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!