Given two
random variables that are defined on the same
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
, the joint probability distribution is the corresponding
probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered for any given number of random variables. The joint distribution encodes the
marginal distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the varia ...
s, i.e. the distributions of each of the individual random variables. It also encodes the
conditional probability distribution
In probability theory and statistics, given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the ...
s, which deal with how the outputs of one random variable are distributed when given information on the outputs of the other random variable(s).
In the formal mathematical setup of
measure theory, the joint distribution is given by the
pushforward measure In measure theory, a pushforward measure (also known as push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a measure from one measurable space to another using a measurable function.
Definition
Given meas ...
, by the map obtained by pairing together the given random variables, of the sample space's
probability measure.
In the case of real-valued random variables, the joint distribution, as a particular multivariate distribution, may be expressed by a multivariate
cumulative distribution function, or by a multivariate
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 ...
together with a multivariate
probability mass function. In the special case of
continuous random variable
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 phenomenon ...
s, it is sufficient to consider probability density functions, and in the case of
discrete 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 po ...
s, it is sufficient to consider probability mass functions.
Examples
Draws from an urn
Suppose each of two urns contains twice as many red balls as blue balls, and no others, and suppose one ball is randomly selected from each urn, with the two draws independent of each other. Let
and
be discrete random variables associated with the outcomes of the draw from the first urn and second urn respectively. The probability of drawing a red ball from either of the urns is 2/3, and the probability of drawing a blue ball is 1/3. The joint probability distribution is presented in the following table:
Each of the four inner cells shows the probability of a particular combination of results from the two draws; these probabilities are the joint distribution. In any one cell the probability of a particular combination occurring is (since the draws are independent) the product of the probability of the specified result for A and the probability of the specified result for B. The probabilities in these four cells sum to 1, as it is always true for probability distributions.
Moreover, the final row and the final column give the
marginal probability distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the varia ...
for A and the marginal probability distribution for B respectively. For example, for A the first of these cells gives the sum of the probabilities for A being red, regardless of which possibility for B in the column above the cell occurs, as 2/3. Thus the marginal probability distribution for
gives
's probabilities ''unconditional'' on
, in a margin of the table.
Coin flips
Consider the flip of two
fair coin
In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair coin. In the ...
s; let
and
be discrete random variables associated with the outcomes of the first and second coin flips respectively. Each coin flip is a
Bernoulli trial
In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is ...
and has a
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 ...
. If a coin displays "heads" then the associated random variable takes the value 1, and it takes the value 0 otherwise. The probability of each of these outcomes is 1/2, so the marginal (unconditional) density functions are
:
:
The joint probability mass function of
and
defines probabilities for each pair of outcomes. All possible outcomes are
:
Since each outcome is equally likely the joint probability mass function becomes
:
Since the coin flips are independent, the joint probability mass function is the product
of the marginals:
:
Rolling a dice
Consider the roll of a fair
dice and let
if the number is even (i.e. 2, 4, or 6) and
otherwise. Furthermore, let
if the number is prime (i.e. 2, 3, or 5) and
otherwise.
Then, the joint distribution of
and
, expressed as a probability mass function, is
:
:
These probabilities necessarily sum to 1, since the probability of ''some'' combination of
and
occurring is 1.
Marginal probability distribution
If more than one random variable is defined in a random experiment, it is important to distinguish between the joint probability distribution of X and Y and the probability distribution of each variable individually. The individual probability distribution of a random variable is referred to as its marginal probability distribution. In general, the marginal probability distribution of X can be determined from the joint probability distribution of X and other random variables.
If the joint probability density function of random variable X and Y is
, the marginal probability density function of X and Y, which defines the
Marginal distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the varia ...
, is given by:
where the first integral is over all points in the range of (X,Y) for which X=x and the second integral is over all points in the range of (X,Y) for which Y=y.
Joint cumulative distribution function
For a pair of random variables
, the joint cumulative distribution function (CDF)
is given by
where the right-hand side represents 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 speakin ...
that the random variable
takes on a value less than or equal to
and that
takes on a value less than or equal to
.
For
random variables
, the joint CDF
is given by
Interpreting the
random variables as a
random vector
In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value ...
yields a shorter notation:
:
Joint density function or mass function
Discrete case
The joint
probability mass function of two
discrete 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 po ...
s
is:
or written in terms of conditional distributions
:
where
is 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 speakin ...
of
given that
.
The generalization of the preceding two-variable case is the joint probability distribution of
discrete random variables
which is:
or equivalently
:
.
This identity is known as the
chain rule of probability.
Since these are probabilities, in the two-variable case
:
which generalizes for
discrete random variables
to
:
Continuous case
The joint
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 ...
for two
continuous random variable
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 phenomenon ...
s is defined as the derivative of the joint cumulative distribution function (see ):
This is equal to:
:
where
and
are the
conditional distribution
In probability theory and statistics, given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the ...
s of
given
and of
given
respectively, and
and
are the
marginal distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the varia ...
s for
and
respectively.
