8.
Assessing Product Reliability
8.1. Introduction 8.1.6. What are the basic lifetime distribution models used for nonrepairable 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: $$ \begin{array}{ll} \mbox{PDF:} & f(t, \lambda) = \lambda e^{\lambda t} \\ & \\ \mbox{CDF:} & F(t) = 1e^{\lambda t} \\ & \\ \mbox{Reliability:} & R(t) = e^{\lambda t} \\ & \\ \mbox{Failure Rate:} & h(t) = \lambda \\ & \\ \mbox{Mean:} & \frac{1}{\lambda} \\ & \\ \mbox{Median:} & \frac{\mbox{ln} 2}{\lambda} \cong \frac{0.693}{\lambda} \\ & \\ \mbox{Variance:} & \frac{1}{\lambda^2} \end{array} $$ Note that the failure rate reduces to the constant \(\lambda\) 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 = \(1/\lambda\). The cumulative hazard function for the exponential is just the integral of the failure rate or \(H(t) = \lambda t\). 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 (\(\lambda\) = 0.01)
is shown below.
We can calculate the exponential PDF and CDF at 100 hours for the case where \(\lambda\) = 0.01. The PDF value is 0.0037 and the CDF value is 0.6321. Functions for computing exponential PDF values, CDF values, and for producing probability plots, are found in both Dataplot code and R code. 