The "random" sample is obviously very, very skewed: If you would be asked to name a random conjecture, it probably will be a "famous" conjecture, and the longer a conjecture stands, the more famous it tends to be.
But that is not my question. Has anybody tried yet a reliable statistic on truely "random" conjectures and the time they need for solving? One would need to pick random math manuscripts (already a challenge - how to avoid inherent skewing?), check whether they contain some open conjectures and check when they were solved later on. This means work, work, work (you would need some expertise in all part of math - or "outsource" the job).
My conjecture :-) is that conjecture longevity follows a Poisson statistic, and the open parameter is rather of the order years than centuries.

Addendum: The entirely non-random example :-) by Kuperberg about the G2 knot polynomial (hey, I just like the paper :-) lists six questions "open to the author" (incidentally showing that the longevity might also be zero, in the case that another mathematician already proved your question - happens all the time - I think such cases should better be excluded from the statistic). Judging from the many papers I DLed "0" is a much more probable number for open conjectures in a paper, though. (Needs a statistic, too.)

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    $\begingroup$ This reminds me of "Some Conjectures of Graffiti.pc." Graffiti is a computer program that generated hundreds of conjectures in graph theory. $\endgroup$ – Joseph O'Rourke Dec 25 '14 at 12:57
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    $\begingroup$ Relevant article: link.springer.com/article/10.1007/s00283-013-9383-7 $\endgroup$ – Dag Oskar Madsen Dec 25 '14 at 13:21
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    $\begingroup$ Joseph - Indeed, a sufficient savvy computer program can generate conjectures faster than any mathematician can prove. (In fact I can do that, and I'm not sufficient savvy :-) Dag Oskar: And while I was writing the question, I thought: "This would be a theme for AMM or the Intelligencer". 1 year? Good grief, where's my memory :-) I'll look the article up. Again. Alexander: While this is a limited area (and possibly not representative - but who knows), have an uppie for your efforts. I was thinking of exactly such a progress report. $\endgroup$ – Hauke Reddmann Dec 26 '14 at 16:25

In some areas of mathematics, there are published lists of unsolved problems. Sometimes, progress surveys on these problems are published later.

One example I am familiar with is "Hayman's collection" in classical Function theory. It started with a book by Hayman, Unsolved problems in Function theory, Athlone press, London 1967, which was continued in 6 papers containing new problems and progress in old problems. The problems were proposed by participants of various conferences and edited by Hayman, Barth, Brannan, Campbell and others. These lists were quite popular among the late 20-th century classical complex analysis.

The latest progress report is available here: http://www.math.purdue.edu/~eremenko/dvi/progr.pdf

It contains the references on the old progress reports. Altogether these lists contain several hundred problems, which can be used for some statistics, I suppose.

At this time, I have no access to all these papers because I am traveling but when I come home I can make some statistics.

I know of similar lists in other areas, but the main problem is that the authors of this lists usually do not care publishing progress reports, and it is difficult to determine which problems are solved and when.

My experience with these lists shows that there are no reasons to expect any kind of stochastic stability or any universal distribution. I mean the distribution of solved problems in time will strongly depend on the area. But this a superficial impression, not conformed with any numerical data.

Remark. I have my own list of unsolved problems (I am not the author of most problem, I collected them from other people). The list is availkable on the web, and I give links to solutions whenever I learn that a problem is solved:



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