Joint probability distribution

Definition

Let $X_1, \dots, X_n$ be real random variables on one probability space $(\Omega, \mathcal{F}, P)$. Their joint distribution is the law 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, 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.