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Useful DefinitionsTable of Contents:
This section presents a number of mathematical definitions considered useful for performing various types of applied reliability studies. Cumulative Distribution FunctionFor continuous random variables, this is expressed by [7]
where у is a continuous random variable. f(y) is the probability density function. F(t) is the cumulative distribution function. For t = oo, Equation (2.19) yields
It simply means that the total area under the probability density function curve is equal to unity. It is to be noted that normally in reliability studies, Equation (2.19) is simply written as Example 2.4 Assume that the probability (i.e., failure) density function of a transportation system is
where t is a continuous random variable (i.e., time). A is the transportation system failure rate. fit) is the probability density function (normally, in the area of reliability engineering, it is referred to as the failure density function). Obtain an expression for the transportation system cumulative distribution function by using Equation (2.21). By substituting Equation (2.22) into Equation (2.21), we obtain
Thus, Equation (2.23) is the expression for the transportation system cumulative distribution function. Probability Density FunctionFor a continuous random variable, the probability density function is expressed by [7] where fit) is the probability density function. F(t) is the cumulative distribution function. Example 2.5 With the aid of Equation (2.23), prove that Equation (2.22) is the probability density function. By inserting Equation (2.23) into Equation (2.24), we obtain
Equations (2.22) and (2.25) are identical. Expected ValueThe expected value of a continuous random variable is expressed by [7] where E(t) is the expected value (i.e., mean value) of the continuous random variable t. Similarly, the expected value, E(t), of a discrete random variable t is expressed by
where n is the number of discrete values of the random variable t. Example 2.6 Find the mean value (i.e., expected value) of the probability (failure) density function defined by Equation (2.22). By inserting Equation (2.22) into Equation (2.26), we obtain Thus, the mean value of the probability (failure) density function expressed by Equation (2.22) is given by Equation (2.28). Laplace TransformThe Laplace transform (named after a French mathematician, PierreSimon Laplace (17491827) of a function, say fit), is defined by [1,9,10].
where t is a variable. 5 is the Laplace transform variable. f(s) is the Laplace transform of function,/(f) Example 2.7 Obtain the Laplace transform of the following function: where в is a constant. By inserting Equation (2.30) into Equation (2.29), we obtain
Thus, Equation (2.31) is the Laplace transform of Equation (2.30). Laplace transforms of some commonly occurring functions used in applied reliabilityrelated analysis studies are presented in Table 2.1 [9,10]. TABLE 2.1 Laplace transforms of some functions.
Laplace Transform: FinalValue TheoremIf the following limits exist, then the finalvalue theorem may be expressed as
Example 2.8 Prove by using the following equation that the lefthand side of Equation (2.32) is equal to its righthand side:
where Aj and A, are the constants. By inserting Equation (2.33) into the lefthand side of Equation (2.32), we obtain
Using Table 2.1, we get the following Laplace transforms of Equation (2.33):
By substituting Equation (2.35) into the righthand side of Equation (2.32), we obtain:
The righthand sides of Equations (2.34) and (2.36) are identical. Thus, it proves that the lefthand side of Equation (2.32) is equal to its righthand side. Probability DistributionsThis section presents a number of probability distributions considered useful for performing various types of studies in the area of applied reliability [11]. Binomial DistributionThis discrete random variable probability distribution is used in circumstances where one is concerned with the probabilities of the outcome, such as the number of occurrences (e.g., failures) in a sequence of, say, n trials. More clearly, each trial has two possible outcomes (e.g., success or failure), but the probability of each trial remains constant or unchanged. This distribution is also known as the Bernoulli distribution, named after its founder Jakob Bernoulli (16541705) [1]. The binomial probability density function, /(y), is defined by
where у is the number of nonoccurrences (e.g., failures) in n trials. p is the single trial probability of occurrence (e.g., success). q is the single trial probability of nonoccurrence (e.g., failure). The cumulative distribution function is given by where F(y) is the cumulative distribution function or the probability of у or fewer nonoccurrences (e.g., failures) in n trials. Using Equations (2.27) and (2.37), we get the mean or the expected value of the distribution as Exponential DistributionThis is one of the simplest continuous random variable probability distributions that is widely used in the industrial sector, particularly in performing reliability studies. The probability density function of the distribution is defined by [12]
where t is the time t (i.e., a continuous random variable). a is the distribution parameter. fit) is the probability density function. By inserting Equation (2.40) into Equation (2.21), we obtain the following equation for the cumulative distribution function:
Using Equations (2.26) and (2.40), we obtain the following expression for the distribution mean value (i.e., expected value): where m is the mean value. Example 2.9 Assume that the mean time to failure of a transportation system is 1500 hours and its times to failure are exponentially distributed. Calculate the transportation system’s probability of failure during an 800 hours mission by using Equations (2.41) and (2.42). By inserting the specified data value into Equation (2.42), we obtain
By substituting the calculated and the specified data values into Equation (2.41), we get
Thus, the transportation system’s probability of failure during the 800 hours mission is 0.4133. Rayleigh DistributionThis continuous random variable probability distribution is named after its founder, John Rayleigh (18421919) [1]. The probability density function of the distribution is defined by
where fi is the distribution parameter. By substituting Equation (2.43) into Equation (2.21), we obtain the following equation for the cumulative distribution function: By inserting Equation (2.43) into Equation (2.26), we obtain the following equation for the distribution mean value:
where Г(.) is the gamma function and is defined by Weibull DistributionThis continuous random variable probability distribution is named after Walliodi Weibull, a Swedish mechanical engineering professor, who developed it in the early 1950s [13]. The distribution probability density function is expressed by
where b and ц are the distribution shape and scale parameters, respectively. By inserting Equation (2.47) into Equation (2.21), we obtain the following equation for the cumulative distribution function:
By substituting Equation (2.47) into Equation (2.26), we obtain the following equation for the distribution mean value (expected value):
It is to be noted that exponential and Rayleigh distributions are the special cases of this distribution for b = 1 and b = 2, respectively. Bathtub Hazard Rate Curve DistributionThe bathtubshape hazard rate curve is the basis for reliability studies. This continuous random variable probability distribution can represent bathtubshape, increasing, and decreasing hazard rates. This distribution was developed in 1981 [14], and in the published literature by other authors around the world, it is generally referred to as the Dhillon distribution/ law/model [1534]. The probability density function of the distribution is expressed by [14] where fi and b are the distribution scale and shape parameters, respectively. By substituting Equation (2.50) into Equation (2.21), we obtain the following equation for the cumulative distribution function: It is to be noted that this probability distribution for b = 0.5 gives the bathtubshaped hazard rate curve, and for b = 1 it gives the extreme value probability distribution. More specifically, the extreme value probability distribution is the special case of this probability distribution at b = 1. 
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