Template:Ipl lognormal

IPL-Lognormal

The pdf for the Inverse Power Law relationship and the lognormal distribution is given next.

The pdf of the lognormal distribution is given by:

[math]\displaystyle{ f(T)=\frac{1}{T\sigma_{T'}\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{T'-\overline{T'}}{\sigma_{T'}})^2} }[/math](6)

where:

[math]\displaystyle{ T'=ln(T) }[/math] = ln(T).

[math]\displaystyle{ T }[/math] = times-to-failure.

[math]\displaystyle{ \overline{T}' }[/math] = mean of the natural logarithms of the times-to-failure.

[math]\displaystyle{ \sigma_{T'} }[/math] = standard deviation of the natural logarithms of the times-to-failure.

The median of the lognormal distribution is given by:

[math]\displaystyle{ \breve{T}=e^{\overline{T}'} }[/math](7)

The IPL-lognormal model pdf can be obtained first by setting = L(V) in Eqn. ( 30). Therefore:

[math]\displaystyle{ \breve{T}=L(V)=\frac{1}{K*V^n} }[/math]

or:

[math]\displaystyle{ e^{\overline{T'}}=\frac{1}{K*V^n} }[/math]

Thus:

[math]\displaystyle{ \overline{T}'=-ln(K)-n ln(V) }[/math](8)

Substituting Eqn. (8) into Eqn. (6) yields the IPL- lognormal model pdf or:

[math]\displaystyle{ f(T,V)=\frac{1}{T\sigma_{T'} \sqrt{2 \pi} e^{\frac{1}{2}(\frac{t'+ln(K)+ln(V)}{\sigma_{T'}})^2} }[/math]

IPL-Lognormal Statistical Properties Summary

The Mean

The mean life of the IPL-lognormal model (mean of the times-to-failure), , is given by:

(9)

The mean of the natural logarithms of the times-to-failure, , in terms of and is given by:

The Standard Deviation

The standard deviation of the IPL-lognormal model (standard deviation of the times-to-failure), , is given by:

(10)

The standard deviation of the natural logarithms of the times-to-failure, , in terms of and is given by:

The Mode

The mode of the IPL-lognormal model is given by:

IPL-Lognormal Reliability

The reliability for a mission of time T, starting at age 0, for the IPL-lognormal model is determined by:

or:

Reliable Life

The reliable life, or the mission duration for a desired reliability goal, tR is estimated by first solving the reliability equation with respect to time, as follows:

where:

and:

Since = ln(T) the reliable life, tR, is given by:

Lognormal Failure Rate

The lognormal failure rate is given by:

Parameter Estimation

Maximum Likelihood Estimation Method

The complete IPL-lognormal log-likelihood function is:

where:

and:

Fe is the number of groups of exact times-to-failure data points. Ni is the number of times-to-failure data points in the ith time-to-failure data group.

is the standard deviation of the natural logarithm of the times-to-failure (unknown, the first of three parameters to be estimated).

K is the IPL parameter (unknown, the second of three parameters to be estimated). n is the second IPL parameter (unknown, the third of three parameters to be estimated). Vi is the stress level of the ith group. Ti is the exact failure time of the ith group. S is the number of groups of suspension data points.

is the number of suspensions in the ith group of suspension data points.
is the running time of the ith suspension data group.

FIis the number of interval data groups.

is the number of intervals in the ith group of data intervals.
is the beginning of the ith interval.
is the ending of the ith interval.

The solution (parameter estimates) will be found by solving for , , so that = 0, = 0 and = 0:

and:

See Also: