The pdf for the temperature non-thermal relationship and the lognormal distribution is given next.
The pdf of the lognormal distribution is given by:
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:
(9)
The T-NT lognormal model pdf
can be obtained first by setting 
= L(V) in Eqn. ( 1).
Therefore:
or:
Thus:
(10)
Substituting Eqn. (10) into Eqn. (8) yields the T-NT lognormal model pdf or:
T-NT Lognormal Statistical Properties Summary
The Mean
The mean life of the T-NT lognormal model (mean of
the times-to-failure), 
,
is given by:
(11)
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 T-NT lognormal model
(standard deviation of the times-to-failure), ,
is given by:
(12)
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 T-NT lognormal model is given by:
T-NT Lognormal Reliability
For the T-NT lognormal model, the reliability for a mission of time T, starting at age 0, for the T-NT lognormal model is determined by:
or:
Reliable Life
For the T-NT lognormal model, 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 T-NT lognormal failure rate is given by:
Parameter Estimation
Maximum Likelihood Estimation Method
The complete T-NT 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 four parameters to be estimated).
B is the first T-NT parameter (unknown, the second of four parameters to be estimated).
C is the second T-NT parameter (unknown, the third of four parameters to be estimated).
n is the third T-NT parameter (unknown, the fourth of four parameters to be estimated).
Vi is the stress level for the first stress type (i.e. temperature) of the ith group.
Ui is the stress level for the second stress type (i.e. non-thermal) 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 
,
, 
, 
so that 
= 0, 
= 0, 
= 0 and 
= 0.
Example
Twelve electronic devices were put into a continuous accelerated test and the following data were collected.
Using ALTA and the T-NT lognormal model, the following parameters were obtained:
= 0.1825579885,
= 3729.6503028119,
= 0.0352919977,
= 0.7767966480.
A probability plot for the use stress levels of 323K and 2V is shown next.
An acceleration factor plot, in which one of the stresses must be kept constant, can also be obtained. For example, in the following plot, the acceleration factor is plotted versus temperature given a constant voltage of 2V.
See Also:
Temperature-NonThermal Relationship
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