Re: Adding two Weibull distributions

From: Bill Rowe (readnewscix_at_earthlink.net.invalid)
Date: 10/20/04 

Date: Wed, 20 Oct 2004 03:00:28 GMT

In article <da11b7ab.0410191718.38c92766@posting.google.com>,
 prasad1@gmail.com (Ramchandrapuram) wrote:

> I understand that the mean and variance of sum of two independent
> normal distributions is the sum of their means and variances.
> Similarly for a weibull distribution, is there is a closed form
> solution for sum of two independent weibull distributions.

> Specifically, I have 2 independent weibull distributions, X and Y,
> where X is the distribution of test scores for Math1 exam and Y is the
> distribution of test scores for Math2 exam. The scale and shape
> parameters for X are 35 and 1.75 and for Y are 20 and 3.10
> respectively. A combined score of less than 40 is considered a
> failure. I want to estimate the probability of students failing the
> Math test (combining Math1 and Math2 test).

I really do not think having any procedure for finding the distribution
of the sum of two variables from a Weibull distribution will solve your
stated problem. In fact, this is virtually certain to give an invalid
answer to your stated problem.

The problem is the standard techniques for finding the distribution of
the sums of variables assume the variables are uncorrelated. A student's
score on one test is quite unlikely to be uncorrelated with his score on
another test in the same subject.

I think your best bet would be to fit a distribution to data that
reflects the combined scores of students taking both tests in the past.
If no students have actually taken both tests, I doubt there is any
procedure that will result in accurate estimates without making
assumptions about the correlation of scores on both tests. And quite
likely, the prediction will be dominated by the assumptions made about
the correlation.

One other comment, unless you have a fairly large sample of data (more
than say 100 students) you are probably better of using discrete
distributions than fitting a continuous distribution to the data. And if
you do have enough data to warrant fitting a continuous distribution,
you would probably be better off using a beta distribution for data that
is clearly bounded at both ends and likely to have values near both
bounds. 

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