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Joint distribution function

by , PhD

The joint distribution function is a function that completely characterizes the probability distribution of a random vector.

Table of Contents

Synonyms and acronyms

It is also called joint cumulative distribution function (abbreviated as joint cdf).

Joint cdf of X and Y

Let us start with the simple case in which we have two random variables X and Y.

Their joint cdf is defined as[eq1]where x and $y$ are two real numbers.

Note that:

In other words, the joint cdf [eq2] gives the probability that two conditions are simultaneously true:


Suppose that there are only four possible cases:[eq3]

Further assume that each of these cases has probability equal to 1/4.

Let us compute, as an example, the following value of the joint distribution function:[eq4]

The two conditions that need to be simultaneously true are:[eq5]

There are two cases in which they are satisfied:[eq6]

Therefore, we have[eq7]

The formula for discrete variables

In the previous example we have shown a special case.

In general, the formula for the joint cdf of two discrete random variables $X $ and Y is:[eq8]where:

The probabilities in the sum are often written using the so-called joint probability mass function[eq11]

How to compute the formula with a table

The sum in the formula above can be easily computed with the help of a table.

Here is an example.

This table provides an example of how to calculate the joint cdf.

In this table, there are nine possible couples [eq10] and they all have the same probability (1/9).

In order to compute the joint cumulative distribution function, all we need to do is to shade all the probabilities to the left of x (included) and above $y$ (included).

Then, the value of [eq13] is equal to the sum of the probabilities in the shaded area.

The formula for continuous variables

When X and Y are continuous random variables, we need to use the formula[eq14]where $f_{XY}$ is the joint probability density function of X and Y.

The computation of the double integral can be broken down in two steps:

  1. first compute the inner integral[eq15]which, in general, is a function of $lambda $ and $y$;

  2. then calculate the outer integral[eq16]


Let us make an example.

Let the joint pdf be[eq17]

When [eq18] and [eq19], we have [eq20]

This is only one of the possible cases. We also have the two cases:

  1. $xleq 1$ or $yleq 0$, in which case[eq21]

  2. $x>2$ and $y>0$, in which case[eq22]

How to derive the marginal cdfs from the joint

The two marginal distribution functions of X and Y are[eq23]

They can be derived from the joint cumulative distribution function as follows:[eq24]where the exact meaning of the notation is[eq25]

This can be demonstrated as follows:[eq26]because the condition $Yleq infty $ is always met and, as a consequence, the condition [eq27]is satisfied whenever $Xleq x$ is true.

The proof for Y is analogous.

Deriving the joint cdf from the marginals

In general, we cannot derive the joint cdf from the marginals, unless we know the so-called copula function, which links the two marginals.

However, there is an important exception, discussed in the next section.

Joint cdf of two independent variables

When X and Y are independent, then the joint cdf is equal to the product of the marginals:[eq28]

See the lecture on independent random variables for a proof, a discussion and some examples.

A more general definition

Until now, we have discussed the case of two random variables. However, the joint cdf is defined for any collection of random variables forming a random vector.

Definition The joint distribution function of a Kx1 random vector V is a function [eq29] such that:[eq30]where the entries of V and $v$ are denoted by $V_{k}$ and $v_{k}$ respectively, for $k=1,ldots ,K$.

More details

More details about joint distribution functions can be found in the lecture entitled Random vectors.

Keep reading the glossary

Previous entry: Integrable random variable

Next entry: Joint probability density function

How to cite

Please cite as:

Taboga, Marco (2021). "Joint distribution function", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix.

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