ESTIMATION OF SYSTEM RELIABILITY FOR LOG – LOGISTIC DISTRIBUTION

Jamonline / 2(2); 2012 / 182–190 Srinivasa Rao G

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Research Article

Journal of Atoms and Molecules An International Online JournalAn International Online JournalAn International Online JournalAn International Online Journal ISSN ISSN ISSN ISSN –––– 2277 2277 2277 2277 –––– 1247124712471247

ESTIMATION OF SYSTEM RELIABILITY FOR LOG – LOGISTIC DISTRIBUTION

G. Srinivasa Rao

Department of Statistics, School of Mathematics and Computer Sciences, Dilla University, Dilla, PO Box 419, Ethiopia..

Received on: 16-03-2012 Revised on: 10-04-2012 Accepted on: 19–04–2012

Abstract:

A multicomponent system of k components having strengths following k- independently and

identically distributed random variables 1 2 ky , y ,....,y and each component experiencing a random

stress X is considered. The system is regarded as alive only if at least s out of k (s<k) strengths

exceed the stress. The reliability of such a system is obtained when strength, stress variates are

given by log-logistic distribution with different scale parameters. The reliability is estimated using

Moment Method, ML method, Modified ML methods and best linear unbiased method of

estimation in samples drawn from strength and stress distributions. The reliability estimators are

compared asymptotically. The small sample comparison of the reliability estimates is made through

Monte Carlo simulation.

Key Words: System reliability, stress-strength, moment method, ML method, modified ML

method, best linear unbiased estimator.

Introduction:

Let X, Y be two random variables

representing stress and strength respectively.

The survival probability or reliability of the

system is defined as p(x<y). If F (.), f (.), G

(.), g (.) are respectively the cumulative

distribution function (c.d.f.) and probability

density function (p.d.f.) of X and Y then we

know that

* Corresponding author

Srinivasa Rao G,

Email: [email protected]

Tel: 00251 – 923986253



0

p (x < y )= r= F (y )g (y ) d y∝

∫ (1.1)

The R.H.S. of (1.1) is called the survival

probability of a single component stress-

strength system. Let a system consist of k

components whose strengths are given by

i.i.d. random variables with c.d.f. G (.) each

experiencing a random stress X with c.d.f. F

(.). Suppose the system survives only if the

strengths of at least s of the k (s≤ k)

components denoted by the random variables

1 2 ky , y ,....., y exceed the stress X. Then the

reliability of such a system is given by

ki k - i

i= s

R = r (1 )k

ri

−

∑ (1.2)

Where r is given in equation (1.1). The

probability in (1.2) is called reliability in a

multicomponent stress-strength model

(Bhattacharyya and Johnson). Several authors

have taken up the problem of estimating

reliability in stress-strength relationships

assuming various lifetime distributions for the

stress-strength random variates.

Enis and Geisser (1971), Bhattacharyya and

Johnson (1974), Pandey and Borhan Uddin

(1985), Awad and Gharraf (1986), Nandi and

Aich (1994) and the references therein cover

the study of estimating p (x<y) in many

standard distributions like exponential,

gamma, Weibull, Burr, inverse Gaussian, half

normal, half cauchy, half logistic assigned to

one or both of stress, strength variates.

The estimation of survival probability in a

multi component stress-strength system when

the stress, strength variates are following log-

logistic distribution is not paid much

attention. Therefore, an attempt is made here

to study the estimation of reliability in s out of

k stress-strength model with reference to log-

logistic probability distribution and the

findings are presented in Sections 2 and 3.

Estimation of System Reliability

The probability density function (p.d.f) and

cumulative distribution function (c.d.f) in

standard form of log-logistic distribution with

shape parameter β as suggested by

Balakrishnan et al (1987) are given by

1

2( ) ; x 0, >1

(1 )

xh x

x

β

ββ β

−

= ≥+

(2.1)

( ) ; x 0, >1 1

xH x

x

β

β β= ≥+

(2.2)

If a scale parameter σ is introduced we get

probability density function (p.d.f) and

cumulative distribution function (c.d.f.) of

scaled log-logistic distribution as follows.

1

2

( / )( ) ; x 0, >0, >1

1 ( / )

xx

x

β

β

β σφ σ βσ σ

−

= ≥ +

(2.3)

( / )( ) ; x 0, >0, >1

[1 ( / ) ]

xx

x

β

βσ σ β

σΦ = ≥

+ (2.4)

Let X, Y be two independent random

variables following log-logistic distribution

with scale parameters 1 2,σ σ respectively.



