multivariate probability distribution

Given two random variables that are defined on the same probability space, 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 distributions, i.e. the distributions of each of the individual random variables. It also encodes the conditional probability distributions, 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, 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 together with a multivariate probability mass function. In the special case of continuous random variables, it is sufficient to consider probability density functions, and in the case of discrete random variables, 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 $A$ and $B$ 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 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 of the first and second coin flips respectively. Each coin flip is a Bernoulli trial and has a Bernoulli distribution. 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

:$P(A)=1/2 \quad \text \quad A\in \;$
:$P(B)=1/2 \quad \text \quad B\in \.$

The joint probability mass function of $A$ and $B$ defines probabilities for each pair of outcomes. All possible outcomes are
:$(A=0,B=0), (A=0,B=1), (A=1,B=0), (A=1,B=1).$
Since each outcome is equally likely the joint probability mass function becomes
:$P(A,B)=1/4 \quad \text \quad A,B\in\.$

Since the coin flips are independent, the joint probability mass function is the product
of the marginals:
:$P(A,B)=P(A)P(B) \quad \text \quad A,B \in\.$

Rolling a dice

Consider the roll of a fair dice 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.

Then, the joint distribution of $A$ and $B$, expressed as a probability mass function, is
:$\mathrm(A=0,B=0)=P\=\frac,\quad \quad \mathrm(A=1,B=0)=P\=\frac,$
:$\mathrm(A=0,B=1)=P\=\frac,\quad \quad \mathrm(A=1,B=1)=P\=\frac.$

These probabilities necessarily sum to 1, since the probability of ''some'' combination of $A$ and $B$ 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 $f_(x,y)$ , the marginal probability density function of X and Y, which defines the Marginal distribution, is given by:

$f_(x)= \int f_(x,y) \; dy$


$f_(y)= \int f_(x,y) \; dx$

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 $X,Y$, the joint cumulative distribution function (CDF) $F_$ is given by

where the right-hand side represents the probability that the random variable $X$ takes on a value less than or equal to $x$ and that $Y$ takes on a value less than or equal to $y$.

For $N$ random variables $X_1,\ldots,X_N$, the joint CDF $F_$ is given by

Interpreting the $N$ random variables as a random vector $\mathbf = (X_1,\ldots,X_N)^T$ yields a shorter notation:

:$F_(\mathbf) = \operatorname(X_1 \leq x_1,\ldots,X_N \leq x_N)$

Joint density function or mass function

Discrete case

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

or written in terms of conditional distributions
:$p_(x,y) = \mathrm(Y=y \mid X=x) \cdot \mathrm(X=x) = \mathrm(X=x \mid Y=y) \cdot \mathrm(Y=y)$
where $\mathrm(Y=y \mid X=x)$ is the probability of $Y = y$ given that $X = x$.

The generalization of the preceding two-variable case is the joint probability distribution of $n\,$ discrete random variables $X_1, X_2, \dots,X_n$ which is:

or equivalently

:$\begin p_(x_1,\ldots,x_n) & = \mathrm(X_1=x_1) \cdot \mathrm(X_2=x_2\mid X_1=x_1) \\ & \cdot \mathrm(X_3=x_3\mid X_1=x_1,X_2=x_2) \\ & \dots \\ & \cdot P(X_n=x_n\mid X_1=x_1,X_2=x_2,\dots,X_=x_). \end$.

This identity is known as the chain rule of probability.

Since these are probabilities, in the two-variable case

:$\sum_i \sum_j \mathrm(X=x_i\ \mathrm\ Y=y_j) = 1,\,$
which generalizes for $n\,$ discrete random variables $X_1, X_2, \dots , X_n$ to

:$\sum_ \sum_ \dots \sum_ \mathrm(X_1=x_,X_2=x_, \dots, X_n=x_) = 1.\;$

Continuous case

The joint probability density function $f_(x,y)$ for two continuous random variables is defined as the derivative of the joint cumulative distribution function (see ):

This is equal to:
:$f_(x,y) = f_(y\mid x)f_X(x) = f_(x\mid y)f_Y(y)$

where $f_(y\mid x)$ and $f_(x\mid y)$ are the conditional distribution