cumulative distribution function

For a probability space $(\Omega ,\mathcal{F},P)$ and a random variable $X:\Omega \to \mathbb{R}$ the cumulative distribution function is

$$F_X(x)=P(X\le x)=P(X^{-1}(-\infty ,x))$$

using the distribution of random variable $P(X)$.

Properties

• $F_X$ is non-decreasing
• ${\lim }_{x\to \infty }{F}_X(x)=1$ and ${\lim }_{x\to -\infty }{F}_X(x)=0$
• $F_X$ is right continuous.