Then the p.d.f.’s and c.d.f.’s of X and Y are

given by

11

1 121 1

( / )( ; ) ; x 0, >0, >1

1 ( / )

xf x

x

β

β

σβσ σ βσ σ

−

= ≥ +

(2.5)

11 1

1

( / )( ; ) ; x 0, >0, >1

[1 ( / ) ]

xF x

x

β

βσσ σ β

σ= ≥

+ (2.6)

12

2 222 2

( / )( ; ) ; y 0, >0, >1

1 ( / )

yg y

y

β

β

σβσ σ βσ σ

−

= ≥ +

(2.7)

22 2

2

( / )( ; ) ; y 0, >0, >1

[1 ( / ) ]

yG y

y

β

βσσ σ β

σ= ≥

+ (2.8)

If X, Y represents the stress and strength

variates of a component respectively, then we

know that the survival probability of the

component is given by

0

r = p ( X < Y ) = F ( y ) g ( y ) d y∝

∫

2

0

1

(1 )(1 )dz

Z Zβλ

∝

=+ +∫ (2.9)

Where 1 1 2Z=x/ and = /σ λ σ σ

From equation (1.2), the survival probability of

multi component stress-strength model can be

obtained, where r is given by equation (2.9).

If 1 2,σ σ are not known, it is necessary to

estimate 1 2,σ σ to estimate R. In this paper we

estimate 1 2,σ σ by ML method and three

different modifications to ML method,

Method of moment, BLUEs thus giving rise

to six estimates. The estimates are substituted

in λ to get an estimate of R using equation

(1.2). The theory of methods of estimation is

explained below.

It is well known that the method of maximum

likelihood estimation (MLE) has invariance

property. When the method of estimation of

parameter is changed from ML to any other

traditional method, this invariance principle

does not hold good to estimate the parametric

function. However, such an adoption of

invariance property for other optimal

estimators of the parameters to estimate a

parametric function is attempted in different

situations by different authors. Travadi and

Ratani (1990), Kantam and Rao (2002) and

the references therein are a few such

instances. In this direction, we have proposed

some estimators for the reliability of multi

component stress-strength model by

considering the estimators of the parameters

of stress, strength distributions by standard

methods of estimation in log- logistic

distribution.

Method of Maximum Likelihood

Estimation (MLE):

Let 1 2 m 1 2 nx x <....<x ; y <y .... y< < < be two

ordered random samples of size m, n

respectively on stress, strength variates each

following log-logistic distribution with scale

parameters 1 2,σ σ and shape parameter β .

Then the MLEs of 1 2,σ σ are given by the



iterative solutions of the following likelihood

equations.

m

i 1i=11

log L m0 F(x , ) 0

2σ

σ∂ = ⇒ − =

∂ ∑ (2.10)

n

i 2i=12

log L n0 G(y , ) 0

2σ

σ∂ = ⇒ − =

∂ ∑ (2.11)

Where i 1 i 2F(x , ) and G(y , )σ σ are given in

(2.6), (2.8).

The asymptotic variance of the MLE is given by

12 2 2 2( log L/ ) (3/ )( /n) ;i=1,2i iE σ β σ−

− ∂ ∂ = (2.12)

The MLE of survival probability of multi

component stress-strength model is given by

( )ki k -i

ii= s

R = k ( r ) [1 ( r ) ]∧ ∧ ∧

−∑ (2.13)

Where r∧ is given by (2.9) with λ is replaced

by 1 2/λ σ σ∧ ∧ ∧

= .

It can be seen that equations (2.10) and (2.11)

can not be solved analytically for 1 2,σ σ .

Therefore, Rao (2001) approximated F (.) by

a linear function say

i i i i iF(u ) u and G (v ) vi i iγ δ µ ϑ≅ + ≅ + Where i i 1 i i 2u x / and v y /σ σ= = and

, , ,i i i iγ δ µ ϑ are to be suitably found. Here we

present three methods of finding , , ,i i i iγ δ µ ϑ

after using these approximations in equations

(2.10) and (2.11), solutions for 1 2,σ σ are

given by

i i1 1

1 2m n

i ii= 1 i= 1

2 x 2 an d

m -2 n -2

m n

i ii i

yδ ϑσ σ

γ µ

⊕ ⊕= == =∑ ∑

∑ ∑

The above estimators are named MMLE of

1 2,σ σ which are linear estimators in

i ix 's and y 's. Hence its variance can be

computed using the variances and covariances

of standard order statistics provided we have

the values of , , ,i i i iγ δ µ ϑ . We now present the

proposed three methods to get , , ,i i i iγ δ µ ϑ .

