Tuesday, February 2, 2010

Multiple Events from a Single Source

Since an event can occur anywhere within the observing interval it is possible that multiple events will be seen from the same source. Let P be the probability of a single event being observed within the interval and P_k be the probability of observing k events. The P_k are easily calculated in nD, a space with arbitrary number of dimensions. For P_k imagine an space of k dimensions in which the time of observation of each of the k events is represented by an axis. The first observation occurs at t_1, the second at t_2 > t_1, the third at t_3 > t_2, etc. Using the continuous distribution given in terms of the probability for a single observation (P) the P_k can be represented by a multiple integration. Each time the probability that the event happens at time t_k is the probability that it doesn't happen before it times the probability that it happens in a small interval of time Δt. We end up integrating higher powers of the same funtion over and over again and can deduce a general expression for P_k.

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.