# Joint probability distribution

${\displaystyle X}$
${\displaystyle Y}$
${\displaystyle p(X)}$
${\displaystyle p(Y)}$
Many sample observations (black) are shown from a joint probability distribution. The marginal densities are shown as well.

In the study of probability, given at least two random variables X, Y, ..., that are defined on a probability space, the joint probability distribution for X, Y, ... is a probability distribution that gives the probability that each of X, Y, ... falls in any particular range or discrete set of values specified for that variable. In the case of only two random variables, this is called a bivariate distribution, but the concept generalizes to any number of random variables, giving a multivariate distribution.

The joint probability distribution can be expressed either in terms of a joint cumulative distribution function or in terms of a joint probability density function (in the case of continuous variables) or joint probability mass function (in the case of discrete variables). These in turn can be used to find two other types of distributions: the marginal distribution giving the probabilities for any one of the variables with no reference to any specific ranges of values for the other variables, and the conditional probability distribution giving the probabilities for any subset of the variables conditional on particular values of the remaining variables.

## 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. 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. We can present the joint probability distribution as the following table:

A=Red A=Blue P(B)
B=Red (2/3)(2/3)=4/9 (1/3)(2/3)=2/9 4/9+2/9=2/3
B=Blue (2/3)(1/3)=2/9 (1/3)(1/3)=1/9 2/9+1/9=1/3
P(A) 4/9+2/9=2/3 2/9+1/9=1/3

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 is always true for probabaility distributions.

Moreover, the final row and the final column give the marginal probability distribution 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 A gives A's probabilities unconditional on B, in a margin of the table.

### Coin flips

Consider the flip of two fair coins; let A and B be discrete random variables associated with the outcomes first and second coin flips respectively. If a coin displays "heads" then associated random variable is 1, and is 0 otherwise. The joint probability density function of A and B defines probabilities for each pair of outcomes. All possible outcomes are

${\displaystyle (A=0,B=0),(A=0,B=1),(A=1,B=0),(A=1,B=1)}$

Since each outcome is equally likely the joint probability density function becomes

${\displaystyle P(A,B)=1/4}$

when ${\displaystyle A,B\in \{0,1\}}$. Since the coin flips are independent, the joint probability density function is the product of the marginals:

${\displaystyle P(A,B)=P(A)P(B)}$.

In general, each coin flip is a Bernoulli trial and the sequence of flips follows a Bernoulli distribution.

### Roll of a die

Consider the roll of a fair die and let A = 1 if the number is even (i.e. 2, 4, or 6) and A = 0 otherwise. Furthermore, let B = 1 if the number is prime (i.e. 2, 3, or 5) and B = 0 otherwise.

1 2 3 4 5 6
A 0 1 0 1 0 1
B 0 1 1 0 1 0

Then, the joint distribution of A and B, expressed as a probability mass function, is

${\displaystyle \mathrm {P} (A=0,B=0)=P\{1\}={\frac {1}{6}},\quad \quad \mathrm {P} (A=1,B=0)=P\{4,6\}={\frac {2}{6}},}$
${\displaystyle \mathrm {P} (A=0,B=1)=P\{3,5\}={\frac {2}{6}},\quad \quad \mathrm {P} (A=1,B=1)=P\{2\}={\frac {1}{6}}.}$

These probabilities necessarily sum to 1, since the probability of some combination of A and B occurring is 1.

## Density function or mass function

### Discrete case

The joint probability mass function of two discrete random variables ${\displaystyle X,Y}$ is:

{\displaystyle {\begin{aligned}\mathrm {P} (X=x\ \mathrm {and} \ Y=y)=\mathrm {P} (Y=y\mid X=x)\cdot \mathrm {P} (X=x)=\mathrm {P} (X=x\mid Y=y)\cdot \mathrm {P} (Y=y)\end{aligned}},}

where ${\displaystyle \mathrm {P} (Y=y\mid X=x)}$ is the probability of ${\displaystyle Y=y}$ given that ${\displaystyle X=x}$.

