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Suppose I have a long network cable buried into ground and construction workers will sometimes unintentionally damage it when doing excavation works. I need to analyze the probability of breakage assuming it happens randomly. If I try to address the probability of breaking the cable at any specific point then it will be zero because the "length of a point" is zero so it's kind of impossible to unintentionally damage the cable at a specific point.

So I have to divide the cable into sections for the analysis so that I then have sections of non-zero lengths and so non-zero probabilities of damage.

How do I choose lengths of sections and what other ways to address the probability calculation are there that deal with "length of breakage point" being zero?

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    $\begingroup$ You appear to confuse probabilities with probability densities: see stats.stackexchange.com/questions/4220/… (inter alia) for a discussion. Also, what are you really doing in your analysis? Do you have data concerning locations of breakage? If so, those data are likely inconsistent with a hypothesis of random breakage (and testing that hypothesis would not be informative). Wouldn't you want instead to estimate the probability distribution of breaks along the cable? $\endgroup$
    – whuber ♦
    Aug 28, 2013 at 15:27

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Surely you need only assess the probability of the cable being broken at any point. From there you can subdivide the length and the probability in parallel to get the probability for a specific subdivision. You would also need to consider the possibility of there being more than one breakage, but the same principles would apply. For overall breakage rates, there may be industry statistics that indicate how frequently it occurs in particular types of terrain and population density.

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  • $\begingroup$ As the OP has pointed out, the probability of the cable being broken at any specific point is zero. How do you proceed from there? $\endgroup$
    – whuber ♦
    Sep 1, 2013 at 15:54
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    $\begingroup$ Sorry I didn't express that very well, when I said "at any point" I meant along its entirety. $\endgroup$
    – Robert Jones
    Oct 24, 2013 at 9:38
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Dividing the cable in sections is not the only possibility.

A natural approach for this problem is through a stochastic point process indexed by the abscissa $x$ along the cable's path which can be used as a 'time'. The Poisson Process is a good candidate: the breakage rate is given by the the intensity (or rate) $\lambda(x)$ of the process, which here is the probability of break by unit of cable length around $x$. This intensity can be constant or piecewise constant and can be related then to some area predictors as cited by Robert Jones. It can also be related if needed to continuous predictors such as the cable depth. For a fixed period of time, say a year, a PP can describe the locations of the breaks. The intensity function $\lambda(x)$ is not unlike a probability density, and will have to be integrated with respect to $x$ for most computations.

A more sophisticated version would use a spatio-temporal point process with intensity $\lambda(x,\,t)$ related to the time $t$ and to the abscissa $x$.

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