Exponential distributions

The exponential distribution is a continuous random variable distribution that is widely used in the industrial sector, particularly in performing reliability studies [11]. The probability density function of fhe distribution is defined by 

By inserting Equation (2.40) into Equation (2.22), we get the following expression for the cumulative distribution function  

Using Equations (2.40) and (2.25), we obtain the following expression for the distribution mean value  

This is probably the most widely used probability distribution in safety and reliability studies. The probability density function of the distribution is defined by 

X = the distribution parameter. In human reliability studies, it is known as the human error rate. t = time. 

This is one of the simplest continuous random variable distribution frequently used in the industry, particularly in performing reliability studies because many engineering items exhibit a constant hazard rate during their useful life [14]. In addition, it is relatively easy to handle in performing reliability analysis-related studies. 

Assume that the mean time to failure of a system used in the oil and gas industry is 1500 h. Calculate the probability of failure of the system during a 700-h mission with the aid of Equations 2.40 and 2.39. 

By inserting the specified data value into Equation 2.40, we obtain 

The approximations in Fqs. (9.24) and (9.25) result from expanding the exponential function in a Taylor series and tmncating it with the second term. This 

the average lifetime is equal to the inverse of its failure rate in case the failure rate is constant. The important property of the exponential distribution is that the probability of the failure of a component in the time interval [t, t + At] does not depend on its preceding operating time but only on the value of its failure rate X and the duration of the time interval At. This is shown by inserting Eq. (9.24) in Eq. (9.25). Furthermore, it can be proved that the exponential distribution is the only one with a constant failure rate [30]. 

A manual valve fails closed (unwantedly in closed position). The applicable failure rate is = 0.3 x 10 a Calculate  

The valve with a better design is to have a failure probability of F(t = 10,000 h) = 1.5 X 10 Using Eq. (9.25) this gives 

1 Poisson Distribution We now calculate the average molecular weights for some typical distributions of a linear polymer. The examples are a Poisson distribution and an exponential distribution in a discrete space and a log-normal distribution in a continuous space. 

In the Poisson distribution, the number fraction of the tth component is given by 

The constant a is the number-average degree of polymerization. Likewise, 

Thus = (1 + a)Mi. Note that M and differ only by Mj. Therefore, 

Given that the Mean Time Between Failure for an item is 10,000 hours, calculate the failure probabilities of the item at f = 0, 10,000 and 100,000 hours if failures foUow an exponential distribution. 

From the above, it can be seen that at t = 0 the item does not fail and after a considerable time it fails. 

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Analysis of tlie global statistics of protein sequences has recently allowed light to be shed on anotlier puzzle, tliat of tlie origin of extant sequences [170]. One proposition is tliat proteins evolved from random amino acid chains, which predict tliat tlieir length distribution is a combination of the exponentially distributed random variable giving tlie intervals between start and stop codons, and tlie probability tliat a given sequence can fold up to fonii a compact... 

In order to complete an assessment of risk, a probabiUty must be determined. The easiest method for representing failure probabiUty of a device is an exponential distribution (2). 

There are other distributions available to represent equipment failures (10), but these require more detailed information on the device and a more detailed analysis. Eor most situations the exponential distribution suffices. 

Eigure 8 compares the failure probabiUty and reflabiUty functions for an exponential distribution. Whereas the reflabiUty of the device is initially unity, it falls off exponentially with time and asymptotically approaches zero. The failure probabiUty, on the other hand, does the reverse. Thus new devices start Life with high reflabiUty and end with a high failure probabiUty. 

A considerable assumption in the exponential distribution is the assumption of a constant failure rate. Real devices demonstrate a failure rate curve more like that shown in Eigure 9. Eor a new device, the failure rate is initially high owing to manufacturing defects, material defects, etc. This period is called infant mortaUty. EoUowing this is a period of relatively constant failure rate. This is the period during which the exponential distribution is most apphcable. EinaHy, as the device ages, the failure rate eventually increases. 

In usiag these failure rates an exponential distribution for time to failure was assumed. Such an assumption should be made with caution. Parallel Systems. A parallel (or redundant) system is not considered to be ia a failed state unless all subsystems have failed. The system rehabihty is calculated as... 

For example, if the time to failure is given as an exponential distribution, then... 

