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| ==Exponential Statistical Properties==
| | #REDIRECT [[Exponential Distribution Functions]] |
| ===The Mean or MTTF===
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| The mean, <math>\overline{T},</math> or mean time to failure (MTTF) is given by:
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| ::<math>\begin{align}
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| \bar{T}= & \int_{\gamma }^{\infty }t\cdot f(t)dt \\
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| = & \int_{\gamma }^{\infty }t\cdot \lambda \cdot {{e}^{-\lambda t}}dt \\
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| = & \gamma +\frac{1}{\lambda }=m
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| \end{align}</math>
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| 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.
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| {{Median}}
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| {{Mode}}
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| {{Standard Deviation}}
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| {{Exponential Reliability Function}}
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| ===The Exponential Reliability Function===
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| The equation for the two-parameter exponential cumulative density function, or <math>cdf,</math> is given by:
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| ::<math>F(T)=Q(T)=1-{{e}^{-\lambda (T-\gamma )}}</math>
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| 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:
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| ::<math>R(T)=1-Q(T)=1-\int_{0}^{T-\gamma }f(T)dT</math>
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| ::<math>R(T)=1-\int_{0}^{T-\gamma }\lambda {{e}^{-\lambda T}}dT={{e}^{-\lambda (T-\gamma )}}</math>
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| {{One-Parameter Exponential Reliability Function}}
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| ====One-Parameter Exponential Reliability Function====
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| The one-parameter exponential reliability function is given by:
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| ::<math>R(T)={{e}^{-\lambda T}}={{e}^{-\tfrac{T}{m}}}</math>
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| {{Exponential Conditional Reliability}}
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| ===The Exponential Conditional Reliability===
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| 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:
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| ::<math>R(t|T)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-\lambda (T+t-\gamma )}}}{{{e}^{-\lambda (T-\gamma )}}}={{e}^{-\lambda t}}</math>
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| 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.
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| {{Exponential Reliable Life}}
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| ===The Exponential Reliable Life===
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| The reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}}</math>, for the one-parameter exponential distribution is:
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| ::<math>R({{t}_{R}})={{e}^{-\lambda ({{t}_{R}}-\gamma )}}</math>
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| ::<math>\ln [R({{t}_{R}})]=-\lambda ({{t}_{R}}-\gamma )</math>
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| :or:
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| ::<math>{{t}_{R}}=\gamma -\frac{\ln [R({{t}_{R}})]}{\lambda }</math>
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| {{Exponential Failure Rate Function}}
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| ===The Exponential Failure Rate Function===
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| The exponential failure rate function is:
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| ::<math>\lambda (T)=\frac{f(T)}{R(T)}=\frac{\lambda {{e}^{-\lambda (T-\gamma )}}}{{{e}^{-\lambda (T-\gamma )}}}=\lambda =\text{constant}</math>
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| 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.
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