4.2.2 Using F(x) to Compute Probabilities

The importance of the cdf here, just as for discrete rv’s, is that probabilities of various intervals can be computed from a formula for or table of $F\left(x\right)$ .

cdf induces probability

Let $X$ be a continuous rv with $\mathrm{pdf}f\left(x\right)$ and $\mathrm{cdf}F\left(x\right)$ . Then for any number $a$,

$$P\left(X>a\right)=1-F\left(a\right)$$

and for any two numbers $a$ and $b$ with $a<b$,

$$P\left(a\le X\le b\right)=F\left(b\right)-F\left(a\right)$$

Figure 4.8 Computing $P\left(a\le X\le b\right)$ from cumulative probabilities

Figure 4.8 illustrates the second part of this proposition; the desired probability is the shaded area under the density curve between $a$ and $b$, and it equals the difference between the two shaded cumulative areas. This is different from what is appropriate for a discrete integer-valued random variable (e.g., binomial or Poisson): $P\left(a\le X\le b\right)=F\left(b\right)-F\left(a-1\right)$ when $a$ and $b$ are integers.

EX 4.7 dynamic load on bridge

Once the cdf has been obtained, any probability involving $X$ can easily be calculated without any further integration.