Bathtub Hazard Rate Curve Distribution

This continuous random variable probability distribution was developed in 1981 [16]. In the published literature by other authors around the world, it is 

By substituting Equation 2.48 into Equation 2.19, we obtain the following equation for the cumulative distribution function  

For a = 0.5, this probability distribution gives the bathtub-shape hazard rate curve, and for a = 1, it gives the extreme value probability distribution. In other words, the extreme value probability distribution is the special case of this probability distribution at a = 1. 

6 Solving First-Order Linear Differential Equations Using Laplace Transforms 

Usually, Laplace transforms are used in finding solutions to first-order differential equations in reliability and safety analysis-related studies of systems used in the oil and gas industry. 

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Write down probability density function for the bathtub hazard rate curve distribution. 

The following equation obtained from a probability distribution presented in Chapter 2 can be used to represent the bathtub hazard rate curve, shown in Figure 3.5, mathematically [23] ... 

This probability distribution can be used to represent the entire bathtub hazard rate curve presented in Chapter 3. The probability density function of the distribution is defined by [13,14]... 

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