Cumulative distribution function

Let $\mathbb{T}$ be a time scale. Let $f \colon \mathbb{T} \rightarrow \mathbb{R}$ be a probability density function. The following function is called the cumulative distribution function (or cdf) of $f$: $$F(x) = \displaystyle\int_0^x f(t) \Delta t.$$

Let $f(x,y)$ be a joint time scales probability density function, then we have a joint time scales cumulative distribution function by $$F_{X,Y}(x,y)=P(X<x,Y<y)=\displaystyle\int_0^x\int_0^y f_{X,Y}(s,t) \Delta t \Delta s.$$

References

Probability theory on time scales and applications to finance and inequalities by Thomas Matthews