EX 4.8 EX 4.6 continued

(EX 4.6 thickness of metal sheet continued)

When $X$ has a uniform distribution, $F\left(x\right)$ is differentiable except at $x=A$ and $x=B$, where the graph of $F\left(x\right)$ has sharp corners. Since

• $F\left(x\right)=0$ for $x<A$
• $F\left(x\right)=1$ for $x>B$,

$F^{\prime }\left(x\right)=0=f\left(x\right)$ for such $x$.

For $A<x<B$ ,

$$F^{\prime }\left(x\right)=\frac{d}{dx}\left(\frac{x-A}{B-A}\right)=\frac{1}{B-A}=f\left(x\right)$$