91Ó°ÊÓ

$\left(a\right)$A fire station is to be located along a road of length$A,A<∞$. If fires occur at points uniformly chosen$(0,A)$, That is, choose a so as to minimize$E\left[\left|X-a\right|\right]$

$\left(b\right)$) Now suppose that the road is of infinite length— stretching from point$0$outward to$∞$. If the distance of fire from a point$0$ is exponentially distributed with the rate$\mathrm{λ}$.

We have that

$$\begin{gathered} E\left[\left|X-a\right|\right]={∫}_{-∞}^{∞}\left(x-a\right)f\left(x\right)dx \\ ={∫}_0^a\left(a-x\right)\frac{1}{A}dx+{∫}_a^A\left(x-a\right)\frac{1}{A}dx \\ ={\left[\frac{a}{A}x-\frac{1}{2A}{x}^2\right]}_0^a+{\left[\frac{1}{2A}{x}^2-\frac{a}{A}x\right]}_a^A \\ =\left(\frac{a^2}{A}-\frac{a^2}{2A}\right)+\left(\left(\frac{A^2}{2A}-\frac{Aa}{A}\right)-\left(\frac{a^2}{2A}-\frac{a^2}{A}\right)\right) \\ =\frac{2a^2-a^2}{2A}+\frac{A^2-2Aa-a^2+2a^2}{2A} \\ =\frac{2a^2-2Aa+A^2}{2A} \end{gathered}$$

Since we want to minimize this, we take the derivative and set it equal to zero:

$$\begin{gathered} \frac{d}{da}\left(\frac{2a^2-2Aa+A^2}{2A}\right)=\frac{1}{2A}\left(4a-2A\right) \\ =\frac{2a}{A}-1=\frac{A}{2} \end{gathered}$$

. Thus, we can minimize the expected value by choosing the midpoint of the interval$(0,A)$

Using integration by parts we can find that

$$\begin{gathered} E\left[\left|X-a\right|\right]={∫}_0^{∞}\left(x-a\right)f\left(x\right)dx \\ ={∫}_0^a\left(a-x\right)\mathrm{λ}{e}^{-\mathrm{λ}x}dx+{∫}_0^{∞}\left(x-a\right)\mathrm{λ}{e}^{-\mathrm{λ}x}dx \\ =a\left(1-e^{\mathrm{λ}a}\right)+a{e}^{-\mathrm{λ}a}+\frac{e^{-\mathrm{λ}a}}{\mathrm{λ}}-\frac{1}{\mathrm{λ}}+a{e}^{-\mathrm{λ}a}+\frac{e^{-\mathrm{λ}a}}{\mathrm{λ}}-a{e}^{-\mathrm{λ}a} \end{gathered}$$

After differentiating and setting equal to zero, we discover that$e^{-\mathrm{λ}a}-\frac{1}{2}=0$which gives the minimum value at$a=\ln \left(\frac{2}{\mathrm{λ}}\right)$