The basic work for our three methods is from

Tikku (1967), Balakrishnan (1992), Tiku and

suresh (1992) respectively.

Method I is based on linearisation of certain

portion of log likelihood equation on lines of

Tiku (1967), Method II is based on

linearization through Taylor’s series

expansion of certain terms of likelihood

equation around population quantiles, Method

III is on Taylor’s series expansion of certain

terms of log likelihood equation around

expected values of sample order statistics. All

these methods would result in linear

estimators requiring certain constants such as

, , ,i i i iγ δ µ ϑ tabulated by Rao (2001) for n=

1(1) 10, 3(1)6β = . Borrowing these tabulated

values of MMLEs we can find estimates of

1 2,σ σ and have estimate of λ and hence

estimate of r. Thus for a given pair of samples

on stress, strength variates we get three

estimates of r by the above three different



methods as$ $$ $$$r , r , r . Thus three MMLEs of

survival probability of multi component

stress-strength model is given by

$ $$ $$$R ,R ,R where λ is replaced

$ $ $1 2/ ,λ σ σ= $$ $$ $$

1 2/ ,λ σ σ=

$$$ $$$ $$$1 2/λ σ σ= .

Method of moment estimation (MOM):

We know that, if x ,y are the sample mean of

samples on stress, strength variates then

moment estimators of 1 2,σ σ are

#1 x/ (1 1/ ) (1 1/ )σ β β= Γ + Γ − and

#2 y/ (1 1 / ) (1 1 / )σ β β= Γ + Γ − respectively.

The fifth estimator, we propose here is

# # #R (r ) (1 r )k k

i k i

i s

i −

=

= −

∑ where #r is given

by (2.9) with λ is replaced by # # #1 2/λ σ σ= .

Best linear unbiased estimation (BLUE)

Rao (2001) has developed the coefficients to

get the BLUEs of σ in a scaled log-logistic

distribution. Hence the BLUEs of 1 2,σ σ are

* *1 2 i i

1 1

l x and c ym n

i ii i

σ σ= =

= =∑ ∑

Where 1 2 m 1 2 n(l ,l ,...,l ) and (c ,c ,....,c ) are to be

borrowed from Rao (2001).

The sixth estimator that we propose be

* * *R (r ) (1 r )k k

i k i

i s

i −

=

= −

∑

where *r is given by (2.9) with λ is replaced

by * * *1 2/λ σ σ= .

Thus for a given pair of samples on stress,

strength variates we get 6 estimates of R by

the above 6 different methods. The

asymptotic variance (AV) of an estimate of R

which a function of two independent statistics

(say) 1 2t ,t is given by Rao (1973).

2 2

1 21 2

R RˆAV(R)=AV(t ) AV(t )σ σ

∂ ∂+ ∂ ∂ (2.14)

Where 1 2t ,t are to be taken in six different

ways namely, exact MLE, TMMLE on lines

of Tiku (1967), BMMLE on lines of

Balakrishnan (1992), SMMLE on line of Tiku

and Suresh (1992), MOM moment estimator

and BLUEs.

In the present case using (2.9) we get

1 2

221 20

r Z(1+Z) dZ

(1+ Z)(1+Z)

β

β

βλσ σλ

∝ −∂ −=∂

∫

1 2

12 22

2 20

dZr Z(1+Z)

(1+ Z)(1+Z)

β

β

σβλσ σλ

∝ −∂ −=∂

∫

Which can be used to get iR/ , i 1,2.σ∂ ∂ =

Because we are using three linear estimators

as MMLEs, which are obtained through

admissible approximation to log likelihood

function, all the three estimators are

asymptotically as efficient as exact MLE.

