Joint Density Function

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.