Template:Ipl ex stat prop sum

IPL-Exponential Statistical Properties Summary

Mean or MTTF

The mean, [math]\displaystyle{ \overline{T}, }[/math] or Mean Time To Failure (MTTF) for the IPL-exponential relationship is given by:

[math]\displaystyle{ \begin{align} & \overline{T}= & \mathop{}_{0}^{\infty }t\cdot f(t,V)dt=\mathop{}_{0}^{\infty }t\cdot K{{V}^{n}}{{e}^{-K{{V}^{n}}t}}dt \\ & = & \frac{1}{K{{V}^{n}}} \end{align} }[/math]

Note that the MTTF is a function of stress only and is simply equal to the IPL relationship (which is the original assumption), when using the exponential distribution.

Median

The median, [math]\displaystyle{ \breve{T}, }[/math] for the IPL-exponential model is given by:

[math]\displaystyle{ \breve{T}=0.693\frac{1}{K{{V}^{n}}} }[/math]

Mode

The mode, [math]\displaystyle{ \tilde{T}, }[/math] for the IPL-exponential model is given by:

[math]\displaystyle{ \tilde{T}=0 }[/math]

Standard Deviation

The standard deviation, [math]\displaystyle{ {{\sigma }_{T}} }[/math] , for the IPL-exponential model is given by:

[math]\displaystyle{ {{\sigma }_{T}}=\frac{1}{K{{V}^{n}}} }[/math]