4.2.1 The Cumulative Distribution Function

The cumulative distribution function (cdf) $F\left(x\right)$ for a discrete rv $X$ gives, for any specified number $x$ , the probability $P\left(X\le x\right).$

• It is obtained by summing the $\mathrm{pmf}p\left(y\right)$ over all possible values $y$ satisfying $y\le x$.

The cdf of a continuous rv gives the same probabilities $P\left(X\le x\right)$ and is obtained by integrating the pdf $f\left(y\right)$ between the limits $-\infty$ and $x$.

cumulative distribution function

The cumulative distribution function $F\left(x\right)$ for a continuous rv $X$ is defined for every number $x$ by

$$F\left(x\right)=P\left(X\le x\right)={\int }_{-\infty }^{x}f\left(y\right)dy$$

For each $x$, $F\left(x\right)$ is the area under the density curve to the left of $x$. This is illustrated in Figure 4.5, where $F\left(x\right)$ increases smoothly as $x$ increases.

Figure 4.5 A pdf and associated cdf

EX 4.6 thickness of metal sheet