Joint probability distribution

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 ${\displaystyle f_{X,Y}(x,y)}$ , the marginal probability density function of X and Y, which defines the marginal distribution, is given by:

$${\displaystyle \begin{array}{cc}{f}_X(x)& =\int f_{X,Y}(x,y)dy\\ f_Y(y)& =\int f_{X,Y}(x,y)dx\end{array}}$$

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

${\displaystyle F_{X,Y}(x,y)=\mathrm{P}(X\le x,Y\le y)}$ ()

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

For ${\displaystyle N}$ random variables ${\displaystyle X_1,\dots ,X_N}$, the joint CDF ${\displaystyle F_{X_1,\dots ,X_N}}$ is given by

${\displaystyle F_{X_1,\dots ,X_N}(x_1,\dots ,x_N)=\mathrm{P}(X_1\le x_1,\dots ,X_N\le x_N)}$ ()

Interpreting the ${\displaystyle N}$ random variables as a random vector ${\displaystyle \mathbf{X}=(X_1,\dots ,X_N{)}^T}$ yields a shorter notation:

$${\displaystyle F_{\mathbf{X}}(\mathbf{x})=\mathrm{P}(X_1\le x_1,\dots ,X_N\le x_N)}$$

Joint density function or mass function

Discrete case

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

${\displaystyle p_{X,Y}(x,y)=\mathrm{P}(X=x\mathrm{a}\mathrm{n}\mathrm{d}Y=y)}$ ()

or written in terms of conditional distributions $${\displaystyle p_{X,Y}(x,y)=\mathrm{P}(Y=y\mid X=x)\cdot \mathrm{P}(X=x)=\mathrm{P}(X=x\mid Y=y)\cdot \mathrm{P}(Y=y)}$$ 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:

${\displaystyle p_{X_1,\dots ,X_n}(x_1,\dots ,x_n)=\mathrm{P}(X_1=x_1\text{ and }\dots \text{ and }{X}_n=x_n)}$ ()

or equivalently

$${\displaystyle \begin{array}{cc}{p}_{X_1,\dots ,X_n}(x_1,\dots ,x_n)=& \mathrm{P}(X_1=x_1)\\ & \cdot \mathrm{P}(X_2=x_2\mid X_1=x_1)\\ & \cdot \mathrm{P}(X_3=x_3\mid X_1=x_1,X_2=x_2)\\ & \cdots \\ & \cdot \mathrm{P}(X_n=x_n\mid X_1=x_1,X_2=x_2,\dots ,X_{n-1}=x_{n-1}).\end{array}}$$

This identity is known as the chain rule of probability.

Since these are probabilities, in the two-variable case

$${\displaystyle \sum _i\sum _j\mathrm{P}(X=x_i\mathrm{a}\mathrm{n}\mathrm{d}Y=y_j)=1,}$$ 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 ${\displaystyle f_{X,Y}(x,y)}$ for two continuous random variables is defined as the derivative of the joint cumulative distribution function (see Eq.1):

${\displaystyle f_{X,Y}(x,y)=\frac{{\mathrm{\partial }}^2{F}_{X,Y}(x,y)}{\mathrm{\partial }x\mathrm{\partial }y}}$ ()

This 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 ${\displaystyle f_{Y\mid X}(y\mid x)}$ and ${\displaystyle f_{X\mid Y}(x\mid y)}$ are the conditional distributions of ${\displaystyle Y}$ given ${\displaystyle X=x}$ and of ${\displaystyle X}$ given ${\displaystyle Y=y}$ respectively, and ${\displaystyle f_X(x)}$ and ${\displaystyle f_Y(y)}$ are the marginal distributions for ${\displaystyle X}$ and ${\displaystyle Y}$ respectively.

The definition extends naturally to more than two random variables:

${\displaystyle f_{X_1,\dots ,X_n}(x_1,\dots ,x_n)=\frac{{\mathrm{\partial }}^n{F}_{X_1,\dots ,X_n}(x_1,\dots ,x_n)}{\mathrm{\partial }{x}_1\dots \mathrm{\partial }{x}_n}}$ ()

Again, since these are probability distributions, one has $${\displaystyle {\int }_x{\int }_y{f}_{X,Y}(x,y)dydx=1}$$ respectively $${\displaystyle {\int }_{x_1}\dots {\int }_{x_n}{f}_{X_1,\dots ,X_n}(x_1,\dots ,x_n)d{x}_n\dots d{x}_1=1}$$

Mixed case

The "mixed joint density" may be defined where one or more random variables are continuous and the other random variables are discrete. With one variable of each type $${\displaystyle f_{X,Y}(x,y)=f_{X\mid Y}(x\mid y)\mathrm{P}(Y=y)=\mathrm{P}(Y=y\mid X=x)f_X(x).}$$ 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 ${\displaystyle X}$. One must use the "mixed" joint density when finding the cumulative distribution of this binary outcome because the input variables ${\displaystyle (X,Y)}$ 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, ${\displaystyle f_{X,Y}(x,y)}$ is the probability density function of ${\displaystyle (X,Y)}$ with respect to the product measure on the respective supports of ${\displaystyle X}$ and ${\displaystyle Y}$. Either of these two decompositions can then be used to recover the joint cumulative distribution function: $${\displaystyle F_{X,Y}(x,y)=\sum _{t\le y}{\int }_{-\mathrm{\infty }}^x{f}_{X,Y}(s,t)ds.}$$ 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 ${\displaystyle X}$ and ${\displaystyle Y}$ are independent if and only if the joint cumulative distribution function satisfies $${\displaystyle F_{X,Y}(x,y)=F_X(x)\cdot F_Y(y)}$$

Two discrete random variables ${\displaystyle X}$ and ${\displaystyle Y}$ are independent if and only if the joint probability mass function satisfies $${\displaystyle P(X=x\text{and}Y=y)=P(X=x)\cdot P(Y=y)}$$ for all ${\displaystyle x}$ and ${\displaystyle y}$.

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 $${\displaystyle f_{X,Y}(x,y)=f_X(x)\cdot f_Y(y)}$$ for all ${\displaystyle x}$ and ${\displaystyle y}$. 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 probability mass function of the joint distribution is ${\displaystyle \mathrm{P}(X_1,\dots ,X_n)}$. ${\displaystyle \mathrm{P}(X_1,\dots ,X_n)}$ 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.

Covariance

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 variables ${\displaystyle X}$ and ${\displaystyle Y}$ is $${\displaystyle \mathrm{cov}(X,Y)={\sigma }_{XY}=\mathrm{E}\left[(X-{\mu }_x)(Y-{\mu }_y)\right]=\mathrm{E}(XY)-{\mu }_x{\mu }_y.}$$

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 coefficient between the random variables ${\displaystyle X}$ and ${\displaystyle Y}$ is $${\displaystyle {\rho }_{XY}=\frac{\mathrm{cov}(X,Y)}{\sqrt{V(X)V(Y)}}=\frac{{\sigma }_{XY}}{{\sigma }_X{\sigma }_Y}.}$$

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.