Q. 5.31 - A First Course in Probability

Question

(a) A fire station is to be located along a road of length $A, A<\infty$. If fires occur at points uniformly chosen on $\left(0,A\right)$, where should the station be located so as to minimize the expected distance from the fire? That is,

choose a so as to minimize $E\left[\left|X-a\right|\right]$ when X is uniformly distributed over $\left(0,A\right)$.

(b) Now suppose that the road is of infinite length— stretching from point $0$outward to $\infty$. If the distance of a fire from point $0$ is exponentially distributed with rate $\lambda$, where should the fire station now be located? That is, we want to minimize $E\left[\left|X-a\right|\right]$, where X is now exponential with rate $\lambda$.

Step-by-Step Solution

Answer

(a) The fire station should be located at the mid point of the length of the road to minimize the expected distance.

(b) The fire station should be located at $a=\frac{\log 2}{\lambda }$ so as to minimize the expected distance.

Part (a) Step 1. Given information.

Here, it is given that a fire station is to be located along a road of length $A, A < \infty$.

Fires occur at points uniformly chosen on $(0, A)$.

Part (a) Step 2. Find the value of E X - a .

Let $A$ be the fire station and $X$ be the place where the fire has occurred.

$X$ is uniformly distributed over $\left(O,A\right)$.

$\Rightarrow f_X\left(x\right)=\left\{\begin{array}{l}\frac{1}{A} 0\le x\le A\\ 0 \text{Otherwise}\end{array}\right.$

$$\begin{gathered} \left|X-a\right| = X-a, if a\le X\le A \\ \left|X-a\right| = a-X, if 0\le X\le a \end{gathered}$$

Now,

$$\begin{gathered} E\left[\left|X-a\right|\right]={\int }_0^a\left(a-x\right) f_X\left(x\right) dx + {\int }_a^A\left(X-a\right) f_X\left(x\right) dx \\ =\frac{1}{A}\left[{\left(ax-\frac{x^2}{2}\right)}_0^a+{\left(\frac{x^2}{2}-ax\right)}_a^A\right] \\ =\frac{1}{A}\left[a^2+\frac{A^2}{2}-aA\right] \end{gathered}$$

Part (a) Step 3. Find the location of fire station so as to minimize the expected distance from the fire.

Differentiating $E\left[\left|X-A\right|\right]$ w.r.t $a$ and equating it with zero, we get

$$\begin{gathered} \frac{d}{da}\left(E\left|X-a\right|\right)=0 \\ \frac{2a}{A}+0-1=0 \\ \frac{2a}{A}=1 \\ a=\frac{A}{2} \end{gathered}$$

Therefore, the  fire station should be located at the mid point of the length of the road to minimize the expected distance. 

 

Part (b). Step 1. Given information.

Here, it is given that the road is of infinite length stretching from point $0$ outward to $\infty$.

The distance of a fire from point $0$ is exponentially distributed with rate $\lambda$.

Part (b) Step 2. Find the value of E X - a .

The road is of infinite length. $X$ is exponentially distributed with $\lambda$ as parameter.

$$\begin{gathered} f_X\left(x\right)=\left\{\begin{array}{l}\lambda e^{-\lambda x} x......0\\ 0 \text{Otherwise}\end{array}\right. \\ \left|X-a\right|=\left\{\begin{array}{l}X-a a\le X\le \infty \\ a-X 0\le X\le a\end{array}\right. \end{gathered}$$

$$\begin{gathered} E\left[\left|X-a\right|\right]={\int }_0^a\left(a-x\right) \left(\lambda e^{-\lambda x}\right) dx + {\int }_a^{\infty }\left(x-a\right) \left(\lambda e^{-\lambda x}\right) dx \\ =\lambda \left[a{\int }_0^a \left(e^{-\lambda x}\right) dx -{\int }_0^a x\left(e^{-\lambda x}\right) dx+ {\int }_a^{\infty }x \left(e^{-\lambda x}\right) -a{\int }_a^{\infty } \left(\lambda e^{-\lambda x}\right) dx\right] \\ =\lambda \left[a {\left({\frac{e}{-\lambda }}^{-\lambda x}\right)}_0^a - {\left({\frac{xe}{-\lambda }}^{-\lambda x}-{\frac{e}{{\lambda }^2}}^{-\lambda x}\right)}_0^a+ {\left({\frac{xe}{-\lambda }}^{-\lambda x}-{\frac{e}{{\lambda }^2}}^{-\lambda x}\right)}_a^{\infty } -a {\left({\frac{e}{-\lambda }}^{-\lambda x}\right)}_a^{\infty }\right] \\ =\lambda \left[\frac{a}{\lambda }\left(1-e^{-\lambda a}\right)+\left(\frac{a{e}^{-\lambda a}}{\lambda }+\frac{e^{-\lambda a}}{{\lambda }^2}-\frac{1}{{\lambda }^2}\right)+\left(\frac{a{e}^{-\lambda a}}{\lambda }+\frac{e^{-\lambda a}}{{\lambda }^2}\right)-\frac{a{e}^{-\lambda a}}{\lambda }\right] \\ =\lambda \left[\frac{a}{\lambda }\left(1-2e^{-\lambda a}\right)+\left(\frac{2a{e}^{-\lambda a}}{{\lambda }^2}+\frac{2e^{-\lambda a}}{{\lambda }^2}-\frac{1}{{\lambda }^2}\right)\right] \\ =\lambda \left[\frac{a}{\lambda }+\frac{2e^{-\lambda a}}{{\lambda }^2}-\frac{1}{{\lambda }^2}\right] \\ =\left[a+\frac{2e^{-\lambda a}}{\lambda }-\frac{1}{\lambda }\right] \end{gathered}$$

Part (b) Step 3. Find the location of fire station so as to minimize the expected distance from the fire.

Differentiating $E\left|X-a\right|$ w.r.t $a$ and equating it with $0$, we get

$$\begin{gathered} \frac{d}{da}\left(E\left|X-a\right|\right)=0 \\ \Rightarrow 1-2e^{-\lambda a}=0 \\ \Rightarrow 1=2e^{-\lambda a} \\ \Rightarrow e^{-\lambda a}=\frac{1}{2} \\ \Rightarrow e^{\lambda a}=2 \\ \therefore a=\frac{\log 2}{\lambda }. \end{gathered}$$

Therefore, the  fire station should be located at  $a=\frac{\log 2}{\lambda }$ to minimize the expected distance.