The following problem is about optimization. It is not a homework, but rather a natural question to ask to oneself afterwards. Here it is.
Consider a road of length $L$ between two cities $A$ and $B$. Whenever a car runs out of fuel on this road, the distance between the car and the city $A$ is uniformely distributed on the interval $[0,L]$. There are three gas stations on the road. Now the question of the exercise is to compare two different distributions of the set of gas stations along the road. The first distribution is to put a station in A, another at distance $L/2$ from $A$ and the third in $B$. The second distribution is to put the stations at distance $L/4$, $L/2$ and $3L/4$.
Clearly the second distribution is better. However, the reference (from which I took the exercise) admits that the second distribution is not optimal!
Question: Given $n$ gas stations, where to place them on the road is such a way that the expectancy of the distance from one station to the place of breakdown is minimal?