EX 4.4 probability of angle

The direction of an imperfection with respect to a reference line on a circular object such as a tire, brake rotor, or flywheel is, in general, subject to uncertainty. Consider the reference line connecting the valve stem on a tire to the center point, and let $X$ be the angle measured clockwise to the location of an imperfection. One possible pdf for $X$ is

$$f\left(x\right)=\{\begin{array}{cc}\frac{1}{360}& 0\le x<360\\ 0& \text{ otherwise }\end{array}$$

The pdf is graphed in Figure 4.3.

Figure 4.3 The pdf and probability from Example 4.4

Clearly $f\left(x\right)\ge 0$ . The area under the density curve is just the area of a rectangle: $(\text{height})(\text{base})=\left(1/360\right)\left(360\right)=1.$ The probability that the angle is between ${90}^{\circ }$ and ${180}^{\circ }$ is

$$\begin{array}{rl}P\left(90\le X\le 180\right)& ={\int }_{90}^{180}\frac{1}{360}dx\\ & ={\frac{x}{360}|}_{x=90}^{x=180}\\ & =\frac{1}{4}=.25\end{array}$$

The probability that the angle of occurrence is within ${90}^{\circ }$ of the reference line is

$$\begin{array}{rl}& P\left(0\le X\le 90\right)+P\left(270\le X<360\right)\\ & =.25+.25=.50\end{array}$$