Cumulative Distribution Function

Basically just a cumulative PMF/PDF, denoted by $F(x)$.

Definition: $F(x)=P(X\le x)$

Properties:

1. $F(X)\ge 0$, non-decreasing
2. $li{m}_{x\to -\infty }F(x)=0$, ${\lim }_{x\to \infty }F(x)=1$
3. $F$ is a step function in the discrete case

Conversions

• See Distribution for conversions PMF to CDF: ${\sum }_{y\le x}f(y)=F(x)$ CDF to PMF: $f(x)=F(x)-F(x-1)$

Uses Cases

Example Four die are rolled simultaneously. Suppose the r.v. $X=\max \{X_1,X_2,X_3,X_4\}$. Find the p.m.f. of $X$.

• In this case, it is much easier to just use a CDF