# Life Distributions Notes

# Life Distributions Notes

We use the term *life distributions* to describe the collection of statistical probability distributions that we use in reliability engineering and life data analysis. A statistical distribution is fully described by its *pdf* (or probability density function). In the previous sections, we used the definition of the *pdf* to show how all other functions most commonly used in reliability engineering and life data analysis can be derived; namely, the reliability function, failure rate function, mean time function and median life function, etc. All of these can be determined directly from the *pdf *definition, or . Different distributions exist, such as the normal, exponential, etc., and each one of them has a predefined form of *lifetime distributions*. One of the simplest and most commonly used distributions (and often erroneously overused due to its simplicity) is the exponential distribution. The *pdf* of the exponential distribution is mathematically defined as:

In this definition, note that *parameter* of the distribution. Depending on the value of

Given the mathematical representation of a distribution, we can also derive all of the functions needed for life data analysis, which again will depend only on the value of *pdf* is given by:

Thus, the exponential reliability function can be derived as:

The exponential failure rate function is:

The exponential mean-time-to-failure (MTTF) is given by:

This exact same methodology can be applied to any distribution given its *pdf*, with various degrees of difficulty depending on the complexity of

## Parameter Types

Distributions can have any number of parameters. Do note that as the number of parameters increases, so does the amount of data required for a proper fit. In general, the lifetime distributions used for reliability and life data analysis are usually limited to a maximum of three parameters. These three parameters are usually known as the *scale parameter*, the *shape parameter* and the *location parameter*.

**Scale Parameter** The scale parameter is the most common type of parameter. All distributions in this reference have a scale parameter. In the case of one-parameter distributions, the sole parameter is the scale parameter. The scale parameter defines where the bulk of the distribution lies, or how stretched out the distribution is. In the case of the normal distribution, the scale parameter is the standard deviation.

**Shape Parameter** The shape parameter, as the name implies, helps define the shape of a distribution. Some distributions, such as the exponential or normal, do not have a shape parameter since they have a predefined shape that does not change. In the case of the normal distribution, the shape is always the familiar bell shape. The effect of the shape parameter on a distribution is reflected in the shapes of the *pdf*, the reliability function and the failure rate function.

**Location Parameter** The location parameter is used to shift a distribution in one direction or another. The location parameter, usually denoted as

This means that the inclusion of a location parameter for a distribution whose domain is normally

## Most Commonly Used Distributions

There are many different lifetime distributions that can be used to model reliability data. Leemis presents a good overview of many of these distributions. In this reference, we will concentrate on the most commonly used and most widely applicable distributions for life data analysis, as outlined in the following sections.

### The Exponential Distribution

The exponential distribution is commonly used for components or systems exhibiting a *constant failure rate*. Due to its simplicity, it has been widely employed, even in cases where it doesn’t apply. In its most general case, the 2-parameter exponential distribution is defined by:

Where

If the location parameter,

For a detailed discussion of this distribution, see The Exponential Distribution.

### The Weibull Distribution

The Weibull distribution is a general purpose reliability distribution used to model material strength, times-to-failure of electronic and mechanical components, equipment or systems. In its most general case, the 3-parameter Weibull *pdf* is defined by:

where

If the location parameter,

One additional form is the 1-parameter Weibull distribution, which assumes that the location parameter,

For a detailed discussion of this distribution, see The Weibull Distribution.

#### Bayesian-Weibull Analysis

Another approach is the Weibull-Bayesian analysis method, which assumes that the analyst has some prior knowledge about the distribution of the shape parameter of the Weibull distribution (beta). There are many practical applications for this model, particularly when dealing with small sample sizes and/or when some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data or physics-of-failure.

Note that this is not the same as the so called “WeiBayes model,” which is really a one-parameter Weibull distribution that assumes a fixed value (constant) for the shape parameter and solves for the scale parameter. The Bayesian-Weibull feature in Weibull++ is actually a true Bayesian model and offers an alternative to the one-parameter Weibull by including the variation and uncertainty that is present in the prior estimation of the shape parameter.

This analysis method and its characteristics are presented in detail in Bayesian-Weibull Analysis.

### The Normal Distribution

The normal distribution is commonly used for general reliability analysis, times-to-failure of simple electronic and mechanical components, equipment or systems. The *pdf* of the normal distribution is given by:

where

The normal distribution and its characteristics are presented in The Normal Distribution.

### The Lognormal Distribution

The lognormal distribution is commonly used for general reliability analysis, cycles-to-failure in fatigue, material strengths and loading variables in probabilistic design. When the natural logarithms of the times-to-failure are normally distributed, then we say that the data follow the lognormal distribution.

The *pdf* of the lognormal distribution is given by:

where

For a detailed discussion of this distribution, see The Lognormal Distribution.