Definition (CDF)

Given a RV $X$, the function $F:\mathbb{R}\to[0,1]$ defined by $$F(x)=P(X\le x), x\in\mathbb{R}$$ is called the distribution function (DF) of X.

Definition (Joint Distribution Function)

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