Yes, you described a case ("homogeneous P.P.") where the intensity, lambda, is constant over time. In the inhomogeneous P.P., there is a time-varying rate, lam

Yes, you described a case ("homogeneous P.P.") where the intensity, lambda, is constant over time. In the inhomogeneous P.P., there is a time-varying rate, lambda(t). The derivation of the properties in the inhomogeneous case is almost the same because the key property is independent increments. When you add up the number of catastrophic events over an 11-year cycle, you get a sequence of "numbers of catastrophic CMEs" N1, N2, ..., Nk each of which is Poisson with intensity \int_{cycle} lambda(t) dt [*] If you ignore the larger-than-a-cycle variations in intensity, this integral is the same for each cycle. So the N's above are iid Poisson with some common parameter lambda_0, say. Observing the N's for many cycles would allow you to estimate lambda_0. That's pretty much where the 1 in 15 comes from (indeed, lambda_0 is so low that there are only 0 or 1 events per cycle, so "estimation" is trivial). The "easy" problem is that, even averaging over a cycle, the activity ([*] above) is not constant, e.g. this graph of sunspot numbers vs. time: http://blogs.nicholas.duke.edu/thegreengrok/files/2012/07/sunspotnumbers.jpg http://blogs.nicholas.duke.edu/thegreengrok/files/2012/07/su... The "hard" problem is that the counts are not, in real life, independent, even across weeks-to-months time scales. This would invalidate the Poisson model at the ground level (see the talk referenced by @gone35 above). There are reasons (e.g., Maunder minimum, see the plot above) to believe that the counts are also not independent at the decadal time scale. Lack of independence would be a huge problem for any Poisson model. You'd have to go up to a doubly stochastic process, for example, in which lambda(t) is itself a random process.