure of dispersion.
*
Quantile
In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin wi ...

: the q-quantile is the value
such that
.
*
Symmetry: a property of some distributions in which the portion of the distribution to the left of a specific value (usually the median) is a mirror image of the portion to its right.
*
Skewness
In probability theory
Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these ...

: a measure of the extent to which a pmf or pdf "leans" to one side of its mean. The third
standardized moment of the distribution.
*
Kurtosis
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 expres ...
: a measure of the "fatness" of the tails of a pmf or pdf. The fourth standardized moment of the distribution.
*
Continuity: a property of some distributions whose values do not change abruptly.
Discrete probability distribution

A discrete probability distribution is the probability distribution of a random variable that can take on only a countable number of values. In the case where the range of values is countably infinite, these values have to decline to zero fast enough for the probabilities to add up to 1. For example, if
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
In probability theory and statistics, the Poisson distribution (; ), named after France, French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a f ...
, the
Bernoulli distribution
In probability theory
Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these ax ...

, the
binomial distribution
In probability theory
Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically thes ...

, the
geometric distribution
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 expres ...
, and the
negative binomial distribution
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 expre ...
.
Additionally, the
discrete uniform distribution
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 expre ...
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
that is discrete, and which provides information about the population distribution.
Cumulative distribution function
Equivalently to the above, a discrete random variable can be defined as a random variable whose
cumulative distribution function
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 exp ...
(cdf) increases only by
jump discontinuities—that is, its cdf increases only where it "jumps" to a higher value, and is constant between those jumps. Note however that the points where the cdf jumps may form a dense set of the real numbers. The points where jumps occur are precisely the values which the random variable may take.
Delta-function representation
Consequently, a discrete probability distribution is often represented as a generalized
probability density function
and probability density function of a normal distribution .
Image:visualisation_mode_median_mean.svg, 150px, Geometric visualisation of the mode (statistics), mode, median (statistics), median and mean (statistics), mean of an arbitrary probabilit ...
involving
Dirac delta function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is n ...
s, which substantially unifies the treatment of continuous and discrete distributions. This is especially useful when dealing with probability distributions involving both a continuous and a discrete part.
Indicator-function representation
For a discrete random variable ''X'', let ''u''
0, ''u''
1, ... be the values it can take with non-zero probability. Denote
:
These are
disjoint set
Image:Disjunkte Mengen.svg, Two disjoint sets.
In mathematics, two Set (mathematics), sets are said to be disjoint sets if they have no element (mathematics), element in common. Equivalently, two disjoint sets are sets whose intersection (set theor ...
s, and for such sets
:
It follows that the probability that ''X'' takes any value except for ''u''
0, ''u''
1, ... is zero, and thus one can write ''X'' as
:
except on a set of probability zero, where
is the indicator function of ''A''. This may serve as an alternative definition of discrete random variables.
One-point distribution
A special case is the discrete distribution of a random variable that can take on only one fixed value; in other words, it is a
deterministic distribution
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
. Expressed formally, the random variable
has a one-point distribution if it has a possible outcome
such that
All other possible outcomes then have probability 0. Its cumulative distribution function jumps immediately from 0 to 1.
Continuous probability distribution
A continuous probability distribution is a probability distribution whose support is an uncountable set, such as an interval in the real line. They are uniquely characterized by a
cumulative distribution function
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 exp ...
that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions:
,
uniform
A uniform is a variety of clothing
A kanga, worn throughout the African Great Lakes region
Clothing (also known as clothes, apparel, and attire) are items worn on the body. Typically, clothing is made of fabrics or textiles, but over ti ...
,
chi-squared, and
others.
A random variable
has a continuous probability distribution if there is a function