CONTEXT OF WEIBULL: The Weibull variable is used in reliability test, i.e. life of a system. The value recorded is generally an independent and "nominal" in a sense that each arrival value (run time) is specific for a particular reading. The first reading and the second reading could not be added because the objective is to verify the system that produced these number. This could be demonstrated by series of readings:

X1: 23

X2: 20

X3: 25

X4: 24

X6: 25

These Xi must be ranked into ascending order X(i): (20, 23, 24, 25, 25). We see right away why Xi could not be summed because Xi are identifier of independent events generated by a system. The objective of Weibull is to use the data as the basis to generate a set of "quantitative values" that could be subjected to mathematical operations. Since we put Xi into X(i), their arrival time now is all mixed. The first step is to turn the newly ranked set into a plottable series by:

(1)   F(t) = (X(i) - 0.03) / (n + 0.40)        where n = sample size < 100

This is known as Weibull plot position. The next step is to generate X and Y coordinate from F(t), thus:

(2)   X = ln(ln (1 / 1 - F(t)))       and its corresponding Y, thus

(3)   Y = ln(X(i))

It is at this step where we have X and Y that a the value could be subjected to mathematical operations: +, - , x and %. thus, in Weibull distribution analysis, the X becomes the independent variable and Y as dependent variable. At this point X and Y are Weibull values, they could be summed. In fact, X and Y will be subjected to OLS in order to obtained Weibull linear regression equation in a form:

(4)   Ywei = a + bX

From this point one can calculate Weibull parameters: shape and location.

THE STATEMENT: The statement that "no results (not even approximations) have been known for sums of Weibull random variables" is only true for the first half concerning the "raw data" of unranked Xi. However, after the ranked X(i) have been fitted into the Weibull plot and X and Y generated, the data now is quantified and ready for any form of mathematical operation. It the statement intended to say that each Weibull random variable is an independently generate set with its own PDF and CDF, scale and location, etc,; therefore, could not be summed because each should be treated independently. This is :true" but it is superfluous because by its nature Weibull is to test one set of fact at a time. Each set of data is taken as an independent reading. For that reason, they cannot be added because the reliability of one reading is generated by a system for that particular time period. Taking a second reading, a second set of data is collected which is independent and different different from the first; therefore, both should be treated independently. Weibull distribution came out of the context of machine reliability testing at Boeing (an airplane producer). One reading of reliability of one air plane cannot be added to the reading of a second airplane. Each airplane must be evaluated on its own. That would be the logic of that quotation. Even several reading from one airplane should not be added if the condition under which the plane (system) running was different.

REFERENCES: See attached lab sheet with step-by-step instruction for exercise. Other articles are also attached---including the original Weibull from Boeing. I hope this is helpful in giving clarity to some points of your query. Cheers.

Thank you so much Paul!

Why would you want to do this?  If, strictly,  all the requirements are met to allow you to sum two random Weibull variables, it means that you don't have two independent random variables.  Read very carefully the excellent answer by Paul Louangrath.

Can anyone please explain the sum of large number of Weibull random variables using CL theorem? I mean the mean and variance of the resulted Gaussian random variable??