The generalization of the preceding two-variable case is the joint probability distribution of ${\displaystyle n\,}$ discrete random variables ${\displaystyle X_{1},X_{2},\dots ,X_{n}}$ which is:

{\displaystyle {\begin{aligned}\mathrm {P} (X_{1}=x_{1},\dots ,X_{n}=x_{n})&=\mathrm {P} (X_{1}=x_{1})\times \mathrm {P} (X_{2}=x_{2}\mid X_{1}=x_{1})\\&\times \mathrm {P} (X_{3}=x_{3}\mid X_{1}=x_{1},X_{2}=x_{2})\\&\dots \\&\times P(X_{n}=x_{n}\mid X_{1}=x_{1},X_{2}=x_{2},\dots ,X_{n-1}=x_{n-1}).\end{aligned}}}

This identity is known as the chain rule of probability.

Since these are probabilities, we have in the two-variable case

${\displaystyle \sum _{i}\sum _{j}\mathrm {P} (X=x_{i}\ \mathrm {and} \ Y=y_{j})=1,\,}$

which generalizes for ${\displaystyle n\,}$ discrete random variables ${\displaystyle X_{1},X_{2},\dots ,X_{n}}$ to

${\displaystyle \sum _{i}\sum _{j}\dots \sum _{k}\mathrm {P} (X_{1}=x_{1i},X_{2}=x_{2j},\dots ,X_{n}=x_{nk})=1.\;}$

### Continuous case

The joint probability density function fX,Y(xy) for two continuous random variables is equal to:

${\displaystyle f_{X,Y}(x,y)=f_{Y\mid X}(y\mid x)f_{X}(x)=f_{X\mid Y}(x\mid y)f_{Y}(y)\;}$

where fY|X(y|x) and fX|Y(x|y) are the conditional distributions of Y given X = x and of X given Y = y respectively, and fX(x) and fY(y) are the marginal distributions for X and Y respectively.

Again, since these are probability distributions, one has

${\displaystyle \int _{x}\int _{y}f_{X,Y}(x,y)\;dy\;dx=1.}$

### Mixed case

The "mixed joint density" may be defined where one or more random variables are continuous and the other random variables are discrete, or vice versa. With one variable of each type we have

{\displaystyle {\begin{aligned}f_{X,Y}(x,y)=f_{X\mid Y}(x\mid y)\mathrm {P} (Y=y)=\mathrm {P} (Y=y\mid X=x)f_{X}(x).\end{aligned}}}

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 predicting the probability of a binary outcome Y conditional on the value of a continuously distributed outcome X. One must use the "mixed" joint density when finding the cumulative distribution of this binary outcome because the input variables (X, Y) 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, fX,Y(x, y) is the probability density function of (X, Y) with respect to the product measure on the respective supports of X and Y. Either of these two decompositions can then be used to recover the joint cumulative distribution function:

{\displaystyle {\begin{aligned}F_{X,Y}(x,y)&=\sum \limits _{t\leq y}\int _{s=-\infty }^{x}f_{X,Y}(s,t)\;ds.\end{aligned}}}

The definition generalizes to a mixture of arbitrary numbers of discrete and continuous random variables.

### Joint distribution for independent variables

Two discrete random variables ${\displaystyle X}$ and ${\displaystyle Y}$ are independent if the joint probability mass function satisfies

${\displaystyle \ P(X=x\ {\mbox{and}}\ Y=y)=P(X=x)\cdot P(Y=y)}$

for all x and y.

Similarly, two absolutely continuous random variables are independent if

${\displaystyle \ f_{X,Y}(x,y)=f_{X}(x)\cdot f_{Y}(y)}$

for all x and y. 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 ${\displaystyle A}$ of the variables ${\displaystyle X_{1},\cdots ,X_{n}}$ is conditionally dependent given another subset ${\displaystyle B}$ of these variables, then the joint distribution ${\displaystyle \mathrm {P} (X_{1},\ldots ,X_{n})}$ is equal to ${\displaystyle P(B)\cdot P(A\mid B)}$. Therefore, it can be efficiently represented by the lower-dimensional probability distributions ${\displaystyle P(B)}$ and ${\displaystyle P(A\mid B)}$. Such conditional independence relations can be represented with a Bayesian network or copula functions.

### Cumulative distribution

The joint probability distribution for a pair of random variables can be expressed in terms of their cumulative distribution function ${\displaystyle F(x,y)=P(X\leq x,Y\leq y).}$

## Important named distributions

Named joint distributions that arise frequently in statistics include the multivariate normal distribution, the multivariate stable distribution, the multinomial distribution, the negative multinomial distribution, the multivariate hypergeometric distribution, and the elliptical distribution.