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Chapter 4: Q10E (page 247)

If n houses are located at various points along a straight road, at what point along the road should a store be located in order to minimize the sum of the distances from the n houses to the store?

Short Answer

Expert verified

The store should be located at the median of the various points along a straight road where the n houses are located.

Step by step solution

01

Given information

There are n houses located at various points along a straight road.

02

Minimize the sum of the distances

Let a random variable X denote the locations of n houses along a straight road.

Let d be the location of the store.

The sum of the distances from the n houses to the store can be written as \(\sum\limits_{i = 1}^n {\left| {{X_i} - d} \right|} .\)

Now, in order to minimize this, we need to minimize the expected value of each of the distances, which is \(E\left( {\left| {X - d} \right|} \right)\).

Now, it is already known that \(E\left( {\left| {X - d} \right|} \right)\) is minimized when d takes the value of the median of X.

Thus, \(\sum\limits_{i = 1}^n {\left| {{X_i} - d} \right|} \) will be minimized when d is the median of X.

Hence, the store should be located at the median location of the n houses along a straight road.

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Most popular questions from this chapter

Suppose that X and Y are random variables such that

\(E\left( {Y|X} \right) = aX + b\)Assuming that\(Cov\left( {X,Y} \right)\)exists and that\(0 < Var\left( X \right) < \infty \), determine expressions for a and b in terms of\(E\left( X \right)\),\(E\left( Y \right)\)and\(Cov\left( {X,Y} \right)\).

Suppose that X and Y have a continuous joint distribution for which the joint p.d.f. is as follows:

\({\bf{f}}\left( {{\bf{x,y}}} \right){\bf{ = }}\left\{ {\begin{align}{}{{\bf{x + y}}}&{{\bf{for}}\,\,{\bf{0}} \le {\bf{x}} \le {\bf{1}}\,\,{\bf{and}}\,\,{\bf{0}} \le {\bf{y}} \le {\bf{1,}}}\\{\bf{0}}&{{\bf{otherwise}}{\bf{.}}}\end{align}} \right.\).

Find\({\bf{E}}\left( {{\bf{Y}}\left| {\bf{X}} \right.} \right)\)and\({\bf{Var}}\left( {{\bf{Y}}\left| {\bf{X}} \right.} \right)\).

Suppose that the random variableXhas the uniform distribution on the interval [0,1], that the random variableYhas the uniform distribution on the interval [5,9],and thatXandYare independent. Suppose also that a

rectangle is to be constructed for which the lengths of two adjacent sides areXandY. Determine the expected value of the area of the rectangle.

Suppose that a class contains 10 boys and 15 girls, and suppose that eight students are to be selected randomly from the class without replacement. Let X denote the number of boys selected, and let Y denote the number of girls selected. Find E(X 鈭� Y ).

Suppose that a fire can occur at any one of five points along the road. These points are located at -3, -1, 0, 1, and 2 in Fig. 4.9. Suppose also that the probability that each of these points will be the location of the next fire that occurs along the road is as specified in Fig. 4.9.

  1. At what point along the road should a fire engine wait in order to minimize the expected value of the square of the distance that it must travel to the next fire?
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