Useful Definitions

This section presents a number of mathematical definitions considered useful for performing various types of applied reliability studies.

Cumulative Distribution Function

For 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 Function

For 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 Value

The 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 Transform

The Laplace transform (named after a French mathematician, Pierre-Simon Laplace (1749-1827) 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 reliability-related analysis studies are presented in Table 2.1 [9,10].

TABLE 2.1

Laplace transforms of some functions.

f(t)

f(s)

Laplace Transform: Final-Value Theorem

If the following limits exist, then the final-value theorem may be expressed as

Example 2.8

Prove by using the following equation that the left-hand side of Equation (2.32) is equal to its right-hand side:

where

Aj and A, are the constants.

By inserting Equation (2.33) into the left-hand 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 right-hand side of Equation (2.32), we obtain:

The right-hand sides of Equations (2.34) and (2.36) are identical. Thus, it proves that the left-hand side of Equation (2.32) is equal to its right-hand side.

Probability Distributions

This section presents a number of probability distributions considered useful for performing various types of studies in the area of applied reliability [11].

Binomial Distribution

This 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 (1654-1705) [1]. The binomial probability density function, /(y), is defined by

where

у is the number of non-occurrences (e.g., failures) in n trials. p is the single trial probability of occurrence (e.g., success). q is the single trial probability of non-occurrence (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 Distribution

This 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 Distribution

This continuous random variable probability distribution is named after its founder, John Rayleigh (1842-1919) [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 Distribution

This 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 Distribution

The bathtub-shape hazard rate curve is the basis for reliability studies. This continuous random variable probability distribution can represent bathtub-shape, 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 [15-34].

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 bathtub-shaped 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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