For a univariate random variable we have \begin{align} F(x) = P(X< x) = \int_{-\infty}^{\infty} p(x) dx \end{align} and \( \frac{dF}{dx} = p(x) \) the probability density function.
We can extend this definition to more than one random variable \begin{align} F(x,y) = \int_{-\infty}^y \int_{-\infty}^x p(x,y) dx y \end{align} \( F(x,y) = P(X < x, Y < y) \) \( \frac{\partial F}{\partial x \partial y} = p(x,y) \) the probability density function. If we can write \( p(x,y) = f(x) g(y) \) then we say \( X \) and \( Y \) are independent.