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## Homework Statement

Let ##U_1, U_2, U_3## be independent uniform on ##[0,1]##.

a) Find the joint density function of ##U_{(1)}, U_{(2)}, U_{(3)}##.

b) The locations of three gas stations are independently and randomly placed along a mile of highway. What is the probability that no two gas stations are less than 1/3 mile apart?

## Homework Equations

## The Attempt at a Solution

Let ##U_{(1)}, U_{(2)}, U_{(3)}## take on values ##x,y,z## respectively

a) The answer was 6, with ##0<x<y<z<1##

b) I defined the limits as such. ##0<x<1/3##, since this is the maximum the first station can be placed before the other 2 get too close to each other. Then ##x+1/3 < y <z - 1/3##, and finally ##2/3<z<1##, so ##y##'s possible positions are constrained by the positions of its neighbours.

However, integrating the joint density function over these limits returned a 0, which was clearly wrong. I know the right limits to integrate over were ##0<x<y-1/3## then ##1/3 < y < z - 1/3##, and ##2/3<z<1##. What was wrong with my reasoning?

Many thanks in advance!