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Very interesting! So, the basic point being made is that if you know that a set of events are random and independent and you know their mean value, then you can predict their spread? (or aggregation)

edit: Hmm...another question that comes to mind: is the converse true? If the spread of values of these events do not match the poisson distribution, then can we presume them to be nonrandom? Or nonindependent? Or both?

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The Poisson distribution is just one random number distribution; there are several others for situations where the events are correlated, or have other properties. Half the fun of probability is figuring out which distribution is the right one to apply to the question at hand. So if your measurements don't match up to Poisson, it doesn't mean they're not random - they could just be interdependent.

So yes, for a Poisson process, the spread (standard deviation) is equal to the square root of the mean; as the number of events gets large, the Poisson distribution approaches the normal distribution, but the relationship between the standard deviation and the mean continues to hold.

If you have enough data and to determine the distribution and it's characteristics (eg mean and s.d. for normal etc) then you can use it to predict spread.

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