From the asymptotic optimum properties of

MLEs (Kendall and Stuart [1979]) and of

linear unbiased estimators (David [1981]), we

know that MLEs and BLUEs are

asymptotically equally efficient having the



Cramer-Rao lower bound as their asymptotic

variance as given in (2.12). Thus from

Equation (2.14), the asymptotic variance of

R̂ when ( 1 2t ,t ) are replaced by MLE,

TMMLE, BMMLE, SMMLE and BLUE in

succession, we get same result. In the log-

logistic distribution the moment estimator of

the scale parameter is sample mean divided

by (1 1/ ) (1 1/ )β βΓ + Γ − . Under central limit

property for i.i.d. variates the asymptotic

distribution of the moment estimator is

normal with the asymptotic variance is

[ ]2V/n (1 1/ ) (1 1/ ) β βΓ + Γ − where V is the

variance of scaled log-logistic variate and is

equal to

[ ]2 2{ (1 2/ ) (1 2/ ) - (1 1/ ) (1 1/ ) }β β β β σΓ + Γ − Γ + Γ − .

Hence the asymptotic variance of moment

estimator is

[ ]{ }[ ]

2

22

(1 2/ ) (1 2/ ) - (1 1/ ) (1 1/ )/n

(1 1/ ) (1 1/ )

β β β βσ

β β

Γ + Γ − Γ + Γ −

Γ + Γ −.

As exact variances of our estimates of R are

not analytically tractable, the small sample

comparisons are studied through simulation in

Section 3.

Small Sample Comparison

3000 random sample of size 3(1) 10 each

from stress population, strength population

are generated for λ = 1, 2 and 3 on lines of

Bhattacharyya and Johnson (1974). The scale

parameters 1 2,σ σ of the variates are estimated

by MLE, three MMLEs, Moment and BLUE

and are used in estimate λ . These six

estimators of λ are used to get the

multicomponet reliability for (s, k) = (1, 3),

(2, 4). The sampling bias and mean square

error (MSE) of the reliability estimates over

the 3000 such samples are given in Tables 1

and 2. With respect to bias MMLE of

Balakrishnan’s approach has given the

minimum bias in most of the parametric and

sample combinations. With respect to

variance and MSE the choice is equally likely

between MMLE by Tiku and Suresh (1992)

approach and Balakrishnan’s approach.

Specifically it is Tiku and Suresh (1992)

approach that gives minimum MSE or

minimum variance over the other suggested

methods of estimation. In any case it is one of

the MMLEs that is giving a minimum

empirical characteristic over the other

methods of estimation like exact MLE, BLUE

and Method of Moments. Hence we conclude

that in order to estimate the multi component

stress- strength reliability one of the three

MML methods of estimation is having the

least value for bias, variance and MSE. Hence

the suggested MML methods of estimation

are preferable then the classical methods like

exact MLE, BLUE and Method of Moments.

References

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statistical Inference with applications in

log-logistic distribution, Ph.D. thesis

awarded by Nagarjuna University.

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Table 1 Results of the Simulation Study of Bias for Estimates of Reliability ( 3β = )

(s,k) (m,n) λ mle tmmle bmmle smmle mom blue (1,3) (3,3) 1 -0.18846 -0.17573 -0.17693 -0.17622 -0.21421 -0.21396

(4,4) 1 -0.17254 -0.16727 -0.16214 -0.16197 -0.19451 -0.19067 (5,5) 1 -0.16372 -0.16242 -0.15366 -0.15442 -0.18100 -0.17805 (6,6) 1 -0.15622 -0.15622 -0.14720 -0.14789 -0.17166 -0.16736 (7,7) 1 -0.15319 -0.15429 -0.14471 -0.14568 -0.16587 -0.16272 (8,8) 1 -0.14967 -0.15151 -0.14189 -0.14298 -0.16055 -0.15722 (9,9) 1 -0.14681 -0.14900 -0.13981 -0.14082 -0.15642 -0.15412 (10,10) 1 -0.14487 -0.14726 -0.13839 -0.13943 -0.15288 -0.15142 (3,3) 2 -0.03651 -0.03183 -0.02843 -0.02996 -0.04529 -0.05043 (4,4) 2 -0.02829 -0.02664 -0.02281 -0.02368 -0.03554 -0.03692 (5,5) 2 -0.02460 -0.02423 -0.02011 -0.02087 -0.02998 -0.03065 (6,6) 2 -0.02178 -0.02190 -0.01810 -0.01869 -0.02643 -0.02618 (7,7) 2 -0.02000 -0.02047 -0.01676 -0.01735 -0.02317 -0.02354 (8,8) 2 -0.01839 -0.01902 -0.01566 -0.01612 -0.02150 -0.02100 (9,9) 2 -0.01716 -0.01790 -0.01476 -0.01520 -0.01950 -0.01966 (10,10) 2 -0.01619 -0.01700 -0.01412 -0.01450 -0.01833 -0.01840 (3,3) 3 -0.01078 -0.00228 0.00047 -0.00277 -0.01394 -0.02367 (4,4) 3 -0.00884 -0.00709 -0.00124 -0.00373 -0.00963 -0.01642 (5,5) 3 -0.00866 -0.00827 -0.00349 -0.00515 -0.00886 -0.01347 (6,6) 3 -0.00799 -0.00804 -0.00448 -0.00554 -0.00811 -0.01128 (7,7) 3 -0.00783 -0.00806 -0.00482 -0.00581 -0.00480 -0.01014 (8,8) 3 -0.00699 -0.00735 -0.00488 -0.00545 -0.00655 -0.00865 (9,9) 3 -0.00655 -0.00696 -0.00481 -0.00530 -0.00468 -0.00802 (10,10) 3 -0.00595 -0.00639 -0.00455 -0.00491 -0.00458 -0.00722

