# How do I analyze a case of breaking a cable at any point given that the breakage area length is zero?

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?

• 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?
– whuber
Aug 28 '13 at 15:27

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 $x$.