The exponential distribution has proved to be a reasonable failure model for electronic equipment (8—13). Since the field of reUabiUty emerged, owing to problems encountered with military electronics during World War II, exponential distribution has had considerable attention and apphcation. However, like any failure model, it has limitations which should be well understood. 

Basic Statistical Properties. The PDF for an exponentially distributed random variable t is given by... 

On complex systems, which are repaired as they fail and placed back in service, the time between system failures can be reasonably well modeled by the exponential distribution (14,15). 

From equation 8 it was shown that the chance of surviving the mean life was 36.8% for the exponential distribution. However, this fact must be used with some degree of rationaHty in appHcations. For example, in the above situation the longest surviving MPU that was observed survived for 291.9 hours. The failure rate beyond this time is not known. What was observed was only a failure rate of A = 1.732 x lO " failures per hour over approximately 292 hours of operation. In order to make predictions beyond this time, it must be assumed that the failure rate does not increase because of wearout and... 

The Nonzero Minimum-Life Case. In many situations, no failures are observed during an initial period of time. For example, when testing engine bearings for fatigue life no failures are expected for a long initial period. Some corrosion processes also have this characteristic. In the foUowing it is assumed that the failure pattern can be reasonably weU approximated by an exponential distribution. 

In the specific case of an MSMPR exponential distribution, the fourth moment of the distribution may be calculated as... 

This linear exponential distribution has also been used to represent dynamic fragmentation data (Grady and Kipp, 1985). 

Unfortunately, fragmentation by construction does not appear to be independent of the random construction algorithm. An alternative method of successive segmentation (Grady and Kipp, 1985) of the surface, as illustrated in Fig. 8.24 leads to a fragment distribution which agrees well with a linear exponential distribution ((8.59)) and differs significantly from the Mott distribution ((8.58)). 

Figure 8.32. Cumulative number distribution data for lead impact experiment and comparison with bilinear exponential distribution.

Figure 2.5-2 depicts the force of mortality as a bathtub curve for the life-death history of a component without repair. The reasons for the near universal use of the constant X exponential distribution (which only applies to the mid-life region) are mathematical convenience, inherent truth (equation 2.5-19), the use of repair to keep components out of the wearout region, startup testing to eliminate infant mortality, and detailed data to support a time-dependent X. 

WASH-1400 treated the probability of failure with time as being exponentially distributed with constant X. It treated X itself as being lognormally distributed. There are better a priori reasons... 

Suppose N identical components with an exponential distribution (constant A) are on test the test is terminated at T, with M failures. What is the confidence that A is the true failure rate. 

A generating function is defined by equation 2.5-47. To illustrate it use. Table 2.> 2 gives the generating function for an exponential distribution as -A/(0-X). Each moment i.s obtained by successive differentiations. Equation 2.5-48 shows how to obtain the first moment. By taking the limit of higher derivatives higher moments are found. 

Similarly, prior information distributed as a gamma lunctiou with an exponentially distributed update gives a posterior that also is gamma distributed. 

A type of time dependence that is available in most codes evaluates the exponential distribution at specified times. This is the constant failure rate - constant repair rate approximation ( Section 2.5.2). This may not be realistic as indicated by Figure 2.5-2 in which the failure rate is not constant. Furthermore, Lapides (1976) shows that repair rates are not constant but in many casc. appear to be lognormally distributed. 

Solution of equation 6.4 is the now familiar exponential distribution which is linear in a semi-logarithmic plot In n L) versus L (Figure 6.11a)... 

Statistical Methods for Nonelectronic Reliability, Reliability Specifications, Special Application Methods for Reliability Prediction Part Failure Characteristics, and Reliability Demonstration Tests. Data is located in section 5.0 on Part Failure Characteristics. This section describes the results of the statistical analyses of failure data from more than 250 distinct nonelectronic parts collected from recent commercial and military projects. This data was collected in-house (from operations and maintenance reports) and from industry wide sources. Tables, alphabetized by part class/ part type, are presented for easy reference to part failure rates assuminng that the part lives are exponentially distributed (as in previous editions of this notebook, the majority of data available included total operating time, and total number of failures only). For parts for which the actual life times for each part under test were included in the database, further tables are presented which describe the results of testing the fit of the exponential and Weibull distributions. 

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