6.1 WEIBULL DISTRIBUTION CHARACTERISTICS

The Weibull distribution is one of the most commonly used distributions in reliability engineering because of the many shapes it attains for various values of ?. The Weibull pdf is given by

(6.1)

where

f( T) ? 0, T ? ?, ? > 0, ? > 0, - ? < ? < ?,

? = shape parameter,

? = scale parameter,

and

? = location parameter.

Figure 6.1 shows how the shape of the Weibull pdf changes as ? changes from ? = 1/5 to ? = 1/2, ? = 1, ? = 1-1/2, ? = 3, and ? = 5. Some of the specific characteristics of the Weibull pdf are the following:

1. For 0 < ? < 1 as T ? ? then f( T) ? ?, as T ? ? then f( T) ? 0. f( T) decreases monotonically and is convex as T increases beyond the value of ?, as may be seen in Fig. 6.1. The mode is nonexistent [1, p. 33].

2. For ? = 1 it becomes the two-parameter exponential distribution, as a special case, or

   (6.2)

   which is illustrated in Fig. 6.2. Equation (6.2) may also be written as

   (6.3)

   where

   
   

   and

3. T = m = ? + ?.

   f