Chap4- 1 Time to Failure

7/29/2019 Chap4- 1 Time to Failure

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Chapter 4

TIME TO FAILURE

DISTRIBUTIONS

4.1 Introduction

This chapter considers probability distributions which are most often used inreliability as time to failure distributions. These distributions include the dis-tributions which have well-known probability density functions for describingtime-to-failure (TTF) in many situations. The chapter also considers other

less used distributions of TTF, and presents characteristics and properties ofthe distributions considered.

The probability distribution of TTF indicates beliefs about the likelihoodof failure times. These beliefs can be based on the frequency of data valuesand have a frequency interpretation or they can be based on judgment andhave a frequency or a subjective interpretation. Basically, the probability dis-tribution is a model for knowledge about the time to failure or the reliabilityof a device.

In the first sections of this chapter, probability distributions of TTF areconsidered from the point of view of their corresponding hazard functions.

These sections also subscribe to the relatively simple point of view that asingle hazard function is sufficient for describing the behavior of the TTF overthe entire range of the failure times. In the last section of this chapter, morecomplex models are introduced where the use of several hazard functions orseveral distributions is necessary to model the situation.

121


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122 CHAPTER 4. TIME TO FAILURE DISTRIBUTIONS

4.2 Monotone hazard functions

The hazard function of a distribution was defined earlier but it is importantto consider such functions here since they describe the relationship betweenthe instantaneous probability of failure and time. Sometimes there is phys-ical or data information about the hazard function and not the probabilitydistribution itself and then the hazard is useful in determining the distribu-tion of TTF. There may also be more general information about the hazardfunction, such as the fact that it is monotone increasing or decreasing. In thiscase, the information is also useful because there are techniques of analysisof failure times that are based solely on the monotonicity information.

In this chapter, however, the type of hazard function and its shape is

considered only from the point of view of determining the distribution of theTTF. Models with monotone increasing hazards are often used because oneis often interested in the life of a device in the period of its life when an agingprocess is in force. It is then useful to consider which distributions have thistype of hazard function. Models with monotone decreasing hazard functionsare used less often but can have application in the study of early lifetimesof devices. Models with constant hazard functions are unique and are oftenuseful as baseline distributions to which other distributions are compared oras simple models for failure modes resulting in random failures.

4.2.1 Constant hazard rate-the exponential distribu-tion

In selecting a pdf to describe the TTF for a particular situation, it seems ap-propriate to require that the properties of the distribution do not contradictwhat is known about the failure behavior of the component of system understudy. For example, for most manufactured devices beyond the burn-inperiod, the hazard rate function h(t) is monotone increasing, or constant. Inthis case, it is important that the hazard rate function of the distributionunder consideration as a model for the failure times of the device incorporatethis property of monotone increasing or constant hazard.

In terms of constant hazard, the exponential distribution can be charac-terized as the only distribution with a constant hazard function. Thus, ifa unit has failure times that follow the exponential distribution, it can besaid that the unit is not subject to burn-in nor is it subject to wear.Conversely, if a unit has survived to time t and its probability of survival,


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4.2. MONOTONE HAZARD FUNCTIONS 123

that is, its reliability, over an additional time period ta is the same regard-

less of the present age t, then the time to failure of the unit must followan exponential distribution. Thus, the exponential distribution is central inreliability studies, in terms of its usage and in terms of this figurative way ofhaving a hazard function which is not increasing or decreasing.

For the exponential distribution, the density function f(t) is:

f(t) =1

e

t = et, , > 0 (4.1)

where is the mean time to failure and is the hazard rate. It followsthat the cdf F(t), the reliability function R(t) and the hazard function h(t)are respectively:

F(t) = 1 e t = 1 et, R(t) = e t = et and h(t) = 1

= (4.2)

If the failure time of a unit follows an exponential distribution,

P(T t) = R(t) = et

and

P(T ta|T t) = R(t + ta)R(t)

= eta.

That is, the conditional probability of failure conditioned on an earliertime period is independent of the earlier time period. This is usually referred

to as the memory-less property of the exponential distribution. It meansthat devices whose failure times follow the exponential distribution model arenot subject to wear out. That is, a device that has survived to time t is asgood as new in terms of its remaining life. In contrast, with devices thatare subject to wear out, the conditional probability of failure conditioned onan earlier time period is a decreasing function of the length of the earliertime period.

Also, it is true that if a unit is functioning at time t and the probabilityof survival over an additional time period is the same regardless of t, then

P(T ta|T t) =R(t + ta)

R(t) = R(ta) or R(t + ta) = R(t)R(ta)

and necessarily,R(t) = et and h(t) = .

That is, the distribution of the time to failure is exponential.


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4.2.2 The exponential distribution and the geometric

distributionNow, consider the time line being segmented into intervals of a length tand let the probability of failure in any interval be p, where p is constantfor all intervals. Also, assume that the occurrence of a failure in an intervalis independent of that occurrence in any other interval. The occurrenceor non-occurrence of a failure in an interval can be called a trial and thetrials in this example make up a Bernoulli process. (These assumptions arealso equivalent to the assumptions or characteristics underlying the binomialdistribution.) Then the number of intervals, R, until the first failure has ageometric distribution and

P{R = r;p} = (1p)r1p, r = 1, 2, 3,...

Thus, the geometric distribution is the discrete analog of the exponentialdistribution. Notice also that the assumptions on the constancy of p and theindependence of the occurrences in the intervals cause the process to operatesimilarly to a continuous process with a constant failure rate. It can alsobe shown that if the length of the interval t goes to zero, the geometricdistribution becomes the exponential distribution.

