Consider a system in which most of the time, nothing happens, but every once in a while, an event occurs which catches our interest. Mathematics requires that we specify precisely what we're talking about, so we list the following properties.

  • The fact that one event happens does not change the probability that another event will happen later.
  • If we look at a very narrow time interval, the probability that an event happens during that slot is a number. We can look at any time slot this size (say a tenth of a second), and the number will be the same for as long as the system continues to operate.
  • Looking at narrower and narrower time slots, we can eventually find one where the probability of two events happening is basically zero.

    • We might be talking about radioactive nuclei decaying inside a lump of uranium metal or about snowflakes landing on a park bench. If we have any system which obeys these three rules, we can make definite mathematical statements about it. For example, we can answer questions like "How many radioactive nuclei will we expect to see decay in 10 seconds?" or "What's the longest time we can go without seeing a snowflake land?" A process which obeys these rules is called a Poisson process, and the study of their behavior forms a branch of mathematics called Poisson statistics.

    Siméon Denis Poisson (1781-1840) introduced the probability concept which bears his name in the book Recherchés sur la probabilité des jugements en matière criminelle et matière civile (Researches on the Probability of Judgments in Criminal and Civil Matters), which was published in 1837. Many mathematicians built upon this work, the first notable one being the Russian Pafnuty Chebyshev (1821-1894).

    In years past, one of the classic examples statistics textbooks gave to describe Poisson probabilities had a rather macabre touch. During 1944 and 1945, Nazi Germany fired 1,358 V-2 rockets at the city of London. The V-2, more fully known as Vergeltungswaffe 2 or "Reprisal Weapon 2", was fired from mobile launch sites and had a range of about 300 kilometres (200 miles). Until the Nazi engineers invented a way to guide the V-2 with radio waves, each rocket used a self-contained guidance system. A primitive computer made from analog ciruit parts adjusted the rocket's angle of flight, and when the rocket ran out of fuel--a time called Brennschluss, from German words meaning "burnout"--it continued under gravity's influence, tracing a parabolic trajectory which ended in a crater on Allied soil. This guidance system wasn't accurate enough to ensure pinpoint impacts, but it did destroy lives.

    If we examine the distribution of V-2 impacts upon London, we notice it has several interesting properites. First, each impact is independent of the others; a V-2 landing on the city doesn't change the probability that another would land some time later. Furthermore, because the Nazis launched their weapons at a roughly constant rate (a rate they tried to make as fast as possible), we have a well-defined average rate of impacts. Finally, the Nazis had a limited number of launch sites, each one of which could only be used every so often, so given a small enough time interval, the probability of two or more V-2 impacts is just about zero. These are the precise three qualifications we listed earlier for a Poisson process.

    We can change our perspective slightly and look at intervals in space instead of in time. Suppose we lay a grid upon a map of London, dividing the city into squares. We can then count the number of V-2 landings in each square. Some sections will have no impacts (the lucky ones), while others will have one or more. Thinking carefully about the Nazi's inaccurate guidance system, you will recognize that a rocket landing in one square doesn't make it either more or less likely that another rocket will land in the same square later. Put another way, the presence of one "dot" in a square doesn't affect the likelihood you'll find another "dot" beside it in the same space. Poisson statistics let us calculate how many squares we expect to have one "bombie", how many we can expect to have two or more, and how many we can reasonably say will have none at all. Of course, we can't say which parts of the city will remain safe; remember, the rockets land randomly.

    Learning statistics in a British classroom after World War II must have been an interesting experience.

    Both the V-2 and the Poissonian description of its destructive behavior became central themes in Thomas Pynchon's postmodern epic novel Gravity's Rainbow. Researched, written and maintained by Blake Stacey.