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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?

user2153235
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  • The paper is terribly written, so I can only guess that they appear to be truncating the distribution to the interval $t\in[0,1].$ – whuber Feb 12 '21 at 17:07
  • 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,... – user2153235 Feb 12 '21 at 17:11
  • 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] $. – user2153235 Feb 13 '21 at 04:10

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