# Chapter 10

# Elements of Reliability Theory

**The focus of this chapter is a discussion of the basic concepts of reliability theory. The topics covered are**:

- The reliability function and the hazard rate function
- Derivation of an appropriate form for the life distribution using the hazard rate function
- Estimation and testing of hypothesis for parameters of different life distributions
- Probability plots and plots of the life distribution function and hazard rate function using MINITAB, EXCEL, and JMP for censored and non-censored data

**Learning Outcomes**

After studying this chapter, the reader will be able to

- Determine the reliability and hazard rate functions for different life probability models.
- Fit different models such as exponential, normal, lognormal, and Weibull to a censored or non-censored data set.
- Determine estimates of parameters of different life probability models.
- Perform testing of hypotheses concerning various parameters of life probability models.
- Use various statistical packages to construct survival plots using censored and non-censored data.

## 10.1 The Reliability Function

Let *T* represent the random variable equal to the time to failure of an item and let *f*(*t*) be its probability density function, where *T* ≥ 0. Then we define the *reliability function R*(*t*) at time *t* as

It follows from (10.1.1) that and *R*(0) = *R*(∞) = 0. Note that the reliability function *R*(t) is a non-increasing ...

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