In the previous chapters, burn-in testing has been quantified and optimized using various life distributions, such as the mixed Weibull, various bathtub models, etc. As a special case of the mixed-Weibull distribution, the mixed-exponential life distribution has its unique mathematical features. It is simple and easy to manipulate. More importantly, it always has a decreasing failure rate ( DFR) function which makes it mathematically qualified to describe the times-to-failure behavior during burn-in.
The pdf of a general mixed-exponential life distribution is given by
| (10.1) |  |
where
| n | = | total number of subpopulations, n ? 2, |
| f i( T) | = | pdf of the ith subpopulation, i = 1, 2, , n, |
| f i( T) | = | ? ie |
| ? i | = | failure rate of the ith subpopulation, i = 1, 2, , n, |
| ? i | ? | ? j, for i ? j, i = l,2, ..., n, and j = 1, 2, , n, |
| p i | = | proportion of the i th subpopulation, i = 1, 2, , n, |
and
The cdf and the reliability functions are given by
| (10.2) |  |
and
| (10.3) |  |
respectively. The failure rate function is given by
or
| (10.4) |  |
which is always a decreasing function of T, as shown next.
The first derivative of Eq. (10.4) with respect to T is
TABLE OF CONTENTS 
