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# Properties of the Cumulative Distribution Function

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- The CDF relates the probability that the true (but unknown) value of the random variable $T$ is less than or equal to some observed value $t$ 

$$F(t)=Pr(T\le t), \quad t\in (0,\infty)$$

- The CDF is strictly positive 

$$F(t) \ge 0, \forall t \in \mathbb{R}^{+}$$

- The CDF is non-decreasing 

$$\begin{aligned}F(0)&=0\\\\\\\\
\displaystyle \lim_{t \to \infty} F(t)&=1\end{aligned}$$

- Alternate names for the CDF

    + The distribution function
    
    + The probability function
