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 ...
and
statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, a probability distribution is the mathematical
function 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 hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs whe ...
.
It is a mathematical description of a
random
In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual ran ...
phenomenon in terms of its
sample space and the
probabilities of
events (
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s 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 to be conducted, etc.
Introduction
A probability distribution is a mathematical description of the probabilities of events, subsets of the
sample space. The sample space, often denoted by
, is the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of all possible
outcomes of a random phenomenon being observed; it may be any set: a set of
real numbers, a set of
vectors, 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 (so the sample space can be seen as a numeric set), it is common to distinguish between discrete and absolutely continuous
random variables. In the discrete case, it is sufficient to specify a
probability mass function 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 semicondu ...
, 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, 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 ev ...
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.
Absolutely continuous probability distributions can be described in several ways. 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) ca ...
describes the
infinitesimal 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, which describes the probability that the random variable is no larger than a given value (i.e.,
for some
). The cumulative distribution function is the area under 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) ca ...
from
to
, as described by the picture to the right.
General probability 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 absolutely continuous and discrete variables, is by means of a probability function
whose input space
is related to the
sample space, and gives a
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
probability as its output.
The probability function
can take as argument subsets of the sample space itself, as in the coin toss example, where the function
was defined so that and . However, because of the widespread use of
random variables, 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 value of the variable
belongs to a certain event
.
The above probability function only characterizes a probability distribution if it satisfies all the
Kolmogorov axioms, 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 , 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 usually belong to one of two classes. A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is
discrete (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. On the other hand, absolutely 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 absolutely continuous case, probabilities are described by a
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) ca ...
, and the probability distribution is by definition the integral of the probability density function.
The
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
is a commonly encountered absolutely continuous probability distribution. More complex experiments, such as those involving
stochastic processes defined in
continuous time, may demand the use of more general
probability measures.
A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called
univariate, while a distribution whose sample space is a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of dimension 2 or more is called
multivariate
Multivariate may refer to:
In mathematics
* Multivariable calculus
* Multivariate function
* Multivariate polynomial
In computing
* Multivariate cryptography
* Multivariate division algorithm
* Multivariate interpolation
* Multivariate optical c ...
. A univariate distribution gives the probabilities of a single
random variable taking on various different values; a multivariate distribution (a
joint probability distribution) gives the probabilities of a
random vector – 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, the
hypergeometric distribution, and the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
. A commonly encountered multivariate distribution is the
multivariate normal distribution.
Besides the probability function, the cumulative distribution function, the probability mass function and the probability density function, the
moment generating function and the
characteristic function 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.
Basic terms
*''
Random variable'': takes values from a sample space; probabilities describe which values and set of values are taken more likely.
*''
Event
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 ev ...
'': set of possible values (outcomes) of a random variable that occurs with a certain probability.
*''
Probability function'' or ''probability measure'': describes the probability
that the event
occurs.
[Chapters 1 and 2 of ]
*''
Cumulative distribution function'': function evaluating the
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
that
will take a value less than or equal to
for a random variable (only for real-valued random variables).
*''
Quantile function'': the inverse of the cumulative distribution function. Gives
such that, with probability
,
will not exceed
.
Discrete probability distributions
*Discrete probability distribution: for many random variables with finitely or countably infinitely many values.
*''
Probability mass function'' (''pmf''): function that gives the probability that a discrete random variable is equal to some value.
*''
Frequency distribution'': a table that displays the frequency of various outcomes .
*''
Relative frequency distribution'': a
frequency distribution where each value has been divided (normalized) by a number of outcomes in a
sample
Sample or samples may refer to:
Base meaning
* Sample (statistics), a subset of a population – complete data set
* Sample (signal), a digital discrete sample of a continuous analog signal
* Sample (material), a specimen or small quantity of ...
(i.e. sample size).
*''
Categorical distribution'': for discrete random variables with a finite set of values.
Absolutely continuous probability distributions
*Absolutely continuous probability distribution: for many random variables with uncountably many values.
*''
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) ca ...
