It is easily seen that the probability distribution for multiple observations is the Poisson distribution.

The P_k can be used to determine the expected number of observations from a single source. The number will range from 0 to e, 2.71828... .

The expected number of events is an increasing function of P. It is not a linear function of the probability. So it is possible that more than one earthquake will happen on the same fault in a given interval if they occur at random. Like lightning don't count on it not happening in the same place twice.

edit (3 Feb): The regions of integration for P_k are "triangular" sections of kD polytopes which are extentions of the square and cube. The P_k are the extentions of the area of a triangle (1/2 x altitude x base) and the volume of the triangular pyramid (1/3 x altitude x base), i. e., 1/k x altitude x base in kD. Each base is previous P_k. The origin of the axes and the points on the the axes at P(ΔT) mark the boundaries of an equivalent section and all the points on the lines joining these extreme points are within the equivalent region of integration.

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