Cumulative distribution function

Another way to specify the distribution of a Random variable. The difference between PMFs and CDFs is that PMFs are only defined for discrete variables, while CDFs are defined for all random variables. Therefore, it is usually only used in the context of continuous variables.

For discrete variables, the CDF would simply be a stepwise function.

Given a random variable $X$, the CDF is given by $F_X$ where $F_X(x)=P(X\le x)$.

Also see Integrating the PDF to get the CDF.

Properties of valid CDFs

A CDF $F$ obeys the following properties:

• Increasing: if $u_1\le u_2$, then $F(u_1)\le F(u_2)$. Basically, the CDF is always increasing with respect to $u$.
• Right-continuous: for any real number $a\in \mathbb{R}$, ${\lim }_{u\to a^{+}}F(u)=F(a)$.
• ${\lim }_{u\to -\infty }F(u)=0$ and ${\lim }_{u\to \infty }F(u)=1$.