cumulative distribution function

A cumulative distribution function is a probability distribution which returns the probability that the random variable will take a value of at most the argument of the function.

Formally,
a random variable $X$ has a cumulative distribution function $F:\mathbb{R}\to [0,1]$ such that

i.e. $F(x)$ is the probability that $X$ will take a value of at most $x$.

A continuous distribution function $F(x)$ must satisfy

• $F(x)$ is strictly increasing
• $\underset{x\to \mathrm{\infty }}{lim}F(x)=1$
• $\underset{x\to -\mathrm{\infty }}{lim}F(x)=0$

For a discrete random variable with probability mass function $p(x)$, the cumulative distribution function is given by

For a continuous random variable with probability density function $f(x)$, the cumulative distribution function is given by

When $X$ is a continuous random variable, $P(X=x)=0$ for any particular $x$; therefore, the probabilities of open and closed intervals are identical. Thus,

• $P(X\ge a)=1-P(X<a)=1-F(a)$
• $P(a\le X\le b)=P(X\le b)-P(X<a)=F(b)-F(a)$