---
tag: 0QAR0XDZH4
kind: definition
name: Joint probability distribution
msc: [62H05]
defines: 0DR81R1F2B
uses: [0DKRR22AEQ, 0TST4MHA3A, 0C81YBVTWH]
sources:
  - url: https://en.wikipedia.org/wiki/Joint_probability_distribution
    locator: Joint cumulative distribution function
---

## Definition

Let $X_1, \dots, X_n$ be real [[random variables|0DKRR22AEQ]] on one probability space $(\Omega, \mathcal{F}, P)$.
Their joint distribution is the [[law|0TST4MHA3A]] of the random vector $(X_1, \dots, X_n) \colon \Omega \to \mathbb{R}^n$. It is the probability measure on the Borel sets of $\mathbb{R}^n$ with
$$\mu(B) = P((X_1, \dots, X_n) \in B).$$
Their joint cumulative distribution function is $F(x_1, \dots, x_n) = P(X_1 \le x_1, \dots, X_n \le x_n)$.

## Remarks

For discrete variables, the joint probability mass function $p(x_1, \dots, x_n) = P(X_1 = x_1, \dots, X_n = x_n)$ determines the joint distribution.
The variables have a joint density $f$ if $\mu(B) = \int_B f(x) \, dx$ for each Borel set $B \subseteq \mathbb{R}^n$.
The joint distribution determines the [[marginal distributions|0AXP2Z5N9H]], but the marginal distributions do not determine the joint distribution.
For two rolls of a fair die, the joint mass function of the two results is $1/36$ at each pair $(i, j)$ with $1 \le i, j \le 6$.

## Explanations

### L1 human

The joint distribution describes several random quantities together, not one by one.
For example, it says how likely it is that a person is tall and also heavy.
It gives more than the chance of each of the two alone.