4.2.3 The exponential distribution and the Poisson pro-

cess

Consider, for example, that failures represent any time related occurrencesand the random variable of interest is the number of failures, X(t), that occurin the interval [0, t]. Then, assume:

a) X(0)=0

b) The numbers of failures in non-overlapping intervals are independentrandom variables

c) the distribution of the number of failures that occur in a given intervaldepends only on the length of the interval and not on the location ofthe failures

d) limh0

p{X(h) = 1}h

=


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4.3. INCREASING AND DECREASING HAZARD FUNCTIONS 125

e) limh0p{X(h)2}

h = 0

Under these assumptions, the random variable or counting process, X(t), thenumber of failures in the interval [0, t] is said to be a Poisson process andthe number of failures in any interval of length t has a Poisson distributionwith parameter = t and the times between failures have an exponentialdistribution with parameter . That is,

P{X(t) = r} = et( t)r

r!

and

P{Xi+1 > ti+1|Xi = ti} = P{0 events in (ti, ti + ti+1)|Xi = ti}= P{0 events in (ti, ti+1)} = exp[(ti+1 ti)]

4.3 Increasing and decreasing hazard func-

tions

If a unit has a conditional survival probability R(ta|t) that decreases as afunction of the time t, the time period previous to the present time periodta, then the device is wearing out or aging and

R(ta|t) = R(t + ta)R(t)

is decreasing in t for each ta. It follows that

limta0

1

ta

1 R(t + ta)

R(t)

= lim

ta0

R(t) R(t + ta)

ta R(t)

= h(t)

is increasing in t and the distribution of t is called an increasing failurerate (IFR) distribution. If R(ta|t) is increasing in t for each ta, h(t) is de-creasing in t and R(t) is called a decreasing failure rate (DFR) distribution.The exponential distribution has constant failure rate and is the only dis-tribution that has. Thus much can be learned about a distribution of thetime to failure if it is either IFR or DFR by comparing it to the exponen-tial. For example, the Weibull distribution is IFR when its shape parameter


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> 1 and DFR for 0 < < 1. The gamma distribution is IFR when its

shape parameter r > 1 and DFR for 0 < r < 1. Both the Weibull and thegamma are exponential for r = = 1. There are also distributions withnon-monotone failure rates. For example, the log normal and the log-logisticare distributions used in reliability which have non-monotone failure rates.

4.4 Time to failure distributions with mono-

tone failure rate

The popular choices for time to failure (TTF) distributions with monotonehazard or failure rates are: the exponential, the Weibull, the gamma and thenormal. The density functions of these distributions are frequently chosenas models for the frequency of occurrence of TTF values. Often, this choiceis made because either their theoretical properties are consistent with theconditions of use and the physics of failure of the device, or because thedensity adequately describes the failure history of the device. The abovedistributions are outlined first in this chapter. The Type I Extreme Valuedistribution, a distribution with a monotonically increasing hazard functionand less popular in its use than the above distributions, is outlined later inthe chapter.

Other distributions of time to failure with non-monotone hazard func-tions, including the log-normal distribution, will also be presented later inthe chapter. A relatively recently proposed choice for the distribution of timeto failure, the Burr distribution, is outlined also later in this chapter. TheBurr and the extreme value distributions are less known and used, especiallythe Burr, but are nevertheless important in the list of available choices formodels and fill important gaps in the span of possible hazard or failure rateconsiderations.

In an earlier chapter, the bathtub curve and its use in reliability wasrecognized. Note that the burn-in portion of the curve represents distri-

butions with DFR and that the wear-out portion represents distributionswith IFR. The constant hazard portion is often called the useful life portionof the hazard curve. For electronic components, the useful life portion isusually longer than for mechanical components. See, for example, Figures3.9 and 3.10


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4.5. EXPONENTIAL DISTRIBUTION 127

4.5 Exponential distribution

In Section 4.2.3, the exponential distribution was characterized by its rela-tionship with the Poisson process. But the gamma distribution is also relatedto the Poisson process. The Poisson process is a model for the number ofevents that occur in a time interval with mild assumptions that failure eventsoften meet. The model then specifies that the distribution of the number ofevents is Poisson and that the distribution of the time between events is ex-ponential. The time to the first failure is also exponentially distributed butthe time to the rth failure has a gamma distribution with shape parameterr.

The exponential distribution with rate parameter or mean has prop-

erties:

R(t) = et, f(t) = et, h(t) = , H(t) = t

E(T) = = =1

, V(T) = 2 = 2 =

2, 3 = 23, 4 = 9

4,

coefficient of skewness =

1 = 2, coefficient of excess = 2 3 = 6,coefficient of variation CV =

V(T)

E(T)= 1,

and tR =

1 lnR, where tR is the reliable life for the value R.

The exponential density (pdf) and cdf are illustrated in Figures 4.1 and4.2 for means = 2, 10, 20, 50 or rates = 0.5, 0.1, 0.05, 0.02.

The exponential distribution is widely used in its own right as a modelof the times to failure in some situations and is also widely used as the stan-dard to which other distributions are compared. It can be chosen for itsconstant hazard property or because it models the observed times. Note,however, that the coefficients of skewness and excess are fixed values for allexponential distributions. This fact indicates that the shape of the exponen-tial distribution remains the same for all values of its parameter. This factalso indicates a lack of flexibility in fitting different models of failure time

data. There is agreement among practitioners that there must exist stronginformation about a constant hazard in order to justify the assumption andsubsequent use of the exponential distribution as a failure time model. Listedbelow in Table 4.1 are 96 simulated values from an exponential distributionwith mean (MTBF) of 1000 cycles.


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0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 5 10 15 20 25 30 35 40 45 50

t

f(t)

f(t),mean=2,skewness=2,excess=9

f(t),mean=10,skewness=2,excess=9

f(t).mean=20,skewness=2,excess=9

f(t).mean=50,skewness=2,excess=9

Figure 4.1: Exponential Density Function

Table 4.12100 2232 625 1852 299 559 54 660 562 69 91 541427 50 571 306 412 9 1129 749 208 490 792 485140 351 398 610 1463 1237 884 1985 2083 588 1640 363315 631 353 853 767 418 855 726 4707 329 196 160224 29 300 2892 2051 62 312 495 2622 832 555 963

2607 1136 405 263 723 967 1516 1731 1785 2278 163 1476712 44 789 666 2894 116 1513 1112 338 1072 2168 2707788 67 1096 572 1013 225 269 1406 644 181 19 1041

By examining these data, one notices that they are all over the place.

They range from 9 to 4707. Remember that the standard deviation of theexponential distribution is equal to its mean; in this case, 1000 cycles. Onecan also compute that the proportion of values larger than the mean 1000 is30/96 = 0.3125. More than 60% of the values are below the mean; this isone of the characteristics of the exponential distribution. In fact, note that


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4.5. EXPONENTIAL DISTRIBUTION 129

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30 35 40 45 50

t

F(t)

F(t),mean=2,skewness=2,excess=9




Figure 4.2: Exponential Cumulative Function

for any device subject to exponential failures, there is a 0.632 probability offailure before the mean time to failure, or, equivalently, 63.2% of such deviceswill fail before the mean. It is rather common in reliability to set goals anddevelop metrics in terms of the mean time to failure. However, in light ofthe above, it seems questionable to set a reliability goal in terms of an eventin which most (63.2%) of what may happen before the event is unfavorable(failure).

