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Let $X$ be a random variable. Define $F_X\colon R \to [0,1] $ as $F_X(x) = \Prb[X \leq x]$ for all $x$ The function $F_X(x)$ is called the cumulative distribution function of $X$
Every cumulative distribution function satisfies the following properties:
- $\lim_{x \to -\infty}{F_X(x)}=0$ and $\lim_{x \to +\infty}{F_X(x)}=1$
- $F_X$ is a monotonically nondecreasing function,
- $F_X$ is continuous from the right,
- $\Prb[a < X \leq b] = F_X(b) - F_X(a)$
If $X$ is a discrete random variable, then the cumulative distribution can be expressed as $F_X(x) = \sum_{k\leq x} \Prb[X = k]$
Similarly, if $X$ is a continuous random variable, then $F_X(x) = \int_{-\infty}^{x} f_X(y) dy$ where $f_X$ is the density distribution function.
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