This is a general question. I have a task that is composed of 3 independent events: A, B, and C. All are mutually exclusive and don't happen at the same time. So first A, then B, then C. I know the probability of completing each event with respect to time, I have the pdfs. How can I calculate the probability of completing the task at hand with respect to time?


First, a minor terminology correction: You can't really have independent and mutually exclusive events, as mutually exclusive implies that if one event happens, the others cannot happen, which makes them not independent. I think what you mean is that they are sequential, in that they happen one after another.

Since you have the pdf of each, what you want to calculate is the sum of the three random times: $T = T_A+T_B+T_C$. Depending on the exact form of your pdfs, you can try one of three approaches:

  1. Recognize the sum as being equal to a particular distribution (e.g., sum of normals is normal)
  2. Simulate the sum using Monte Carlo simulation
  3. Analytically calculate the distribution of the sum by either multiplying thecharacteristic functions of each pdf and then back-transforming to a denstity, or by directly performing two convolutions: $f_C*(f_A*f_B)$

I would recommend method 2 if you have access to a monte carlo simulator or know R or Matlab or any other numerical package. 1 is also OK...3 is a real pain unless you are lucky to be using simple distributions...in which case you will likely find a solution as per 1.


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