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:

(6)

where:

• = ln(T).

• T = times-to-failure.

• = mean of the natural logarithms of the times-to-failure.

• = standard deviation of the natural logarithms of the times-to-failure.

The median of the lognormal distribution is given by:

(7)

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

or:

Thus:

(8)

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

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.

• FI is 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:
Inverse Power Law Relationship

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