EXAMPLE: 4.5 Suppose that the TTF density of a machine is givenby:

f(t) =1

1000 e(t/1000)

, t > 0,

that is, the TTF has an exponential distribution with mean 1000. The prob-ability of surviving the interval (0,1000) is given by: R(1000) = exp(1) =0.368 or 1 0.632.


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There are several viable alternatives to focusing on the mean in this way.

Among them are the use of distributional properties, such as percentiles andreliable life. The reliable life, tr is defined as the time at which the reliabilityis r. For example, t0.9 = 300 cycles means that after 300 cycles of operation,the reliability is 0.9.

EXAMPLE: 4.6 For the exponential distribution, any value of themean life, determines a life for a specified value of R; i.e., the reliablelife may be established for any mean and desired reliability. Since R(t) =exp(t/), tR = ln R. For example, if = 1000 cycles, then:R = 0.1 0.3 0.5 0.7 0.9 0.95 0.99 0.999

tR =2303 1204 693 357 105 51 10 1

It becomes clear in examining the table that if the mean number of cyclesto failure is 1000, high reliabilities are achieved only for relatively short num-bers of cycles. Thus, it seems more meaningful to set a reliability goal, not interms of, but in terms of a statement that a 0.95 reliability for 1000 cyclesof operation is required. If the exponential distribution is appropriate, the value equivalent to this goal is easily obtained using = t/ ln R(t). In thiscase, = 1000/ ln(.95) = 19496 cycles. The concept of reliable life will bereferred to frequently throughout this text.

EXAMPLE: 4.7 A semiconductor is characterized by an exponentialTTF distribution with = 1 106 failures per hour. The probability of nofailure in the interval (106, 107) given no failure in (0, 106) is:

R(107)

R(106)=

e106(107)

e106(106)=

e10

e1= 0.0001234

The conditional reliability in the interval (67 106, 76 106) is:

R(76

106)

R(67 106) =e10

6(76106)

e106(67106) =

e76

e67 = e9

= 0.0001234

that is, with the exponential distribution, the reliability in an interval ofequal length is the same regardless of where in the time zone that intervaloccurs.


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4.6. WEIBULL DISTRIBUTION 131

4.6 Weibull distribution

The Weibull distribution is perhaps the most widely used of all the failuretime distributions and it is noted for its flexibility as a model. The Weibulldistribution with scale parameter and shape parameter has properties:

R(t) = e(t )

,

f(t) = t1

e(

t)

, h(t) =t1

, H(t) =

t

E(T) = = 1 +1

, V(T) = 2 = 2 = 2 2= 2

1 +

2

2

1 +

1

,

where () =0 x

1 ex dx,3 = 3

1 + 3

3 2 + 23,

where i = i

1 + i

,

4 = 4

1 +

4

4 3+62234, mode =

( 1)

1

, > 1

(4.3)

coefficient of skewness = 1 = 323/2 , coefficient of excess = 23 = 422 3,coefficient of variation =CV =

(1+ 2 )2(1+ 1 )

(1+ 1 ), tR = (lnR)

1 .

The examples of Weibull pdf, cdf and hazard function are shown in Fig-ures 3.4-3.6.

For < 1, the Weibull distribution is DFR, for > 1, the Weibull isIFR and for = 1, the Weibull becomes the exponential. That is, if < 1,h(t) is monotone decreasing, if > 1, h(t) is monotone increasing, and if = 1, h(t) is constant. Also, note that if is approximately 3.5, f(t) isapproximately symmetrical, if 1 < < 3.5, f(t) is positively skewed, and

if > 3.5, f(t) is negatively skewed. Note that the coefficient of skewness and the coefficient of excess 2 3 are both functions only of the shape

parameter since the scale parameter gets cancelled in the ratio.Since the Weibull distribution allows both increasing and decreasing fail-

ure rates, it can be used to represent the times to failure for both the burn-


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in portion of the hazard curve as well as the wear-out portion. It has

been successfully used in both situations.The Weibull distribution also has a justification related to the extremevalue of a group of failure times. This aspect will be discussed as part of thediscussion of the extreme value distribution.

EXAMPLE: 4.8 Failures associated with the grasping function of arobot arm follow a Weibull distribution with parameters = 2.1 and =2200 hours. The probability of the robot arm grasping function survivingpast 1000 hours is

R(1000) = e(10002200)

2.1

= 0.826

EXAMPLE: 4.9 The times to failures for a pick and place machineused in surface mount technology appear to follow a Weibull distribution,with = 2 and = 30 weeks. The probability of surviving 40 weeks is

R(40) = e(4030)

2

= e1.778 = 0.169

This Weibull is IFR, since h(t) = t450

. The mean (MTTF) is 30(1+ 12

) = 26.6weeks.

The probability that the pick and place machine will fail in the interval(40, 50) given survival in (0, 40) is:

F(50) F(40)R(40)

= 1 e(50/40)2

(1 e(40/40)2

)e(40/40)2

=e1 e1.5625

e1= 1 e0.5625 = 1 0.5625 = 0.4302

EXAMPLE: 4.10 Suppose that one knew that a device had a reliabilityof 0.60 at 1000 hours of operation and a reliability of 0.10 at 4000 hours ofoperation and that the failure times followed a Weibull distribution. In thiscase it follows that:

e(1000 )

= 0.60 ande(4000 )

= 0.10

Using logarithms, one can solve the two equations and can find that theappropriate distribution of time to failure is a Weibull with = 1.68 and = 2437.5.


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4.7. GAMMA DISTRIBUTION 133

4.7 Gamma distribution

Another distribution that is closely related to the exponential distributionis the gamma distribution. The gamma distribution has a scale parameter,denoted here as and a shape parameter, denoted here as r. The gammadistribution can also arise from a Poisson process as the distribution of thetime to the rth occurrence. In terms of the Bernoulli process, the number oftrials to the rth occurrence is distributed as negative binomial and the nega-tive binomial is the discrete analog of the gamma. It follows that the time tothe rth occurrence is gamma distributed and, in the same consideration, thesum of r independent exponential random variables is gamma distributed.The gamma distribution with scale parameter and shape parameter r has

properties:

f(t) =r tr1 et

(r), t 0 R(t) = et

r1i=0

(t)i

i!, if r is an integer,

R(t) = 1(r)t

r ur1 eudu = 1(r) (r, t), otherwise.

h(t) =rtr1 et

(t,r)

E(T) = =r

, V(T) = 2 = 2 =

r

2,

4 =r2+6r4 , where, i

= (r+i)(r) ,

mode =r 1

, if r > 1, coefficient of skewness =

1 =

2r

(4.4)

coefficient of excess = 2 3 = 6r , coefficient of variation = CV = 1r , tRis the solution tR of R(tR) = R, or the solution to (r, tr) = 1R.

The gamma pdf is illustrated in Figure 4.3, the gamma cdf is illustratedin Figure 4.4 and the gamma hazard function is illustrated in Figure 4.5 for

pairs of values of the scale parameter and the shape parameter r equal to(0.2, 0.75), (0.333, 1), (0.25, 2) and (1, 3).

The chi-square distribution with degrees of freedom is a special case ofthe gamma distribution, when = 1/2 and r = /2. The gamma distribu-tion is DFR for r < 1, decreasing to a constant value . The gamma is an


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0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 2 4 6 8 10 12 14 16 18 20

t

f(t)

f(t),r=0.75,lam=0.2,mean=3.75,st.dev=4.33,skewness=2.31,excess=8

f(t),r=1,lam=0.333,mean=3,st.dev=3,skewness=2,excess=6

f(t),r=2,lam=0.25,mean=8,st.dev=5.66,skewness=1.41,excess=3

f(t),r=3,lam=1,mean=3,st.dev=1.73,skewness=1.15,excess=2

Figure 4.3: Gamma Density Function

exponential when r = 1. When r > 1, the gamma is IFR, increasing fromzero to a constant value . The primary use of the gamma in reliability isas the distribution of the sum of r exponential random variables when suchis needed in practice. The use of the gamma distribution as the sum of rindependent exponential random variables is also of use in the theoreticaldevelopment of inferences related to the exponential.

EXAMPLE: 4.11 The time to failure of a particular device follows theexponential distribution with mean time to failure 100 hours. A sample ofn=15 such devices are tested to failure and the sample mean is observed. The

sum of the sample times to failure, T, is distributed gamma with = .01and r=15. This sum can be transformed into a chi-squared variable by mul-tiplying by 2; that is, 2T is chi-squared distributed with 30 degrees offreedom. (If X is distributed exponential with hazard rate , then X is ex-ponentially distributed with hazard rate 1. 2X is exponential with hazard


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4.7. GAMMA DISTRIBUTION 135

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16 18 20

t

F(t)

F(t),r=0.75,lam=0.2,mean=3.75,st.dev=4.33,skewness=2.31,excess=8

F(t),r=1,lam=0.333,mean=3,st.dev=3,skewness=2,excess=6

F(t),r=2,lam=0.25,mean=8,st.dev=5.66,skewness=1.41,excess=3

F(t),r=3,lam=1,mean=3,st.dev=1.73,skewness=1.15,excess=2

Figure 4.4: Gamma Cumulative Function

rate 2 and 2T is gamma with parameters = 2 and r=n.) In this example,P{2T < 43.773} = 0.95 and thus P{T < 2188.65} = 0.95.

EXAMPLE: 4.12 A sensitive microswitch with an exponential failurerate of 2 per year has been replicated 4 times to increase system reliability.For example, all 4 switches must fail for unreliability. What is the reliabilityof this switch over 3 years?

R(3) = 1 F(3) = 1k=4

e2(3)

6k

k! =

3k=0

e6

6k

k! = 0.285

Also, note that the MTBF= E(t) = r =42 = 2 years.


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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16 18 20

t

h(t)

h(t),r=0.75,lam=0.2,mean=3.75,st.dev=4.33,skewness=2.31,excess=8

h(t),r=1,lam=0.333,mean=3,st.dev=3,skewness=2,excess=6

h(t),r=2,lam=0.25,mean=8,st.dev=5.66,skewness=1.41,excess=3

h(t),r=3,lam=1,mean=3,st.dev=1.73,skewness=1.15,excess=2

Figure 4.5: Gamma Hazard Function

4.8 Normal distribution

Strictly speaking, since the normal distribution allows negative values andsince the life of an item is always positive, some caution should be observed inusing the normal distribution to represent the time to failure distribution. Insome places, it is simply stated that the normal distribution should be usedin reliability only if it can be assumed that (/) > 3. This would essentiallyensure that only positive values occur. The normal distribution with mean and standard deviation has properties

f(t) = 1

2e

12(

x )

2

, R(t) = 1

t

,

where t

=t

12

ez2

2 dz, h(t) = f(t)R(t) ,


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4.8. NORMAL DISTRIBUTION 137

H(t) = ln

1

t

, E(T) = , V(T) = 2 = 2, 3 = 0, 4 = 3

4


1 = 0, coefficient of excess = 2 3 = 0,coefficient of variation CV = , tR =

1(1R) + .The pdf, cdf and hazard function of the normal distribution are presented

in Figures 4.6, 4.7 and 4.8 respectively, and for mean and standard deviationpairs (2, 0.5), (5, 1), (10, 2) and (12, 0.75).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12 14 16

t

f(t)

f(t),mean=2,st dev=0.5,CV=.25

f(t),mean=5,st dev=1,CV=.20

f(t),mean=10,st dev=2,CV=.20

f(t),mean=12,st dev=0.75,CV=.0625

Figure 4.6: Normal Distribution Density Function

The normal distribution is an increasing failure rate (IFR) distribution,and for values of t larger than the mean, the hazard rate approaches anasymptote (t)

, which increases linearly in t.The use of normal distribution can also be justified as a limit of the

sums (based on Central Limit Theorem); for example, when approximating


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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16 18

t

F(t)

F(t),mean=2,stdev=0.5,CV=.25F(t),mean=5,st dev=1,CV=.20

F(t),mean=10,st dev=2,CV=.20

F(t),mean=12,stdev=0.75,CV=.0625

Figure 4.7: Normal Distribution Cumulative Function

Gamma distribution with large r.

EXAMPLE: 4.13 The refractory lining of an oven used as part of acuring process wears out (must be replaced) periodically. The wear-out dis-tribution is believed to be normal with a mean of 2700 hours and a standarddeviation of 250 hours. After how many hours of operation should the ovenbe relined so the wear out is virtually impossible?

Schedule relining so that it occurs at a point in the extreme left tail ofthe distribution, say at 3. Reline every 2700 3(250) = 1950 hours.

EXAMPLE: 4.14 Suppose one knew that the failure process of a devicewas such that essentially all of the devices will fail before 10000 hours, butthat 95% of the devices will operate beyond 2000 hours. Further, it is knownthat the failure times follow a normal distribution. In this case, it is known


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4.9. EXTREME VALUE DISTRIBUTIONS 139

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16

t

h(t)

h(t),mean=2,st dev=0.5,CV=.25

h(t),mean=5,st dev=1,CV=.20

h(t),mean=10,st dev=2,CV=.20

h(t),mean=12,st dev=0.75,CV=.0625

Figure 4.8: Normal distribution Hazard Function

that: + 3 = 10000 and 1.645 = 2000

These equations result in the determination of the mean = 4833 hours andthe standard deviation = 1722 hours for the normal distribution of failuretimes.

4.9 Extreme value distributions

The distribution of the first order statistic T(1), from a sample of n, is, interms of the reliability function or in terms of the cumulative distributionfunction,

RT(1)(t) = [RT(t)]n or FT(1)(t) = 1 [1 fT(t)]n.

This will be discussed further in a later section on order statistics. It canalso be shown that when n gets large, the random variable nFT(T(1)) = Yn


20/36


goes to the random variable Y which has distribution FY(y) = 1 ey and

T(1) goes to the random variable F1

T (Y /n).Then, if the random variable T is bounded from below and with a weakconditions on FT, the limiting distribution for T(1) is a Weibull distribution.This case is called the Type III asymptotic distribution of the smallest ex-treme and indicates another physical scenario for the usage of the Weibulldistribution as a distribution of the time to failure.

Suppose a complex device consists of parts and material with many modesof failure and each mode for each part or material has a corresponding time tofailure and the failure of any part or material will cause failure of the device.Then the first order statistic from a large sample of n modes of failure is thedetermining factor for the time to failure of the device. This is the weakest

link theory and many of the devices of interest in reliability seem to followthis theory for the failure process. Thus, under the above conditions, theWeibull distribution is the distribution of the time to failure.

The term extreme value distribution is, however, more often associatedwith the Type I asymptotic distribution of the smallest extreme. This typeof asymptotic distribution relates to the situation where the underlying dis-tribution has no lower bound on its values. Then the limiting distribution ofthe first order statistic T(1) is of the form given below with the correspondingproperties.

For the Type I Extreme Value Distribution of random variable Y withlocation parameter and scale parameter :

f(y) =e(y)

ee

(y) , < y < , R(y) = ee

(y) ,

h(y) =e(y)

, H(y) = e

(y) , mode = , E(Y) = = 0.5772,

where 0.5772 . . . is Eulers constant, V(Y) = 2 = 2 =

2

6 2, coefficient of

skewness =

1 = 1.3, coefficient of excess = 2 3 = 2.4,

coefficient of variation = CV = 6 0.5772 , yR = + [ln( ln R)].

(4.5)The extreme value distribution density, cumulative and hazard functions

are illustrated in Figures 4.9, 4.10 and 4.11, respectively, for the scale pa-


21/36


rameter and location parameter pairs (2, 0), (1, 0), (0.5, 0) and (0.2857,

0).

0

0.2

0.4

0.6

0.8

1

1.2

-5 -4 -3 -2 -1 0 1 2 3 4 5

y

f(y

)

f(y),location=0,scale=2,mean=-1.15,stdev=2.57,skewness=1.3,excess=2.4

f(y),location=0,scale=1,mean=-.58,stdev=1.28,skewness=1.3,excess=2.4

f(y),location=0,scale=0.5,mean=-.29,stdev=.64,skewness=1.3,excess=2.4

f(y),location=0,scale=0.2857,mean=-.16,stdev=.37,skewness=1.3,excess=2.4

Figure 4.9: Extreme Value Density Function

The form of the Type I extreme value distribution that is used here isslightly different from the form used in some texts on distribution theory (e.g.,Johnson and Kotz (1970)). The form above that is used here is typical of theform used in reliability texts, theory and applications. Most of the uses of theType I extreme value distribution in reliability is based on its relationship tothe Weibull distribution, and the above form of the distribution makes that

relationship straight-forward and easily accessible. Note that the form here isnegatively skewed with the same value of skewness and excess for all versionsof the distribution. Thus, one should note that the parameters of the Type Iextreme value distribution, and , are purely location and scale parametersrespectively, which means that all of the extreme value distributions have


22/36


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-5 -4 -3 -2 -1 0 1 2 3

y

F(y)

F(y),location=0,scale=2,mean=-1.15,stdev=2.57,skewness=1.3,excess=2.4

F(y),location=0,scale=1,mean=-.58,stdev=1.28,skewness=1.3,excess=2.4

F(y),location=0,scale=0.5,mean=-.29,stdev=.64,skewness=1.3,excess=2.4

F(y),location=0,scale=0.2857,mean=-.16,st dev=.37,skewness=1.3,excess=2.4

Figure 4.10: Extreme Value Cumulative Function

the same shape. As was mentioned earlier with the exponential distribution,this fact diminishes the practical usefulness of the distribution for fittingdata. The extreme value distribution is an IFR distribution with the hazardincreasing from zero exponentially.

The Type I extreme value distribution is also related to the Weibull dis-tribution in the same way that the normal and the lognormal are related,that is, the log base e of the Weibull times are distributed as extreme value.If the Weibull random variable T has scale and shape , the distribution

of Y = ln T is extreme value with = ln , and =1

.

EXAMPLE: 4.15 In an earlier example, the times to failure for a pickand place machine used in surface mount technology were assumed to follow aWeibull distribution with shape parameter = 2 and scale parameter = 30


23/36


0

1

2

3

4

5

6

7

8

9

10

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

y

h(y)

h(y),location=0,scale=2,mean=-1.15,stdev=2.57,skewness=1.3,excess=2.4

h(y),location=0,scale=1,mean=-.58,stdev=1.28,skewness=1.3,excess=2.4

h(y),location=0,scale=0.5,mean=-.29,stdev=.64,skewness=1.3,excess=2.4

h(y),location=0,scale=0.2857,mean=-.16,stdev=.37,skewness=1.3,excess=2.4

Figure 4.11: Extreme Value Hazard Function

weeks. The reliability at 40 weeks was

R(40) = e(4030) = 0.169.

