What's fishy about this post?
P(x) = e-m*mx/x!
is the "Poisson probability distribution function" and gives the probability you'll get x occurrences of something if the average or mean value of the "population" is m. x! stands for x factorial (e.g. 4! = 4*3*2*1) and e = the famous constant 2.71828+ ("Euler's number")
http://www.anesi.com/poisson.htm
As the above link says, there are many things that exhibit a so-called Poisson distribution. This picture of dandelions is one of them. I did a rough count of all the dandelions and came up with about 400. When printed, the image measured roughly 3 3/4" x 8". This works out to about 3 1/3 dandelions per 1/4 square inch (details omitted but check my math). I then cut out a 1/4" square "hole" in a piece of paper and randomly placed this "window" on various places of the printed image. I did this 20 times and each time I counted the number of dandelions in the window. (This is a little tough to do because it's hard to see the flowers in some cases and of course you sometimes get 1/2 a flower etc.)
Despite all the above rough estimating, I came up with the following results:
There were / was
1 "window" that contained 0 dandys 1/20 = .05
3 "windows" that contained 1 dandy 3/20 = .15
5 "windows" that contained 2 dandys 5/20 = .25
4 "windows" that contained 3 dandys 4/20 = .20
2 "windows" that contained 4 dandys 2/20 = .10
1 "window" that contained 5 dandys 1/20 = .05
1 "window" that contained 6 dandys 1/20 = .05
1 "window" that contained 7 dandys 1/20 = .05
2 "windows" that contained 8 dandys 2/20 = .10
These percentages in the right-most column correspond fairly well with the predicted probabilities when you use the Poisson generator in the above link with a value of 3 for the "expected occurrences per trial" (which is as close to 3 1/3 as we can get).
That fancy formula above is what is used by the Poisson generator to get its numbers. The mx part gets big as x gets big but the x! tends to "damp it out" as x gets big (because it's in the denominator). This is why the probability is "skewed." It climbs from 0 occurrences up to the average and a bit beyond and then drops down to nothing pretty quickly. But there *is* a chance you'll see as many as 8 or 9 dandys in a random 1/4".
I find it fascinating that dandelions (and cars arriving at a McDonalds and calls coming into a call center and and ...) all exhibit this kind of "distribution." But then again, I'm tutoring Statistux and there's nothing like teaching to get you interested in such stuff.
2 comments:
looks like my lawn before I started using lawn service!
Am glad you are having so much fun (and sharing ) with your new tutor/teaching position. Are you numerically engaging with some interesting young minds, formulating some new friends, balancing your time with running, equating to your wife's work schedule? And, what is the probability that you will have time for your garden this year? ;)
Ditto on the no math brain! But I still find it interesting none the less.
Post a Comment