3.2 CDF for discrete rvs

The CDF $F_X(x)$ for a discrete rv $X$ with PMF $f_X(x)$ is $F_X(x) = \Pr(T \le x) = \sum_{-\infty}^x f(t).$

Example 3.2 (CDF: discrete rv) Consider the discrete rv $Z$ from Example 2.6, with PMF $f_Z(z) = \frac{|z - 3|}{4} \quad\text{for $z=1, 2, 3, 4$}.$ The CDF is
$F_Z(z) = \left\{ \begin{array}{ll} 0 & \text{for $z<1$}\\ 1/2 & \text{for $1\le z < 2$}\\ 3/4 & \text{for $2\le z < 4$}\\ 1 & \text{for $z\ge4$}. \end{array} \right.$ See Fig. 3.2.

FIGURE 3.2: The cumulative distribution function (CDF) for $Z$ . Open dots mean that the point IS NOT included; solid dots mean the point IS included