Cumulative distribution function

For a real-valued random variable $X$, this is the cumulative distribution function of its law: $F_X(t)=\mathbb P(X\le t)$.

The function is nondecreasing and right-continuous, with limits zero at $-\infty$ and one at $+\infty$. It determines the measure through $P((a,b])=F(b)-F(a)$ for $a<b$.

For a mass function on a countable subset of $\mathbb R$, $F(t)=\sum_{x\le t}p(x)$. If $P$ has a density $f$, then $F(t)=\int_{-\infty}^t f(x)\,dx$. A cumulative distribution function exists even when neither representation applies.