Template:Exponential Distribution Definition

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:

[math]\displaystyle{ \begin{align} f(t)=\lambda e^{-\lambda (t-\gamma)} \end{align}\,\! }[/math]

Where [math]\displaystyle{ \lambda\,\! }[/math] is the constant failure rate in failures per unit of measurement (e.g., failures per hour, per cycle, etc.) and [math]\displaystyle{ \gamma\,\! }[/math] is the location parameter. In addition, [math]\displaystyle{ \lambda =\tfrac{1}{m}\,\! }[/math], where [math]\displaystyle{ {m}\,\! }[/math] is the mean time between failures (or to failure).

If the location parameter, [math]\displaystyle{ \gamma\,\! }[/math], is assumed to be zero, then the distribution becomes the 1-parameter exponential or:

[math]\displaystyle{ \begin{align} f(t)=\lambda e^{-\lambda t} \end{align}\,\! }[/math]

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