# Joint distribution function

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

## 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 and .

Their joint cdf is defined aswhere and are two real numbers.

Note that:

• indicates a probability;

• the comma inside the parentheses stands for AND.

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

• the random variable takes a value less than or equal to ;

• the random variable takes a value less than or equal to .

## Example

Suppose that there are only four possible cases:

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:

The two conditions that need to be simultaneously true are:

There are two cases in which they are satisfied:

Therefore, we have

## 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 and is:where:

• is the support of the vector , that is, the set of all the values of that have a strictly positive probability of being observed;

• we sum the probabilities over the setthat contains all the couples belonging to the support and such that and .

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

## 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.

In this table, there are nine possible couples 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 (included) and above (included).

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

## The formula for continuous variables

When and are continuous random variables, we need to use the formulawhere is the joint probability density function of and .

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

1. first compute the inner integralwhich, in general, is a function of and ;

2. then calculate the outer integral

### Example

Let us make an example.

Let the joint pdf be

When and , we have

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

1. or , in which case

2. and , in which case

## How to derive the marginal cdfs from the joint

The two marginal distribution functions of and are

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

This can be demonstrated as follows:because the condition is always met and, as a consequence, the condition is satisfied whenever is true.

The proof for 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 and are independent, then the joint cdf is equal to the product of the marginals:

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 random vector is a function such that:where the entries of and are denoted by and respectively, for .

## More details

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

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