(2,4) (3,3) 1 -0.38933 -0.36199 -0.36725 -0.36689 -0.42722 -0.42949 (4,4) 1 -0.37167 -0.36076 -0.35179 -0.35222 -0.40439 -0.40231 (5,5) 1 -0.36122 -0.35808 -0.34220 -0.34396 -0.38721 -0.38608 (6,6) 1 -0.35121 -0.35091 -0.33400 -0.33566 -0.37470 -0.37131 (7,7) 1 -0.34848 -0.35038 -0.33227 -0.33436 -0.36720 -0.36581 (8,8) 1 -0.34446 -0.34777 -0.32948 -0.33178 -0.36023 -0.35835 (9,9) 1 -0.34029 -0.34429 -0.32673 -0.32884 -0.35395 -0.35386 (10,10) 1 -0.33800 -0.34240 -0.32536 -0.32750 -0.34901 -0.35019 (3,3) 2 -0.09201 -0.07362 -0.06128 -0.07166 -0.09774 -0.12659 (4,4) 2 -0.07470 -0.06988 -0.05726 -0.06190 -0.08608 -0.09843 (5,5) 2 -0.06870 -0.06763 -0.05520 -0.05825 -0.07831 -0.08518 (6,6) 2 -0.06276 -0.06302 -0.05211 -0.05408 -0.07335 -0.07476 (7,7) 2 -0.05892 -0.06017 -0.04935 -0.05137 -0.06337 -0.06862 (8,8) 2 -0.05498 -0.05673 -0.04713 -0.04850 -0.06228 -0.06225 (9,9) 2 -0.05185 -0.05395 -0.04490 -0.04619 -0.05652 -0.05890 (10,10) 2 -0.04939 -0.05171 -0.04336 -0.04447 -0.05414 -0.05567 (3,3) 3 0.00335 0.05058 0.06701 0.04123 0.02496 -0.04518 (4,4) 3 -0.00538 0.00189 0.03358 0.01681 0.01434 -0.03496 (5,5) 3 -0.01539 -0.01375 0.00845 -0.00122 -0.00055 -0.03271 (6,6) 3 -0.01759 -0.01754 -0.00311 -0.00848 -0.00770 -0.02909 (7,7) 3 -0.02071 -0.02120 -0.00766 -0.01308 0.01252 -0.02817 (8,8) 3 -0.01932 -0.02034 -0.01133 -0.01381 -0.01079 -0.02469 (9,9) 3 -0.01907 -0.02033 -0.01270 -0.01475 -0.00155 -0.02376 (10,10) 3 -0.01746 -0.01889 -0.01247 -0.01392 -0.00462 -0.02157



Table 2 Results of the Simulation Study of MSE for Estimates of Reliability ( 3β = )

(s,k) (m,n) λ mle tmmle bmmle smmle mom blue (1,3) (3,3) 1 0.05701 0.05460 0.05285 0.05239 0.07078 0.07006