The distribution of the logarithm of the failure time, ln T, has the extremevalue distribution with = ln30 = 3.401 and = 0.5. The reliability ofln(40) = 3.689 is

R(3.689) = ee(3.6893.401)

.5 = ee0.576

= e1.77 = 0.169.

The relationship of the extreme value distribution to the Weibull distri-bution is also useful in the theoretical development of analytical proceduresfor the Weibull distribution. But the extreme value distribution often hasuseful practical properties as a model for the failure times in much the same


24/36


way as the lognormal distribution has. A major difference in considering the

usage of a model based on the extreme value distribution and the lognormaldistribution is the fact that the extreme value distribution is IFR and thelognormal distribution has a non-monotone failure rate.

4.10 Non-monotone hazard rate distributions

There are three failure time distributions that are considered in this chap-ter for which the failure rate is non-monotone. These are similar in theirapplication, since in their most interesting cases they all have failure ratesthat are initially increasing to a maximum and then decrease. They are the

lognormal, the log-logistic and the inverse Gaussian; and all have advantagesand disadvantages. The log-logistic is considered in its more general form,usually referred to as the Burr Type XII distribution.

Although the hazard rate of these distributions is non-monotone, thedistributions are all increasing in popularity of use as models of failure time.The families of these distributions are flexible in their use as descriptivemodels, that is, some member of the family can be found to fit or model adistribution which represents the failure time distribution for failure timesfrom a wide variety of failure time models.

The lognormal distribution is the most popular of these non-monotonehazard rate distributions but the Burr distribution is as flexible for fittingmodels of failure time data and has the added advantage of having a cdf thatis a simple closed form function. This advantage makes computations withthe Burr distribution easy and also has a theoretical advantage when oneworks with censored data.

The usual hazard function of these distributions increases over time toa maximum and then decreases. Since this implies that the failure risk de-creases after some time and then continues to decrease as the item of interestages, it is a concern of some practitioners and prospective users of these dis-tributions as models of failure times as they note that for many items therisks of failure do not decrease with increasing time. In spite of this concern,

there are at least three reasons that these distributions with non-monotonehazard functions might be useful as models of time-to-an-event data:1) The failure process might be such that it actually has an non-monotonehazard rate. For example, with modern components with high reliabilities,there may actually be a hardening process at work so that the hazard rate is


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4.10. NON-MONOTONE HAZARD RATE DISTRIBUTIONS 145

non-monotone and decreasing with larger ages. In addition, some analysts of

time to death of cancer patients have noted that the observed hazard func-tion was such that mortality reached a peak after a finite period and thenslowly declined (Farewell and Prentice (1979) and Bennett (1983)).2) The time to an event process might be other than a failure process (suchas the duration of a strike (Lawrence (1984)) with a non-monotone hazardfunction.3) The large values of time may not be of interest or the interest in the timeto failure is only over a finite interval of time in which the physical hazardrate is matched by that of one of the non-monotone hazard distributions.

4.10.1 Lognormal distribution

The lognormal distribution is one of the most widely used distributions oftime to failure. The lognormal is denoted by that name since, if T is therandom variable representing the lognormal time to failure, the random vari-able, Y = ln T, is normally distributed with mean and standard deviation.

The failure rate of the lognormal distribution increases from zero at t = 0to a maximum and then decreases to zero as t increases. The amount ofincrease or decrease of the failure rate depends mostly on the value of andfor a certain range of values the failure rate is essentially constant for animportant range of failure time values. For very small values of the hazardfunction of the lognormal essentially increases over most of the time valuesand behaves similarly to the normal hazard function. For large values of the hazard function essentially decreases over most of the time values. Thelognormal distribution appears to be able to represent the distribution of thetime to failure for many failure processes. It is becoming more popular in itsusage and has become perhaps the second most used distribution of failuretimes, second in use to the Weibull.

In attempting to model the distribution of a value associated with fail-ure in terms of the physics of failure, it seems that the use of the lognormaldistribution can be justified whenever the accumulated causes of failure act

in a multiplicative manner. If the cause of failure is due to a build-up ofaccumulated value from many causes and failure occurs when the accumu-lated build-up passes a threshold value, and, in addition, the accumulatedbuild-up occurs multiplicatively, then the value at failure is approximatelylognormal. In this case, failure depends on the value of a product so that


26/36


the failure value, denoted by T, is such that T = ni=1 Tci, where Tci is the

accumulated value associated with the ith cause. Then the distribution ofT is of interest and, if n is large, the distribution of log T is normal by theCentral Limit Theorem.

For the log normal distribution with parameters and :

Y = ln T N(, ), f(t) = 1 t

2e

(ln t)2

22 , t > 0

R(t) = 1

ln t

, h(t) =

f(t)

R(t), H(t) = ln

1

ln t

,

E(T) = T = e

+

2

2

, V(T) = 2T = e

(2+2)

e2 1 , 2T = e

2e22

e2 1

3T =

e2 3

2

e2 1

2 e

2

+ 2

e3,

4T =

e22

e2 1

2 e

24

+ 2

e23

+ 3

e22 3

e4


1 =

e2 1

12

e2

+ 2

> 0,

coefficient of excess = 2 3 = e2

4

+ 2 e2

3

+ 3 e2

2

6,

mode= e2

, median = e, E(T) = T > median{T} > mode {T},coefficient of variation = CV =

V(T)

E(T) , tR = e[1(1R)+].

The lognormal density, cumulative and hazard function are illustrated inFigures 4.12, 4.13 and 4.14, respectively, for mean and standard deviationpairs: (0, 0.35), (1, 0.5), (2, 0.3) and (2.5, 1).

EXAMPLE: 4.16 The time (in minutes) to repair a tracking malfunc-tion on a robot arm used in circuit card assembly tends to follow a log normaldistribution with parameters = 4.2 and = 1. The probability that a repairaction will be started and completed in 30 minutes is:

ln30 4.21

=

3.4 4.2

1

= (0.8) = 0.2119.

