Reliability HotWire: eMagazine for the Reliability Professional
Reliability HotWire

Issue 13, March 2002

Defining Distributions

Part 2: Deriving Reliability Functions

In last month's Reliability Basics, we took a look at the mathematical function that is the basis for many useful reliability and life data metrics, the probability density function, or pdf. The pdf function is essentially a continuous form of a histogram, which is a bar chart that characterizes the relative frequency of an occurrence. In this month's article, we will look at how the pdf is used to derive other frequently-used life data metrics. 

The probability that the random variable X takes on a value in the interval [a, b] is the area under the pdf from a to b, or:

In the terms of life data analysis, the above equation describes the probability of a failure occurring between two different points in time. However, this sort of information is required infrequently at best. What would be of greater interest would be the probability of a failure occurring before or after a certain time. If a were equal to zero, the above equation would return the probability of a failure occurring before time b. This introduces the concept of the cumulative distribution function, or cdf. As the name implies, the cdf measures the cumulative probability of a failure occurring before a certain time. The equation for the cdf is given by:

Cumulative distribution function equation

Note that the lower limit is given as zero or negative infinity. The value of the lower limit varies from distribution to distribution. For example, the normal or Gaussian distribution has a lower limit of negative infinity, while the Weibull distribution has a lower limit of zero. Note that the value of the cdf always approaches 1 as time approaches infinity. This is because the area under the curve of the pdf is always equal to 1, and the cdf is essentially measuring the area under the pdf curve from zero to the point of interest. The following graphic shows the relationship between the pdf and the cdf.

Plot comparing pdf to cdf

Looking at the definition of the cdf, it should be apparent that this function would have a direct application to life data analysis. This function returns the probability of a failure occurring before a certain time. Another useful function is the one that provides the probability of a failure occurring after a certain time. Note that the cdf measures the area under the pdf curve up to a given time, and that the area under the pdf curve is always equal to 1. Given these concepts, subtracting the cdf from 1 would result in the probability of a failure occurring after a given time. This is the widely-used reliability function. Accordingly, the cdf is also known as the unreliability function, and is represented by the function Q(T)

F(t) = Q(t) = 1 - R(t)

where R(t) is the reliability function. The reliability function can then be related to the pdf in the following manner:

Relationship between cdf and reliability function

Another function that can be derived from the pdf is the failure rate function. The failure rate function (also known as the hazard rate function) gives the instantaneous failure frequency based on accumulated age. Note that the failure rate is constant only for the exponential distribution; in most cases the failure rate changes with time. The failure rate function is defined by:

Failure rate function definition

Thus, the failure rate function is simply the pdf function divided by the reliability function, and has the units of failure per unit time among surviving parts, e.g. one failure per month.

(Note that the gamma (γ) symbol that appears in the lower bound of some of the previous equations represents the location parameter that is found in some distributions. This is a parameter that effectively shifts the entire distribution by a value equal to the parameter value. This can be visualized as sliding the pdf curve along the x-axis of the plot.)

The mean life, or MTTF, is another widely-used function that can be derived directly from the pdf. The arithmetic mean or expected value is defined by:

Mean life function definition

Another function that can be derived from the pdf is that of the BX life. Originated in the ball bearing industry (hence the B), this metric provides the time at which X% of the population will fail, or the time for a corresponding unreliability of X%. For example, the B10 life is the time at which 10% of the population will fail. This metric is determined by solving the following equation for BX:

BX function definition

Where X is the fraction failing or unreliability expressed as a decimal.

The probability density function is the most important mathematical function in life data analysis. This single function fully characterizes the distribution it describes. The area under the pdf curve between two defined points on the x-axis gives the probability of an event occurring between those two points. This function can be used to derive other functions that are important to life data analysis, including the unreliability function, the reliability function, the failure rate function, the mean life function, and the BX function.


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