Birds on a Wire
A number of birds randomly land on a telephone wire. If the intervals between each bird and its closest companion were painted, what would be the expected fraction of wire painted? And what proportion of the number of intervals would be painted? The key to answering these questions is to realize that the probability distribution of interval lengths is (approximately) exponential. Knowing that, these fractions can be calculated to be 7/18 and 2/3.
This problem was inspired by Birds on a Wire.
If the mean density of birds is , then the probability distribution of interval lengths is . Integrating this, the probability that an interval is less than is . Squaring that gives the probability that the two intervals on either side of a chosen interval are both smaller than the chosen interval, in which case it would not be painted. Thus, the probability that an interval of length will be painted is . Thus, the probability that a link will be painted is . Likewise, the average length of an interval weighted by the probability that it is painted is . Dividing by the average length of an interval λ gives the fraction of painted wire as .