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Appendix B Technical Details on Combining r . ~ ~ ~ Information In Estimation: A Treatment of Separate Failure Modes The last section of Chapter 2 indicated an approach to a part of the following general problem: How can one use information from developmental testing to reveal the reliability of a defense system in operational testing when the failure modes of developmental and opera- tional testing may be distinct? The discussion in Chapter 2 outlined the methods for a simple special case, leaving the technical details for this ap- pendix. Consider the following problem. In developmental testing, a system exhibits two failure modes of interest, and the distance to failure (in miles) for each mode is an exponentially distributed random variate, with (un- known) failure rates Al and ~2, respectively. In operational testing, we as- sume that one of four possibilities remains: (1) all failure modes have been removed, (2, 3) either of the individual failure modes remains, or (4) both failure modes remain. We would like to use operational test data to update expert judgment concerning the probability of each of these four possible . . situations. PRIOR INFORMATION Prior to the start of operational testing, we assume that previous engi- neerins, experience. analysis of developmental test data, and redesign ac- __= ---r --------, ~_~' tions support the assessment of an a priori prior probability density func- tion (p.~.f.) for each failure rate: 85

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86 APPENDIX B fA1 (71 ) fA2 ( 2 ) In other words, the values provided by these functions are expert as- sessments of the probable values of the two failure rates. In addition, the same tests and judgments support the assignment of a priori probabilities for four possible scenarios: = probability Ino failure modes remainT = probability Tonly mode 1 remainsT P2 = probability T only mo de 2 remains T pa = 1 - To -P1-P2 = Probability Iboth mode 1 and mode 2 remainT i, For the purpose of this exposition, we assume that ~1 and ~2 are ndependent, and their priors can each be well represented by a Gamma p d f: fA1 (l ) = g(71 1 ~'[1) and fA2 (72) = g(72 1 a2'[2)~ where ~ a a-1 - [x g(X I a, A) = r(.,) In the analysis below, the following properties of the Gamma p.d.f. are applied: Moments of the Gamma p.d.f.: E(X) I,; Var(X) ~2; E(X )=- E( 1 )= ( ). '2 ~ ~ Vary)=- ~2 (a 1) (a 2) E(e-Xt ) = ( ~ ) Var(e-Xt ) = ( ~ ) - ( ~ ) (e-Xt) ]=() ; 1 ( ) (a 1)(~1 2) .

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TECHNICAL DETAILS ON COMBINING INFORMATIONINESTIMATION 87 POSTERIOR COMPUTATION Assume the system has a failure mode with distance to failure being exponentially distributed with parameter ~ (so that the number of failures in a test interval t is Poisson distributed with parameter At), and assume that ~ has prior p.~.f fade = g(1l a',t'). Assuming that testing yields n failures in t miles, then the posterior p.~.f. of ~ is g(1l a",', where the "updated" parameters are a" = a' + n, if'= b'+ t. UNCONDITIONAL DISTRIBUTION OF THE NUMBER OF FAILURES IN A TEST INTERVAL If a failure rate is A, where p.~.f. gull a,[), then n = the number of failures (for a single, unidentified mode) during an exposure of t (miles) and has the following unconditional probability mass function: 00 fN(ft)= ~ ~' e g(XIa,ID) d~=h o where h(n|a,t+~)-( a-~ It+ distribution). t a, t+[ ) (t + ~ | n = 0,1,2 (the negative binomial AN APPROXIMATION FOR THE DISTRIBUTION OF THE SUM OF TWO GAMMA DISTRIBUTED VARIATES Although there is no closed-form expression for the p.~.f. ofthe sum of two gamma distributed random variables, a reasonable approximation can be obtained. In particular, if fx1 (x)= gtx I a1,hl), fx2 (x) = gtx I a2,h2), and X; = X1 + X:, then fX3 ~ g(X I as, b3 ) ~ where ~ +a2 3 ~i ~ ~ an] a3 = [3 ( ~ + ~ ) '2 '2 ~1 ~2

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88 APPENDIX B PRIOR PERFORMANCE MEASURES Without data from operational testing, the probability assessments above enable test evaluators to evaluate typical performance measures of interest. In particular, using the information for the moments of the Gamma distribution given above, and the approximation above for the sum of two independent Gamma random variables, these can be shown to be: A a) X, an estimate of the system's total failure rate, A, and its uncer- tainty (expressed by itS standard deviation, MA = N Var(A) ), = E(A) = ~ Pi( ~ ) CT2 = ~ p ~ *( ) ] _ b) flu, an estimate of the system's mean miles to first failure, and its uncertainty CJ~ / i=l ( al1 ~ ,U ~1 1 (ai-l)(ai-2) J c) r, the expected value of the reliability of the system at t miles (e.g., the probability that there is no failure in the first t miles), and its uncer- tainty CAR. [ ( )] 3 ( ~ ) I ( b, +2M ai ^2 . r

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TECHNICAL DETAILS ON COMBINING INFORMATIONINESTIMATION 89 COMBINING PRIOR INFORMATION WITH DATA FROM OPERATIONAL TESTING . After t test miles in operational testing, if the data show n1 mode 1 failures and n2 mode 2 failures, then the performance measures can be cal- culated as above, but with the parameters "updated" so that ai and bi are replaced by a'i and <, respectively, where: a'=ai+f~i; ~=[i+t, i= 1,2 and the posterior probabilities for the failure mode scenarios are: Po =< P1 =< p, = P3 ='1 A(O, O) O A(n~, O) O else ~ /nl 1 ~ t~ 1, t + bl ) n1 > 0, n2 ' else A(O, n2 ) t2 ~ 2 102' t + b' ) n1 = 0, n2 > 0 o A(nl,n2) ( o else al, +~)h(n2 a2' +~0 n >0 else where A(O,O)=pO+plh(Oal ' b')+p2h(0a2' b')+ p3h(0al, / 2 t + b2 J A(~ ) = pl h! nl al, b' ) + P3h(~l al t + bl ) ( 2 t + b2 J , . ~ A(O, n2) = P2h(n2 a2' t + b' I + p3h A(n1,n2)=p3h~n1 I ~ t | 1 t+ b (n2 a2' b') n2 > o a1 ~ b' )h(n2 a2' b' ) n1 > 0, n2 > 0