EX 4.6 thickness of metal sheet

Let $X$ , the thickness of a certain metal sheet, have a uniform distribution on $\left[A,B\right]$. The density function is shown in Figure 4.6.

Figure 4.6 The pdf for a uniform distribution

• For $x<A$, $F\left(x\right)=0$,
  • since there is no area under the graph of the density function to the left of such an $x$.
• For $x\ge B$, $F\left(x\right)=1$,
  • since all the area is accumulated to the left of such an $x$.
• Finally, for $A\le x\le B,$

$$\begin{array}{rl}F\left(x\right)& ={\int }_{-\infty }^{x}f\left(y\right)dy\\ & ={\int }_A^x\frac{1}{B-A}dy\\ & ={\frac{1}{B-A}\cdot y|}_{y=A}^{y=x}\\ & =\frac{x-A}{B-A}\end{array}$$

The entire cdf is

$$F\left(x\right)=\{\begin{array}{cc}0& x<A\\ \frac{x-A}{B-A}& A\le x<B\\ 1& x\ge B\end{array}$$

The graph of this cdf appears in Figure 4.7.

Figure 4.7 The cdf for a uniform distribution