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 2.10 (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.
The cumulative distribution function (CDF)  for $Z$. Open dots mean that the point IS NOT included; solid dots mean the point IS included

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