I am reading a paper which normalizes a Weibull distribution (equation 2):

$$ y_{ij} = \frac{ 1 - \exp \left[ - \left( \frac{t_j}{\beta} \right)^\alpha \right] }{ 1 - \exp \left[ - \left( \frac{1}{\beta} \right)^\alpha \right] } $$

where $y_{ij}$ is cumulative expenditure in project $i$ at time $t_j$.

I get that this is needed because of the infinite tail, but what is the thinking behind how the denominator is chosen? Is this standard practice? Where can I find (say) an online textbook describing this?

  • $\begingroup$ The paper is terribly written, so I can only guess that they appear to be truncating the distribution to the interval $t\in[0,1].$ $\endgroup$
    – whuber
    Feb 12, 2021 at 17:07
  • $\begingroup$ I would have to agree that this is what they are doing, but I'm puzzled by why choose t=1 for the point of normalization. In a subsequent paper that this author co-authored, the same normalization is used but t=1,2,... $\endgroup$ Feb 12, 2021 at 17:11
  • $\begingroup$ Given that $y_{ij}$ is the proportion of total project budget expended at time $t_j$, it has to reach 1.00 at some final time $t_d$ in the project schedule. Therefore, I believe that the denominator can only be $ 1 - \exp [ - ( t_d / \beta ) ^\alpha] $. $\endgroup$ Feb 13, 2021 at 4:10


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.