EXAMPLE: 4.17 The failure of a certain metal under stress and tem-perature due to fatigue cracks is known to follow a lognormal distribution


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4.11. BURR DISTRIBUTION WITH SPECIAL CASE LOG-LOGISTIC147

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12 14 16t

f(t)

f(t),mu=0=0,sigma=0.35,mean=1.06,st.dev=0.38,skewness=1.13,excess=2.35,mode=0.88

f(t),mu=1,sigma=0.5,mean=3.08,st.dev=1.64,skewness=1.75,excess=5.90,mode=2.12

f(t),mu=2,sigma=0.3,mean=7.73,st.dev=2.37,skewness=0.95,excess=1.64,mode=6.75

f(t),mu=2.5,sigma=1,mean=20.09,st.dev=26.33,skewness=6.18,excess=110.94,mode=4.48

Figure 4.12: Lognormal Density Function

with parameter = 2. What would the value of the parameter have to bein order for the reliable life with 0.95 reliability to be 10000 hours?

t0.95 = 10000 = e(21(0.05)+), so that ln(10000) = 21(0.05) +

9.21034 (1.645(2)) = 12.5 = .That is, one needs a lognormal distribution with mean: e(12.5+2) = 1982759.26.

4.11 Burr distribution with special case log-

logisticA distribution which is not well known in reliability practice at this point,but which has the possibility of effective usage is the Burr distribution. TheBurr distribution has properties which are useful in reliability studies and it


28/36


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16t

F(t)

F(t),mu=0,sigma=0.35,mean=1.06,st.dev=0.38,skewness=1.13,excess=2.35,mode=0.88



F(t),mu=2.5,sigma=1,mean=20.09,st.dev=26.33,skewness=6.18,excess=110.94,mode=4.48

Figure 4.13: Lognormal Cumulative Function

is easy to use since the Burr reliability function can be written in closed form.The Burr distribution is similar to the lognormal distribution in the sensethat it can be thought of as representing a random variable which is the log ofa random variable having distribution similar to a normal distribution. Thatis, the Burr random variable can be thought of as the log of an extension ofthe logistic random variable. The Burr distribution also has a non-monotonefailure rate as does the lognormal distribution. But an advantage of theBurr family of distributions over the lognormal family is that the reliabilityfunction of the Burr family can be written in a simple explicit form and caneasily be computed without the use of tables. In addition, the log-logistic

distribution is becoming more used in reliability studies, especially in thebiological and medical sciences studies, and the log-logistic is a special caseof the general Burr distribution.

The Burr distribution is easier to handle with censored data than thelognormal because the cdf of the Burr can be written in closed form and


29/36


0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

t

h(t)

h(t),mu=0,sigma=0.35,mean=1.06,stdev=0.38,skewness=1.13,excess=2.35,mode=0.88



h(t),mu=2.5,sigma=1,mean=20.09,stdev=26.33,skewness=6.18,excess=110.94,mode=4.48

Figure 4.14: Lognormal Hazard Function

thus the probability statements and likelihoods for the Burr are more easilyviewed and manipulated.

The potential for the use of the Burr distribution in reliability is great(see Zimmer, Keats and Wang (1998) and Gupta, Gupta and Lvin (1996))and lies in several directions. The Burr distribution has been shown toapproximate many important distributions, including the normal, lognormal,gamma, logistic and extreme-value. See Rodriguez (1977). As the log-logisticbecomes more used, it can be seen that the Burr, being more general, canbecome an important alternative to the lognormal.

The original Burr distribution, proposed by Burr (1942) was a two-

parameter distribution, although when Burr used the distribution for approx-imating other distributions, he, by necessity, used four parameters. However,most references to the Burr distribution uses only the two-parameter version.In this text, the 4-parameter Burr distribution will presented in the chap-ter where the Burr distribution is discussed in more detail, that is, Chapter


30/36


6. In this section the 3-parameter Burr distribution will be outlined with

parameters c, k and S and has properties:

f(t) =kc

tS

c1 dtS

1 +tS

ck+1 t > 0

R(T) =

1 +

t

S

ck, S, c, k > 0

h(t) =kcS

tS

c11 +

tS

c , H(t) = k ln

1 +

t

S

c, TR = S(R

1k 1) 1c ,

E(T) = =S

1c

k 1c

c(k),

V(T) = 2 =2S2

2c

k 2c

c(k)

S1c

k 1c

c(k)

2,

if k is an ineger,

E(T) =

k 1 1c

. . .

1 1c

mc(k 1)! sin c, V(T) =

2

k 1 2c

. . .

1 2c

m2c(k 1)! sin 2c[E(T)]2,

The Burr density, cumulative and hazard function are illustrated in Fig-ures 4.15, 4.16 and 4.17, respectively and for c, k and S triples:(3, 2, 0.5), (4.8737, 6.15784, 1), (2.105, 5.0, 1.5267), (10, 5, 2).

Note that the Burr distribution with parameters (4.8737, 6.15784, 1) hasproperties that are essentially the same as the normal distribution. Thiswill be discussed further in Chapter 6 where the Burr approximations to thenormal, lognormal and Weibull will be illustrated.

The Burr density is unimodal ifc > 1 and L-shaped ifc 1. Ifc > 1, thehazard rate increases to a single maximum at mt = (c1) 1c , then decreases.The hazard function is decreasing when c 1. For values of c > 1 but near1, the hazard is almost constant after the maximum. When comparing theBurr hazard to the lognormal hazard, values of c > 1 are more interesting forthe comparison. Also, note that the rth moment for the Burr distributiononly exists if ck > r.


31/36


0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6

t

f(t)

f(t), c=3, k=2, S=0.5, mean=0.40, stdev=0.20,skewness=1.59, excess=7.81, mode=0.33

f(t), c=4.8737, k=6.15784, S=1, mean=0.64, stdev=0.16, skewness=0.00, excess=0.00, mode=0.65

f(t), c=2.105, k=5.0, S=1.5267, mean=0.68, stdev=.38, skewness=1.11, excess=2.36, mode=0.50

f(t), c=10, k=5, S=2, mean=1.64, st dev=0.21,skewness=-0.39, excess=0.40, mode=1.68

Figure 4.15: Burr Probability Density Function

EXAMPLE: 4.18 Suppose that a device will almost always fail before200 hours and the time (in hours) to failure of the device follows the Burrdistribution with parameters: c=3.33, k=4 and S=100. From a Burr table(See Zimmer and Burr (1963)), the mean and standard deviation for t/S arethe values: = 0.62384 and = 0.23406, thus the mean time to failure is62.384 hours with a standard deviation of 23.406 hours. It is of interest tocompute the probability that the device survives past 100 hours:

P{t > 100} = R(100) = 1

1 + ([100/100])3.33

4 =

1

24=

1

16= 0.0625.

