Cumulative Distribution Function (cdf)

Definition

Let $X$ : Random variable

If

$$\begin{array}{rl}{F}_X(x)& =P_X((-\infty ,x])\\ & =P(\{c\in \mathcal{C}:X(c)\le x\})\\ & =P(X\le x)\\ & ={\int }_{-\infty }^x{f}_X(t)dt\end{array}$$

Then we say $F_X(x)$ is the cumulative distribution function (cdf) of $X$.

Remark

We often shorten

• $P(\{c\in \mathcal{C}:X(c)\le x\})$ to $P(X\le x)$
• “Cumulative distribution function” to “distribution function” or “cdf”