Supernovas: Making Astronomical History

Poisson Statistics

What is a Poisson distribution?

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 Recherches sur la probabilité des jugements en matière criminelle et en matière civile (Research 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).

There are many ways we can look at how to actually understand the Poisson distribution, a common way too look at it from an astronomical perspective is to look at the amount of meteors entering Earth's atmosphere. Many of these meteors are too small to be even noticeable at all, Often burning up in the atmosphere. In fact, over 100,000 kg of space debris hits the Earth every day, however, most of this is in the form of stellar dust, which is basically harmless.

To figure out exactly how many meteors hit in a certain time period can be a very difficult process, as there are so many different sizes and forms of meteors that it's difficult to give solid statistics without setting proper size restrictions. For example, if we look at a 1996 study which estimated the number of events by finding meteorites in the desert, they found that by extrapolating to the whole earth, there would be about 18,000 to 84,000 meteorites bigger than 10 grams each year. Poisson distributions prefer much smaller numbers and time intervals, so lets first off average those two values to get about 51,000 events per year, if we bring the time interval down to an hour, we get about 5.8 events per hour.

An example of what a Poisson distribtuon looks like as you increase λ and/or k.

Now we can get into the actual math of the situation. Poissonian events are described by the fairly straight forward equation:

`P = (lambda^k * e^lambda)/(k!)`

Here P is the probability of whichever specific k we choose, where k can be any whole integer we choose, λ is our expected value, so in this situation 5.8 and e is the natural number (2.71828...). This will allow us to get a estimate of the probability that a certain amount of events will occur within the time frame, this is our value k, so if we want to know the probability that 4 events will occur, we plug 4 into our equation, and 5.8 in for λ. We can do this for all of the values surrounding 5.8 until we hit the point of things being fairly unlikely to happen. In this table I have shown all the probabilities of a certain number of events occuring in an hour, I stopped going once the probability dropped below 1%.

Events	Probability	Percentage
1	.018	1.8%
2	.051	5.1%
3	.098	9.8%
4	.143	14.3%
5	.166	16.6%
6	.160	16%
7	.133	13.3%
8	.096	9.6%
9	.062	6.2%
10	.036	3.6%
11	.019	1.9%

As you can see, the probability steadily increases from 1 to 5, with 5 being the max value, and then begin to drop off again after that. All this is saying is that you're more likely to see a value closer to 5.8 than one farther away. This specific example is obviously hard to directly observe, as this is metoers larger than 10 grams across the entire Earth, but it can give you a good idea of what you would see if it we could do something like that.

There are many other example of situations where a Poisson distribution can be useful, looking at the amount of goals scored in soccer games, the number of cars going through a gas station, or people going to the top of the Empire State Building.

References and Further Reading

Introduction to Error Analysis by John R. Taylor, University Science Books (1997), (p. 245-260) gives a good outline of Poissonian distribution and how to apply them.