= Cumulative distribution function
{title2=$F_X(x)$}
{wiki}
= Distribution function
{synonym}
= CDF
{c}
{synonym}
The <cumulative distribution function> of a real <random variable> $X$ is
$$
F_X(x)=\mathbb P(X\leq x).
$$
It records the probability accumulated up to a threshold. It is a <nondecreasing function> and is <right-continuous>; its limits are zero at $-\infty$ and one at $+\infty$. These properties follow from ordered threshold events, <continuity from above of a measure>, and <continuity from below of a measure>. A <cumulative distribution function> determines the entire <probability distribution> on the <real line>. https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/9797310bed4c7f5b5d40d007783eec8d_MIT6_436JF18_lec04.pdf[MIT Lecture 4: properties of cumulative distribution functions]{ref}.
For $a<b$,
$$
\mathbb P(a<X\leq b)=F_X(b)-F_X(a),\qquad
\mathbb P(X=x)=F_X(x)-F_X(x^-),
$$
where $F_X(x^-)=\lim_{t\uparrow x}F_X(t)$ is its left-hand <limit>. The second formula measures a jump. Thus $\mathbb P(a\leq X\leq b)=F_X(b)-F_X(a^-)$; changing an endpoint is harmless only when its point probability is zero.
For the finite example under <probability distribution>, $F_X(x)$ is $0$ below zero, $0.2$ on $[0,1)$, $0.7$ on $[1,3)$, and $1$ on $[3,\infty)$. In particular $F_X(1)=0.7$, but $F_X(1^-)=0.2$, so the jump gives $\mathbb P(X=1)=0.5$. For the density example, $F_Y(y)=y^2$ on $[0,1]$, and its increase from $1/4$ to $3/4$ is $9/16-1/16=1/2$.
When an <absolutely continuous probability distribution> has <probability density function> $f$, its <cumulative distribution function> is $F(x)=\int_{-\infty}^{x}f(t)\,dt$ and $F'=f$ <almost everywhere>. A continuous <cumulative distribution function> by itself does not guarantee such a density: the <Cantor distribution> is an important exception.
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