'' (''pdf'') or ''probability density'': function whose value at any given sample (or point) in the
sample space (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.
Related terms
*
''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, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution. In many applications it is the right tail of the distrib ...
, Long-tailed distribution, fat-tailed distribution 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'' or ''mean'': the
weighted average of the possible values, using their probabilities as their weights; or the continuous analog thereof.
*''
Median'': 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.
*
''Mode'': for a discrete random variable, the value with highest probability; for an absolutely continuous random variable, a location at which the probability density function has a local peak.
*''
Quantile
In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile th ...
'': the q-quantile is the value
such that
.
*''
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 numbe ...
'': 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 variatio ...
of the distribution.
*''
Standard deviation'': the square root of the variance, and hence another measure of dispersion.
*
''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'': 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'': a measure of the "fatness" of the tails of a pmf or pdf. The fourth standardized moment of the distribution.
Cumulative distribution function
In the special case of a real-valued random variable, the probability distribution can equivalently be represented by a cumulative distribution function instead of a probability measure. The cumulative distribution function of a random variable
with regard to a probability distribution
is defined as
The cumulative distribution function of any real-valued random variable has the properties:
*
is non-decreasing;
*
is right-continuous;
*
;
*
and ; and
*
.
Conversely, any function
that satisfies the first four of the properties above is the cumulative distribution function of some probability distribution on the real numbers.
Any probability distribution can be decomposed as the sum of a
discrete, an
absolutely continuous and a
singular continuous distribution, and thus any cumulative distribution function admits a decomposition as the sum of the three according cumulative distribution functions.
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 (
almost surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
) which means that the probability of any event
can be expressed as a (finite or
countably infinite
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
) sum:
where
is a countable set. Thus the discrete random variables are exactly those with a
probability mass function . 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
, the sum of probabilities would be
.
A discrete random variable is a random variable whose probability distribution is discrete.
Well-known discrete probability distributions used in statistical modeling include the
Poisson distribution, the
Bernoulli distribution
In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45 is the discrete probabi ...
, the
binomial distribution, the
geometric distribution, the
negative binomial distribution and
categorical distribution.
When a
sample
Sample or samples may refer to:
Base meaning
* Sample (statistics), a subset of a population – complete data set
* Sample (signal), a digital discrete sample of a continuous analog signal
* Sample (material), a specimen or small quantity of ...
(a set of observations) is drawn from a larger population, the sample points have an
empirical distribution that is discrete, and which provides information about the population distribution. Additionally, the
discrete uniform distribution is commonly used in computer programs that make equal-probability random selections between a number of choices.
Cumulative distribution function
A real-valued discrete random variable can equivalently be defined as a random variable whose cumulative distribution function increases only by
jump discontinuities—that is, its cdf increases only where it "jumps" to a higher value, and is constant in intervals without jumps. The points where jumps occur are precisely the values which the random variable may take.
Thus the cumulative distribution function has the form
Note that the points where the cdf jumps always form a countable set; this may be any countable set and thus may even be dense in the real numbers.
Dirac delta representation
A discrete probability distribution is often represented with
Dirac measures, the probability distributions of
deterministic random variables. For any outcome
, let
be the Dirac measure concentrated at
. Given a discrete probability distribution, there is a countable set
with
and a probability mass function
. If
is any event, then
or in short,
Similarly, discrete distributions can be represented with the
Dirac delta function as a
generalized 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) ca ...
, where
which means
for any event
Indicator-function representation
For a discrete random variable
, let
be the values it can take with non-zero probability. Denote
These are
disjoint sets, and for such sets
It follows that the probability that
takes any value except for
is zero, and thus one can write
as
except on a set of probability zero, where
is the indicator function of
. 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. 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.
Absolutely continuous probability distribution
An absolutely continuous probability distribution is a probability distribution on the real numbers with uncountably many possible values, such as a whole interval in the real line, and where the probability of any event can be expressed as an integral. More precisely, a real random variable
has an
absolutely continuous probability distribution if there is a function