It is also of interest to the practitioners to compute the probability that thedevice fails before the time + 3 :

P{t < +3} = P{t < 132.602} = 1 11 + ([132.602]/100)3.33

4 = 10.00623 = 0.9938


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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6

t

F(t)

F(t), c=3, k=2, S=0.5, mean=0.40, st dev=0.20,skewness=1.59, excess=7.81, mode=0.33

F(t), c=4.8737, k=6.15784, S=1, mean=0.64, stdev=0.16, skewness=0.00, excess=0.00,mode=0.65F(t), c=2.105, k=5.0, S=1.5267, mean=0.68, stdev=0.38, skewness=1.11, excess=2.36,mode=0.50F(t), c=10, k=5, S=2, mean=1.64, st dev=0.21,skewness=-0.39, excess=0.40, mode=1.68

Figure 4.16: Burr Cumulative Function

EXAMPLE: 4.19 Suppose that one has the above information and thatit is of interest to find the probability that the device will survive past 100hours, given that it has survived to 50 hours. This probability is easy tocompute under the Burr distribution assumption and is given by:

R(100|50) = P(T > 100|T > 50) =

1 +

50100

3.334

1 +100100

3.334 = 1.46116 = 0.091

4.12 Inverse Gaussian distributionAnother distribution that is used in reliability in more and more applicationsis the inverse Gaussian (or inverse Normal). The inverse Gaussian is usuallypresented as a two-parameter distribution with parameters which repre-


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4.12. INVERSE GAUSSIAN DISTRIBUTION 153

0

2

4

6

8

10

12

14

16

18

20

0 1 2 3 4 5 6

t

h(t)

h(t), c=3, k=2, S=0.5,mean=0.40, st dev=0.20,skewness=1.59, excess=7.81,mode=0.33h(t), c=4.8737, k=6.15784, S=1,mean=0.64, st dev=0.16,skewness=0.00, excess=0.00,mode=0.65h(t), c=2.105, k=5.0, S=1.5267,mean=0.68, st dev=0.38,skewness=1.11, excess=2.36,mode=0.50h(t), c=10, k=5, S=2, mean=1.64,st dev=0.21, skewness=-0.39,excess=0.40, mode=1.68

Figure 4.17: Burr Hazard Function

sents the mean, and which represents the shape parameter. It has a non-monotonic hazard but has an advantage that the hazard does not necessarilydecrease to zero as the time increases. The inverse Gaussian distribution is aunimodal distribution. Also, the inverse Gaussian is asymptotically normalas . For the inverse Gaussian distribution with parameters and :

f(t) =

2 t3

12

e

22 t(t)2

, t, > 0

R(t) = t

12

1t

e

2

t

12

1 + t ,

where represents the standard normal cdf, h(t) = f(t)R(t)

, E(T) = , V(T) =

2 = 3

,


34/36


3 =35

2, 4 = 15

7

3+3 6

2 , coefficient of skewness =

1 = 3

,

coefficient of excess = 3 = 15 , coefficient of variation=CV= ,mode=322 +

1 + 9

2

42

12

.

The inverse Gaussian density, cumulative and hazard function are illus-trated in Figure 4.23, Figure 4.24 and Figure 4.25, respectively and for and pairs: (1, 1), (2, 0.60), (1, 5), (3, 125).

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5t

f(t)

f(t), mu=1, lam=1, mean=1, stdev=1,skewness=3, excess=15, mode=0.30

f(t), mu=2, lam=0.6, mean=2, st dev=3.65,skewness=5.48, excess=50, mode=0.20

f(t), mu=1, lam=5, mean=1, st dev=0.45,skewness=1.34, excess=3, mode=0.74

f(t), mu=3, lam=125, mean=3, st dev=0.46,skewness=0.46, excess=0.36, mode-2.89

Figure 4.18: Inverse Gaussian Density Function

The mode of the hazard is between tmode and23 , and the asympotote

of the hazard as time increases is22 . Since the hazard function does not

decrease asymptotically to zero as does the lognormal, this is seen by someto be an advantage of the inverse Gaussian to the lognormal. The distributionof the sample mean from a sample of n independent inverse Gaussian randomvariables is also inverse Gaussian with parameters and n.


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4.12. INVERSE GAUSSIAN DISTRIBUTION 155

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t

F(t)

F(t), mu=1, lam=1, mean=1, stdev=1,skewness=3, excess=15, mode=0.30

F(t), mu=2, lam=0.6, mean=2, st dev=3.65,skewness=5.48, excess=50, mode=0.20

F(t), mu=1, lam=5, mean=1, st dev=0.45,skewness=1.34, excess=3, mode=0.74

F(t), mu=3, lam=125, mean=3, st dev=0.46,skewness=0.46, excess=0.36, mode-2.89

Figure 4.19: Inverse Gaussian Distribution Function

EXAMPLE: 4.20 It is assumed that the time to failure of a particulardevice follows an inverse Gaussian distribution with parameters = 4 hoursand = 16. In this case, the mean is 4 hours and the standard deviation is2 hours. The sample mean T from a sample of 10 devices also has an inverseGaussian distribution with mean T = 4 hours, T = 160 and standarddeviation

0.4 = 0.63. For this example, the probability that the mean of a

sample of 10 devices is greater than 3 hours is:

P

{T > 3

}= R(3) = (1.826)

e8 (

12.78) = 0.966

0 = 0.966.

EXAMPLE: 4.21 Suppose it is known that the time to failure of a devicehas an inverse Gaussian distribution with mean 50 hours and a standarddeviation of 20 hours. In this case, one knows that = 50 and one cancompute that:


36/36


0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t

h(t)

h(t), mu=1, lam=1, mean=1, st dev=1,skewness=3,excess=15, mode=0.30

h(t), mu=2, lam=0.6, mean=2, st dev=3.65,skewness=5.48, excess=50, mode=0.20

h(t), mu=1, lam=5, mean=1, st dev=0.45,skewness=1.34, excess=3, mode=0.74

h(t), mu=3, lam=125, m ean=3, st dev=0.46,skewness=0.46, excess=0.36, mode-2.89

Figure 4.20: Inverse Gaussian Hazard Function

400 =125000

or = 312.5

If one is interested in the probability that the device will survive until +2.5 = 100 hours, one finds that:

R(100) = 312.5

100 12

1100

50 e 2(312.5)50 312.5100

12

1100

50

R(100) = 0.0386 0.0153 = 0.0233

Chap4- 1 Time to Failure

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Transcript of Chap4- 1 Time to Failure