Distributions

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. The reliability function, failure rate function, mean time function and median life function can be determined directly from the pdf definition, or f(t). Different distributions exist, such as the normal, exponential etc., and each has a predefined f(t) which can be found in most references.

These distributions were formulated by statisticians, mathematicians and/or engineers to mathematically model or represent certain behavior. For example, the Weibull distribution was formulated by Waloddi Weibull and thus it bears his name. Some distributions tend to better represent life data and are most commonly called lifetime distributions.

The exponential distribution is a very commonly used distribution in reliability engineering. Due to its simplicity, it has been widely employed even in cases to which it does not apply. The pdf of the exponential distribution is mathematically defined as:

In this definition, note that t is our random variable, which represents time, and the Greek letter λ (lambda) represents what is commonly referred to as the parameter of the distribution. For any distribution, the parameter or parameters of the distribution are estimated (obtained) from the data. For example, in the case of the most well known distribution, namely the normal distribution:

where the mean, μ and the standard deviation, σ, are its parameters. Both of these parameters are estimated from the data, i.e. the mean and standard deviation of the data. Once these parameters have been estimated, our function f(t) is fully defined and we can obtain any value for f(t) given any value of t.

Given the mathematical representation of a distribution, we can also derive all of the functions needed for life data analysis, which again will only depend on the value of t after the value(s) of the distribution parameter(s) have been estimated from data.

For example, we know that the exponential distribution pdf is given by:

Thus the reliability function can be derived to be:

The failure rate function is given by:

The mean time to/before failure (MTTF/MTBF) is given by:

Exactly the same methodology can be applied to any distribution given its pdf with various degrees of difficulty depending on the complexity of f(t).

Most Commonly Used Distributions

There are many different lifetime distributions that can be used. ReliaSoft [34] presents a thorough overview of lifetime distributions. Leemis [23] and others also present good overviews of many of these distributions. The three distributions used in ALTA, the 1-parameter exponential, 2-parameter Weibull and the lognormal, are presented in greater detail in the Life Distributions chapter.

Go to Weibull.com
Go to ReliaSoft.com