Predicting Damages to System Elements: A Binomial or Poisson Approach?

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In summary, the conversation discusses the distribution of damages in a system comprised of N=10^10 elements, with a hypothesis that it can be approximated by a poisson distribution. However, the system under study has a non-uniform probability of damage for each element, making it difficult to predict the distribution. The possibility of multiple damages in a single time step is also considered.
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I'm studying a particular system but to ease the understanding let me take a general approach.

1) Suppose I have a system comprised of N=10^10 elements. Assuming now that I deal a damage to a random element for every timestep, I want to figure out how the damages are distributed in the elements. My hypothesis is that this binomial distribution can be approximated very well by an easier poisson distribution.
And this brings me to my first question: Is this a correct hypothesis?

2) Next: Now the above does not really have anything to do with the system I study. Because in my system it is so that the probability of dealing damage to an element is not uniformly distributed but rather something specific to each element (normally distributed).
I want to be able to predict how the damages will be distributed amongst my elements, like in 1) but this time I'm not sure how to approach the problem. Can I predict a distribution and how would I do that?
 
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The system is under-specified, even for (1).

If the "damage" is delivered externally (by a hail of bullets, say) then there is a chance that an already damaged element will receive another hit ... i.e. the probability of getting hit does not change with damage, so the more damaged elements there are the less likely a new element will get damaged and so the damage rate slows down. OR - perhaps damaged elements get removed from consideration?

It sounds like only one element can be damaged in a single time step - i.e. unlike, say, radioactive decay where each element has a change of changing state in a given time step so there can be more than one decay per unit time. Or are you just modelling the system on a timescale sufficiently tiny that the probability of more than one decay in one time-step is vanishingly small?

In the case of radioactive decay, where each element has a characteristic probability of changing state from "undamaged" to "damaged", the poisson approximation works quite well.
 

1. What is the purpose of predicting damages to system elements using a binomial or Poisson approach?

The purpose of predicting damages to system elements using a binomial or Poisson approach is to forecast the likelihood and severity of potential damages to a system. This information can be used to identify potential problem areas and implement preventive measures to minimize the damages.

2. What is a binomial approach and how does it differ from a Poisson approach?

A binomial approach is a statistical method that involves calculating the probability of a certain number of successes in a fixed number of trials. It assumes that each trial is independent and has only two possible outcomes. A Poisson approach, on the other hand, is based on the assumption that the number of occurrences of an event in a fixed time interval follows a Poisson distribution, which is a probability distribution that describes the likelihood of a certain number of events occurring in a given time frame.

3. What factors should be considered when using a binomial or Poisson approach to predict damages to system elements?

When using a binomial or Poisson approach to predict damages, it is important to consider factors such as the type of system being analyzed, the frequency and severity of potential damages, and the accuracy and reliability of the data used in the calculations. Other factors, such as environmental conditions and human error, may also play a role in the prediction of damages.

4. How can the results of a binomial or Poisson approach be used to inform decision-making?

The results of a binomial or Poisson approach can be used to inform decision-making by providing insights into the potential risks and vulnerabilities of a system. This information can help decision-makers prioritize resources and implement preventative measures to mitigate damages and ensure the smooth operation of the system.

5. Are there any limitations to using a binomial or Poisson approach for predicting damages to system elements?

Yes, there are limitations to using a binomial or Poisson approach for predicting damages. These methods rely on certain assumptions and may not accurately predict damages in complex systems with multiple variables. Additionally, the quality of the results depends on the accuracy and reliability of the data used in the calculations.

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