Sad Drowned Earthworms

Lots of rain this past week has saturated the ground and brought up earthworms attempting to escape drowning. These poor ones weren't lucky. Their last appearances were these squiggly shapes on the sidewalk. They resemble strings thrown on the ground, which is actually an established problem in probability called "The Thrown String". First formulated by J.L. Synge in a question in the Mathematical Gazette in 1968:

A perfectly flexible inextensible string of length L is thrown down at
random on a horizontal table. It is assumed that the form of the string
is represented by x =x(s), y =y(s), these functions possessing derivatives
of all orders for 0<s<L. The experiment is repeated many times.
What is the average value of the rectilinear distance between the ends
of the string?

Synge later explored the problem in the Mathematical Gazette in 1970. He collected experiments of actually throwing stings and measuring the distance between the endpoints. Let D be the distance between the endpoints of a string of length L.

Suggesting that a ratio around 1/3 might be possible. But his final conclusion leaves the problem open:

 Conclusion
The problem of the thrown string is not solved. Perhaps we should
say that it has not been adequately formulated.

Others have considered the problem. Two in particular: Clarke in the Mathematical Gazette (1971) represents the the string as a sequence of line segments J.F.C. Kingman in the Journal of the Royal Statistical Society B (1982) considers the string as a chain and models its dynamics on the way to the floor. But before this approach he considers the string as the realization of a stochastic process. The squiggly string is the continuous sample path of the process. He further argues that if the string is cut at a point P, relative to axes, one of which is tangent at P, the two segments are independent. This leaves us with a stochastic process with continuous sample paths and independent increments. This implies the process is Gaussian. The distance measured is then between two points from a bivariate normal distribution and as such twice the square of the distance is proportional to a chi-square distribution with 2 degrees of freedom.

Of course, none of this helps the poor worms!

The First Published Random Walk?

According to Alex Bellos in his new book "Here's Looking at Euclid", this is the world's first graph of a random walk, published by John Venn in his book "The Logic of Chance" in 1888. Venn produces this path using the digits from 0 to 7 from the first 707 digits of π. Venn writes,

"Of these, after omitting 8 and 9, there remain 568; the diagram represents the course traced out by following the direction of these as the clue to our path. Many of the steps have of course been taken in opposite directions twice or oftener. The result seems to me to furnish a very fair graphical indication of randomness. I have compared it with corresponding paths furnished by rows of figures taken from logarithmic tables, and in other ways, and find the results to be much the same."

Here are the directions Venn uses and the results from the first ten decimal digits.

Artificial Population Sampler

I found this brochure in an old file. It dates from 1978. It shows a physical model for teaching sampling and quantitative methods in ecology. Click on the images for larger views. Developed by Arnold M. Schultz, a Professor of Systems Ecology at the University of California, Berkeley and used as a teaching tool beginning in 1958. His 1961 patent application seen here was awarded in 1965.

The model contains two plates of plastic containing arrangements of vinyl discs of various sizes and colors that represent populations in a square region of the Cartesian plane. One plate is called the random population plate made up of six colors that seem to be distributed uniformly over the square. The other plate is called the nonrandom population plate made up of 19 colors that are arranged in rough clusters to illustrate more contiguous, local populations. Instruments are provided for measuring locations and size. Along with color they provide both qualitative and quantitative data for estimating means, variances, density, diversity from various random sampling methods. The population parameters are fixed in each of the plates allowing assessment of measurement error and bias. A paper by Schultz et al. from the Journal of Range Management (1961,v14,n5,pp 236-242) illustrates its use. I've found one listing of the color distribution (from the paper "A comparison of techniques for assessing dispersion patterns," D.W. Goodall and N.E. West, Vegetatio, (40)1:15-27, 1979) but other than a reference to it having six sizes, nothing explicit about the size distribution.

A real hands-on approach to sampling that is very useful for both teaching and research. An example in teaching is illustrated here. Its use in ecological research in 2006 is shown here.