Suppose the pdf of the magnitude of a dynamic load on a bridge (in newtons) is given by
For any number between 0 and 2,$$ \begin{align} F\left( x \right) &= \int_{-\infty}^{x} f\left( y \right) , dy \ &= \int_{0}^{x} \left( \frac{1}{8} + \frac{3}{8}y \right) , dy \ &= \frac{x}{8} + \frac{3}{16}x^{2} \end{align}
F\left( x\right) = \left{ \begin{matrix} 0 & x < 0 \ \frac{x}{8} + \frac{3}{16}{x}^{2} & 0 \leq x \leq 2 \ 1 & 2 < x \end{matrix}\right.
The graphs of $f\left( x\right)$ and $F\left( x\right)$ are shown in Figure 4.9. Figure 4.9 The pdf and cdf for Example 4.7  The probability that the load is between 1 and 1.5 is\begin{align} &P\left( 1 \leq X \leq 1.5 \right) \&= F\left( 1.5 \right) - F\left( 1 \right) \ &= \left[ \frac{1}{8}\left( 1.5 \right) + \frac{3}{16}\left( 1.5 \right)^{2} \right] - \left[ \frac{1}{8}\left( 1 \right) + \frac{3}{16}\left( 1 \right)^{2} \right] \ &= \frac{19}{64} \ &= 0.297 \end{align}
\begin{align} &P\left( X > 1 \right) \ &= 1 - P\left( X \leq 1 \right) \ &= 1 - F\left( 1 \right) \ &= 1 - \left[ \frac{1}{8}\left( 1 \right) + \frac{3}{16}\left( 1 \right)^{2} \right] \ &= \frac{11}{16} \ &= 0.688 \end{align}