Definition

Let $X=(X_1,...,X_n)^T$ be a random vector. The joint distribution function of $X$ is defined by $$\begin{align*} F_X(x)&=P(X_1\le x_1,...,X_n\le x_n)\\ &=P(\{X_1\le x_1\}\cap...\cap\{X_n\le x_n\})\\ &=P((-\infty,x_1]\times\cdots\times(-\infty,x_n])\\ &=P(A_x), \ \text{where} \ A_x=(-\infty,x_1]\times\cdots\times(-\infty,x_n]\subset\mathbb{R}^n \end{align*}$$ ## Proposition (Properties of JDF) 1. $0\le F_X(x)\le 1 \ , \forall x\in\mathbb{R}^n$ 2. $F_X(x)$ is right continuous.