91Ó°ÊÓ

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

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.