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==Exponential Statistical Properties==
#REDIRECT [[Exponential Distribution Functions]]
===The Mean or MTTF===
The mean, <math>\overline{T},</math> or mean time to failure (MTTF) is given by:
 
 
::<math>\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>\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.
 
{{Median}}
 
 
{{Mode}}
 
===The Mode===
The mode, <math>\tilde{T},</math> is:
 
 
::<math>\tilde{T}=\gamma </math>
 
{{Standard Deviation}}
===The Standard Deviation===
The standard deviation, <math>{{\sigma }_{T}}</math>, is:
 
 
::<math>{{\sigma }_{T}}=\frac{1}{\lambda }=m</math>
 
{{Exponential Reliability Function}}
===The Exponential Reliability Function===
The equation for the two-parameter exponential cumulative density function, or <math>cdf,</math> is given by:
 
 
::<math>F(T)=Q(T)=1-{{e}^{-\lambda (T-\gamma )}}</math>
 
 
Recalling that the reliability function of a distribution is simply one minus the <math>cdf</math>, the reliability function of the two-parameter exponential distribution is given by:
 
 
::<math>R(T)=1-Q(T)=1-\int_{0}^{T-\gamma }f(T)dT</math>
 
 
 
::<math>R(T)=1-\int_{0}^{T-\gamma }\lambda {{e}^{-\lambda T}}dT={{e}^{-\lambda (T-\gamma )}}</math>
 
{{One-Parameter Exponential Reliability Function}}
====One-Parameter Exponential Reliability Function====
 
The one-parameter exponential reliability function is given by:
 
 
::<math>R(T)={{e}^{-\lambda T}}={{e}^{-\tfrac{T}{m}}}</math>
 
{{Exponential Conditional Reliability}}
===The Exponential Conditional Reliability===
 
The exponential conditional reliability equation gives the reliability for a mission of <math>t</math> duration, having already successfully accumulated <math>T</math> hours of operation up to the start of this new mission. The exponential conditional reliability function is:
 
 
::<math>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>t</math> duration undertaken after the component or equipment has already accumulated <math>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.
 
{{Exponential Reliable Life}}
===The Exponential Reliable Life===
 
The reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}}</math>, for the one-parameter exponential distribution is:
 
::<math>R({{t}_{R}})={{e}^{-\lambda ({{t}_{R}}-\gamma )}}</math>
 
::<math>\ln [R({{t}_{R}})]=-\lambda ({{t}_{R}}-\gamma )</math>
 
:or:
 
::<math>{{t}_{R}}=\gamma -\frac{\ln [R({{t}_{R}})]}{\lambda }</math>
 
{{Exponential Failure Rate Function}}
===The Exponential Failure Rate Function===
The exponential failure rate function is:
 
::<math>\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.

Latest revision as of 08:08, 10 August 2012

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