Poisson Distribution

There are serious difficulties with the problem.

Explanation

Coliform bacteria, says the problem (due to Kalbfleisch), are randomly distributed in the river, at some average concentration or other. But is this really probable? First, A river is not a lab vessel, it is water in motion, between trees and around rocks. Who has held it still, while somebody else stirred it to mix it evenly and thus randomize it? Second, coliform bacteria are not a component of water; they are a component of soil. They include some harmless stuff, but also the Escherichia coli group (E coli, discovered in 1885 by the pediatrician Theodor Escherich), which derive from human and other warm-blooded animal fecal matter. Such material enters rivers not randomly, as solar neutrinos might, but at farm or urban runoff points. Samples taken above and below such runoff points may be expected to yield wildly different results. Third, coliform organisms themselves either thrive or die in the river depending on temperature and nutrients, and those factors are subject to rapid variation. So, not only does the river not operate to produce a uniform concentration (its rate of flow is different in midstream and along its banks), it would not be likely to maintain a given concentration once it had somehow been reached.

All this implies the need for testing at different points in the same river, and for repeating that set of tests at short intervals. The condition of homogeneity of substance, and thus constancy of rate r, is not met in this problem, despite the presence of the word "randomly." The textbook writer is not entitled to write "randomly." The problem as stated is imaginary. One cannot validly test a river in that way.

Comment

But the problem looks good in the textbook, and it gives nice numerical practice. Numerical practice counts as virtue. For the practicer, if not for the river.

For the record, and in case anyone is interested, 77 E coli bacteria per 100cc (or 15 per 20cc, 15 times the rate specified in our original problem for undifferentiated coliform bacteria) is the official Vermont danger level for swimming water (that is, for water likely to be indirectly ingested by bathing in it). Any E coli at all are considered a sign of danger in drinking water (which will be drunk straight). Vermont recommends weekly testing of swimming areas. Well, it's something.

E coli, and not coliform bacteria in general, are what the public health people really care about. They are distinguished from other anaerobic rod-shaped organisms by their ability to ferment glucose. To test their ability to ferment glucose requires 48 hours at 35 degrees C. The implication of these laboratory facts is that coliform bacteria tests can be done by eye, quickly, while no specific E coli test results will be available for action earlier than three days after the samples were collected. E coli testing is thus intrinsically retrospective. However accurately the samples may have represented the original river, that river has receded three days into the past. It is ancient history. Such are the complications which public health professionals have to think about.