(4,4) 1 0.04479 0.04352 0.04139 0.04096 0.05656 0.05269 (5,5) 1 0.03864 0.03854 0.03523 0.03535 0.04756 0.04446 (6,6) 1 0.03422 0.03435 0.03126 0.03136 0.04195 0.03832 (7,7) 1 0.03170 0.03215 0.02900 0.02922 0.03831 0.03513 (8,8) 1 0.02940 0.03006 0.02698 0.02725 0.03503 0.03200 (9,9) 1 0.02787 0.02863 0.02575 0.02599 0.03295 0.03031 (10,10) 1 0.02664 0.02745 0.02473 0.02500 0.03095 0.02881 (3,3) 2 0.00460 0.00510 0.00670 0.00433 0.01353 0.00706 (4,4) 2 0.00292 0.00273 0.00281 0.00240 0.00565 0.00368 (5,5) 2 0.00180 0.00177 0.00152 0.00148 0.00314 0.00239 (6,6) 2 0.00130 0.00132 0.00102 0.00105 0.00197 0.00170 (7,7) 2 0.00102 0.00106 0.00082 0.00084 0.00189 0.00131 (8,8) 2 0.00083 0.00087 0.00066 0.00068 0.00119 0.00102 (9,9) 2 0.00071 0.00075 0.00057 0.00059 0.00103 0.00087 (10,10) 2 0.00061 0.00065 0.00049 0.00051 0.00084 0.00074 (3,3) 3 0.00757 0.01097 0.01978 0.01005 0.03397 0.00506 (4,4) 3 0.00399 0.00342 0.00843 0.00480 0.01500 0.00254 (5,5) 3 0.00144 0.00138 0.00254 0.00163 0.00604 0.00135 (6,6) 3 0.00085 0.00088 0.00108 0.00087 0.00308 0.00089 (7,7) 3 0.00050 0.00053 0.00106 0.00055 0.01214 0.00060 (8,8) 3 0.00038 0.00040 0.00040 0.00036 0.00127 0.00044 (9,9) 3 0.00028 0.00029 0.00027 0.00024 0.00328 0.00033 (10,10) 3 0.00024 0.00025 0.00021 0.00021 0.00186 0.00027

(2,4) (3,3) 1 0.20036 0.18514 0.18941 0.18511 0.25381 0.23434 (4,4) 1 0.17691 0.17067 0.16404 0.16353 0.20852 0.20114 (5,5) 1 0.16337 0.16211 0.15095 0.15168 0.18826 0.18250 (6,6) 1 0.15172 0.15186 0.14058 0.14118 0.17419 0.16630 (7,7) 1 0.14654 0.14805 0.13592 0.13699 0.16532 0.15903 (8,8) 1 0.14044 0.14289 0.13075 0.13196 0.15721 0.15031 (9,9) 1 0.13617 0.13906 0.12745 0.12858 0.15112 0.14568 (10,10) 1 0.13279 0.13593 0.12479 0.12596 0.14576 0.14132 (3,3) 2 0.03182 0.04146 0.19497 0.05062 0.49703 0.03863 (4,4) 2 0.03466 0.02549 0.05282 0.03034 0.11821 0.02511 (5,5) 2 0.01391 0.01325 0.01513 0.01224 0.04136 0.01649 (6,6) 2 0.00978 0.00990 0.00850 0.00823 0.01973 0.01230 (7,7) 2 0.00793 0.00821 0.00711 0.00669 0.02980 0.00993 (8,8) 2 0.00665 0.00700 0.00542 0.00559 0.01056 0.00803 (9,9) 2 0.00580 0.00616 0.00475 0.00492 0.01075 0.00704 (10,10) 2 0.00514 0.00549 0.00424 0.00441 0.00779 0.00615 (3,3) 3 0.33179 0.49598 0.56205 0.46317 0.62058 0.09497 (4,4) 3 0.15272 0.10607 0.43961 0.20101 0.50398 0.04498 (5,5) 3 0.02987 0.02621 0.08750 0.04214 0.27288 0.01594 (6,6) 3 0.01406 0.01493 0.02721 0.01810 0.11620 0.01044 (7,7) 3 0.00559 0.00626 0.01686 0.00988 0.07398 0.00586 (8,8) 3 0.00475 0.00492 0.00705 0.00563 0.03718 0.00476 (9,9) 3 0.00277 0.00282 0.00395 0.00275 0.02179 0.00304 (10,10) 3 0.00274 0.00270 0.00292 0.00263 0.00839 0.00274

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ESTIMATION OF SYSTEM RELIABILITY FOR LOG – LOGISTIC DISTRIBUTION