The definition extends naturally to more than two random variables:
Again, since these are probability distributions, one has
:
respectively
:
Mixed case
The "mixed joint density" may be defined where one or more random variables are continuous and the other random variables are discrete. With one variable of each type
:
One example of a situation in which one may wish to find the cumulative distribution of one random variable which is continuous and another random variable which is discrete arises when one wishes to use a
logistic regression
In statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear combination of one or more independent variables. In regression a ...
in predicting the probability of a binary outcome Y conditional on the value of a continuously distributed outcome
. One ''must'' use the "mixed" joint density when finding the cumulative distribution of this binary outcome because the input variables
were initially defined in such a way that one could not collectively assign it either a probability density function or a probability mass function. Formally,
is the probability density function of
with respect to the
product measure In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology of tw ...
on the respective
supports of
and
. Either of these two decompositions can then be used to recover the joint cumulative distribution function:
:
The definition generalizes to a mixture of arbitrary numbers of discrete and continuous random variables.
Additional properties
Joint distribution for independent variables
In general two random variables
and
are
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
if and only if the joint cumulative distribution function satisfies
:
Two discrete random variables
and
are independent if and only if the joint probability mass function satisfies
:
for all
and
.
While the number of independent random events grows, the related joint probability value decreases rapidly to zero, according to a negative exponential law.
Similarly, two absolutely continuous random variables are independent if and only if
:
for all
and
. This means that acquiring any information about the value of one or more of the random variables leads to a conditional distribution of any other variable that is identical to its unconditional (marginal) distribution; thus no variable provides any information about any other variable.
Joint distribution for conditionally dependent variables
If a subset
of the variables
is
conditionally dependent given another subset
of these variables, then the probability mass function of the joint distribution is
.
is equal to
. Therefore, it can be efficiently represented by the lower-dimensional probability distributions
and
. Such conditional independence relations can be represented with a
Bayesian network
A Bayesian network (also known as a Bayes network, Bayes net, belief network, or decision network) is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). Bay ...
or
copula functions.
Covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the ...
When two or more random variables are defined on a probability space, it is useful to describe how they vary together; that is, it is useful to measure the relationship between the variables. A common measure of the relationship between two random variables is the covariance. Covariance is a measure of linear relationship between the random variables. If the relationship between the random variables is nonlinear, the covariance might not be sensitive to the relationship, which means, it does not relate the correlation between two variables.
The covariance between the random variable X and Y, denoted as cov(X,Y), is :
Correlation
There is another measure of the relationship between two random variables that is often easier to interpret than the covariance.
The correlation just scales the covariance by the product of the standard deviation of each variable. Consequently, the correlation is a dimensionless quantity that can be used to compare the linear relationships between pairs of variables in different units. If the points in the joint probability distribution of X and Y that receive positive probability tend to fall along a line of positive (or negative) slope, ρ
XY is near +1 (or −1). If ρ
XY equals +1 or −1, it can be shown that the points in the joint probability distribution that receive positive probability fall exactly along a straight line. Two random variables with nonzero correlation are said to be correlated. Similar to covariance, the correlation is a measure of the linear relationship between random variables.
The correlation between random variable X and Y, denoted as
Important named distributions
Named joint distributions that arise frequently in statistics include the
multivariate normal distribution
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One ...
, the
multivariate stable distribution
The multivariate stable distribution is a multivariate probability distribution that is a multivariate generalisation of the univariate stable distribution. The multivariate stable distribution defines linear relations between stable distribution ...
, the
multinomial distribution
In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for each side of a ''k''-sided dice rolled ''n'' times. For ''n'' independent trials each of wh ...
, the
negative multinomial distribution
In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB(''x''0, ''p'')) to more than two outcomes.Le Gall, F. The modes of a negative multinomial distribution ...
, the
multivariate hypergeometric distribution
In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of k successes (random draws for which the object drawn has a specified feature) in n draws, ''without'' ...
, and the
elliptical distribution
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. Intuitively, in the simplified two and three dimensional case, the joint ...
.
See also
*
Bayesian programming
Bayesian programming is a formalism and a methodology for having a technique to specify probabilistic models and solve problems when less than the necessary information is available.
Edwin T. Jaynes proposed that probability could be consider ...
*
Chow–Liu tree
*
Conditional probability
In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occu ...
*
Copula (probability theory)
In probability theory and statistics, a copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform on the interval , 1 Copulas are used to describe/model the ...
*
Disintegration theorem
In mathematics, the disintegration theorem is a result in measure theory and probability theory. It rigorously defines the idea of a non-trivial "restriction" of a measure to a measure zero subset of the measure space in question. It is relate ...
*
Multivariate statistics
Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable.
Multivariate statistics concerns understanding the different aims and background of each of the dif ...
*
Statistical interference
*
Pairwise independent distribution
References
External links
*
*
*''A modern introduction to probability and statistics : understanding why and how''. Dekking, Michel, 1946-. London: Springer. 2005. .
OCLC 262680588.
*
Mathworld: Joint Distribution Function
{{Probability distributions, multivariate
Theory of probability distributions
Types of probability distributions