Mean Value of Weibull Distribution

[Steve]

[Originally Posted: 5/17/00-- Transferred by ReliaSoft Moderator]

I begin with a Weibull Distribution. Next I do some calculations and determine that I'm no longer interested in values below a certain minimum, essentially creating a new min. How do I calculate the mean of what's left of the Weibull Distr?

The total probability is no longer 1, so the function should be weighted somehow, but I'm at a loss.

Thanks, Steve

[Dr. Dave Olwell]

[Originally Posted: 5/23/00-- Transferred by ReliaSoft Moderator]

There are two ways to think about this problem.

First, you can think that you have a Weibull pdf, but you have sliced off a portion of the left side - corresponding to a minimum. You assume that the remaining allowed values have the same pdf shape, but now the remaining doesn't integrate to one. To get an appropriate weighting constant, you integrate the remaining part of the pdf and get a constant (less than one). Dividing the old pdf by that constant will result in a new pdf that now does integrate to one.

Of course, it will no longer be a Weibull pdf but rather a "truncated" Weibull. So all your metrics (mean, BX, etc) will need to be calculated by hand.

In particular, the mean will be the integral from the min value to infinity of t times the new pdf, integrated with respect to time.

The second approach is to introduce a 3 parameter Weibull, with gamma preset as your minimum value or estimated from the data. Then you fit the model to the data using Weibull++, and get the mean from the QCP. This is not a truncated Weibull, but a translated Weibull model: we have shifted by gamma.

I'd need to know more about your specific application to know which approach is best for you.

Best wishes,

Dr. Dave