Exponential Distribution Functions

This article also appears in the Life Data Analysis Reference and Accelerated Life Testing Data Analysis Reference books.

The Mean or MTTF

The mean, [math]\displaystyle{ \overline{T}, }[/math] or mean time to failure (MTTF) is given by:

[math]\displaystyle{ \begin{align} \bar{T}= & \int_{\gamma }^{\infty }t\cdot f(t)dt \\ = & \int_{\gamma }^{\infty }t\cdot \lambda \cdot {{e}^{-\lambda t}}dt \\ = & \gamma +\frac{1}{\lambda }=m \end{align} }[/math]

Note that when [math]\displaystyle{ \gamma =0\,\! }[/math], the MTTF is the inverse of the exponential distribution's constant failure rate. This is only true for the exponential distribution. Most other distributions do not have a constant failure rate. Consequently, the inverse relationship between failure rate and MTTF does not hold for these other distributions.

The Median

The median, [math]\displaystyle{ \breve{T}, }[/math] is:

[math]\displaystyle{ \breve{T}=\gamma +\frac{1}{\lambda}\cdot 0.693 }[/math]

The Mode

The mode, [math]\displaystyle{ \tilde{T}, }[/math] is:

[math]\displaystyle{ \tilde{T}=\gamma }[/math]

The Standard Deviation

The standard deviation, [math]\displaystyle{ {\sigma }_{T}\,\! }[/math], is:

[math]\displaystyle{ {\sigma}_{T}=\frac{1}{\lambda }=m }[/math]

The Exponential Reliability Function

The equation for the 2-parameter exponential cumulative density function, or [math]\displaystyle{ cdf, }[/math] is given by:

[math]\displaystyle{ \begin{align} F(t)=Q(t)=1-{{e}^{-\lambda (t-\gamma )}} \end{align} }[/math]

Recalling that the reliability function of a distribution is simply one minus the [math]\displaystyle{ cdf }[/math], the reliability function of the 2-parameter exponential distribution is given by:

[math]\displaystyle{ R(t)=1-Q(t)=1-\int_{0}^{t-\gamma }f(x)dx }[/math]

[math]\displaystyle{ R(t)=1-\int_{0}^{t-\gamma }\lambda {{e}^{-\lambda x}}dx={{e}^{-\lambda (t-\gamma )}} }[/math]

The 1-parameter exponential reliability function is given by:

[math]\displaystyle{ R(t)={{e}^{-\lambda t}}={{e}^{-\tfrac{t}{m}}} }[/math]

The Exponential Conditional Reliability Function

The exponential conditional reliability equation gives the reliability for a mission of [math]\displaystyle{ t }[/math] duration, having already successfully accumulated [math]\displaystyle{ T }[/math] hours of operation up to the start of this new mission. The exponential conditional reliability function is:

[math]\displaystyle{ R(t|T)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-\lambda (T+t-\gamma )}}}{{{e}^{-\lambda (T-\gamma )}}}={{e}^{-\lambda t}} }[/math]

which says that the reliability for a mission of [math]\displaystyle{ t }[/math] duration undertaken after the component or equipment has already accumulated [math]\displaystyle{ T }[/math] hours of operation from age zero is only a function of the mission duration, and not a function of the age at the beginning of the mission. This is referred to as the memoryless property.

The Exponential Reliable Life Function

The reliable life, or the mission duration for a desired reliability goal, [math]\displaystyle{ {{t}_{R}} }[/math], for the 1-parameter exponential distribution is:

[math]\displaystyle{ R({{t}_{R}})={{e}^{-\lambda ({{t}_{R}}-\gamma )}} }[/math]

[math]\displaystyle{ \begin{align} \ln[R({{t}_{R}})]=-\lambda({{t}_{R}}-\gamma ) \end{align} }[/math]

or:

[math]\displaystyle{ {{t}_{R}}=\gamma -\frac{\ln [R({{t}_{R}})]}{\lambda } }[/math]

The Exponential Failure Rate Function

The exponential failure rate function is:

[math]\displaystyle{ \lambda (t)=\frac{f(t)}{R(t)}=\frac{\lambda {{e}^{-\lambda (t-\gamma )}}}{{{e}^{-\lambda (t-\gamma )}}}=\lambda =\text{constant} }[/math]

Once again, note that the constant failure rate is a characteristic of the exponential distribution, and special cases of other distributions only. Most other distributions have failure rates that are functions of time.