There is an east-west street of length L units. And we park cars of unit length along the north side until we can't place any more cars. Each car is placed randomly (uniformly).
What is the expected number of cars that can be parked (as a function of L)?
I'll start you off...
For 0 <= L < 1, F(L) = 0
For 1 <= L < 2, F(L) = 1
Okay... now the easy ones are out of the way, can you describe the function for L>=2?
...But unlike the bird problem, this seems to be more 'sequential.' By that I mean cars are parked one at a time, and then the problem changes for the next person.
Basically, the expected value of the first car's location is L/2. We then have two identical subproblems of length ((L-1)/2). The expected value of those locations should be ((L-1)/4) taken relative from each subproblem's starting point, leaving four identical subproblems of length ((L-3)/4). It continues on and on in this form, but whether or not a car can be parked there depends on the value of L.
Posted by Eric
on 2004-07-01 14:45:04