Poisson Distribution

Yes.

Explanation

There are some things that the Poisson will not do for us, and one of them is interpet the significance of a given departure from an expect number of occurrences. That sort of evaluation needs to be made by other means. What the Poisson will do for us is to show us the probabilities of all the variant numbers of occurrences. Then it lets us make up our own mind. We here attempt to follow that process.

The general rate r is 1 per 10,000, and for Reno College this translates into 2.5 per 25,000. Here is the table for r = 2.5. If causation is indeed random, the variability in the observed rate will be:

 r = 2.5 p(0) 0.0821 p(1) 0.2052 p(2) 0.2565 p(3) 0.2138 p(4) 0.1336 p(5) 0.0668 p(6) 0.0278 p(7) 0.0099 p(8) 0.0031 p(9) 0.0009 p(10) 0.0002

It is intuitively obvious that we cannot have exactly 2.5 suicides, and instead we might expect 2 or 3 as the normal or unexceptional value. This turns out to be not exactly right. What the table tells us is that there are actually three almost equal and therefore unexceptional values: 1 (21%), 2 (26%), or 3 (again 21%). These three options together exhaust 67%, or two-thirds of the whole frequency profile. There is a considerable step down to 4 suicides (13%, or slightly less than 1 chance in 8). Everything else (0, 5 or anything higher than 5) is much less likely.

Intuitively, then, if Reno records 1, 2, or 3 suicides, they may reasonably be attributed to general causes that show up in other institutions as well. 4 suicides are marginal; 5 or higher would seem to suggest a definite local problem. How to deal with the transitional result 4? This problematic value is exactly what the problem asks about. We may here recall that cultural factors will probably come into play. Student suicide is a high-tension event. One suicide can disarrange a campus community for days; four suicides on the same campus will have a very strong collective impact. Fellow students get involved; the respective parents get involved. For the Dean of Students at Reno, then, the 4 result, though perhaps technically ambiguous as to local responsibility, would probably be reason for investigating those cases, to see if they provided any clues that might require local corrective action. In fact, given the high visibility of even 1 suicide, the Dean will be investigating every single suicide in detail anyway.

To that extent, the present problem is superfluous. Any action it suggests is going to be taken anyway, and any lack of action which it seems to authorize is going to be overridden by public demand for action. Still, it is interesting to see how the numbers work out. They work out as shown above.

Comment

Poisson events must be independent. Are student suicides independent? There would seem to be a high likelihood of dependent behavior, if only in the sense that one student suicide will tend to create suicide as a conceivable option in the minds of many other students; individual stresses can then more easily find their way to that outcome. But there are further possibilities as well. Dual suicides (as of two lovers who cannot marry because of family disapproval) are a virtual topos, a standard event, in Japanese traditional culture as expressed in Japanese traditional literature. Here we have the strongest possible connection between two events; an anthropologist gathering data in this area might well count such a "love suicide" as one person rather than two. In fact, the point of a "love suicide" (and there are Chinese poems on such cases going back to the 3rd century) is to become, despite social disapproval, one person rather than two.

Similar objections to the validity of such a problem would remain if it were transposed from "suicides" to "instances of a particular disease." The medical literature is rich in research complications created by lack of independence among the instances studied. Epidemiologically, five cases of some disease at a given nursing home might plausibly count as one event rather than as five events. Similar questions plague the study of the typology of languages.

Update 2007. For the record, an extensive 1997 study showed that the student suicide rate (which in that year was 7.5 per 100,000, not yet the 2003 rate reported above) was half that of the suicide rate of nonstudents in the same age group (15 per 100,000). More effective prohibition of firearms on campus is thought to have accounted for a sizeable part of the difference. Student suicide, which had long been common in Japan, steadily increased in America during the second half of the 20th century. Suicide in Asian traditional cultures is strongly supported by a sense of honor which goes back into the classical period of those cultures. For a 2003 account of the University of Illinois suicide prevention program, click here. It brings further factors into play.