Notes, Lai & Xie, 2006, stochastic ageing and dependence for reliability

page 48, mixture failure rate r(t):
 r(t) = weighted sum of subpopulation pdf / weighted sum of subpopulation survival function
 This can be found by the definition of r(t).  Failure rate is mortality rate in biology.

page 57, initial failure mixture rate r(0+):
 r(0+) = weighted sum of subpopulation r_i(0+),
shown by Block, Li, Savit, 2003, initial and final behavior of failure rate function for mixtures and systems, Journal of Applied Probability, 40, 721-740
 I am surprised that this basic formula has a 2003 citation.  This formula was in GG91 book, and they cited the Barlow, Proschan, (Hunter) book.