I am trying to understand a paper by Aaron Clauset. How does he come up the the

Question

I am trying to understand a paper by Aaron Clauset. How does he come up the the integrals that are set equal to each other? What is the "identity from probability theory"?

Figure 5), the choice of target, or the availability of resources. If we further assume that the payoff rate on additional planning is proportional to the amount of time already invested, i.e., increasing the severity of a well-planned event is easier than for a more ad hoc event, then the potential severity of the event can be expressed p(t) ox ext, where K> 0 is a constant However, planned events are often prevented, aborted or executed prematurely, possibly as a result of intervention by a state. This process by which some events are carried out, while others are not, can be modeled as a selection mechanism Assuming that the probability of a successful execution is exponentially related to the amount of time invested in its planning, perhaps because there is a small chance at each major step of the planning process that the actors will be incarcerated or killed by the state, or will abandon their efforts, we can relate the severity of a real event to the planning time of a potential event by x e, where

Explanation / Answer

identity from probability theory-We know if A and B are two events over a region (a,b) then integration over A and B over the region (a,b) will be equal if A=B.

now if A is a function of B say B=f(A) then integration over(a,b) on A is equal to integration over (f inverse a,f inverse b) on B.

The same thing has happened here.x i.e.severity is a function of time.over a fixed time period if we want to know the function of p either by integrating with x or t over their respective ranges it's gonna be the same.i.e.integration over x on p(x) is equal to integration over f inverse t on P(f inverse t)..where f(x)=t

Thus we get the identity.

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