Assessing Product Reliability
8.1.6. What are the basic lifetime distribution models used for non-repairable populations?
|All the key formulas for using the exponential model||
Formulas and Plots
The exponential model, with only one unknown parameter, is the simplest of all life distribution models. The key equations for the exponential are shown below:
Note that the failure rate reduces to the constant λ for any time. The exponential distribution is the only distribution to have a constant failure rate. Also, another name for the exponential mean is the Mean Time To Fail or MTTF and we have MTTF = 1/λ.
The cumulative hazard function for the exponential is just the integral of the failure rate or H(t) = λt.
The PDF for the exponential has the familiar shape shown below.
|The Exponential distribution 'shape'|
|The Exponential CDF||
Below is an example of typical exponential lifetime data displayed in Histogram form with corresponding exponential PDF drawn through the histogram.
|Histogram of Exponential Data|
|The Exponential models the flat portion of the "bathtub" curve - where most systems spend most of their 'lives'||Uses
of the Exponential Distribution Model
|Exponential probability plot||
We can generate a probability plot
of normalized exponential data, so that a perfect exponential fit is a
diagonal line with slope 1. The probability plot for 100 normalized random exponential
observations (λ = 0.01) is shown below.
We can calculate the exponential PDF and CDF at 100 hours for the case where λ = 0.01. The PDF value is 0.0037 and the CDF value is 0.6321.