The pdf for the temperature non-thermal relationship and the exponential distribution is given next.
By setting m = L(U, V) as given in Eqn. ( 1), the exponential pdf becomes:
T-NT Statistical Properties Summary
Mean or MTTF
The mean, 
, or mean time
to failure (MTTF) for the T-NT exponential model is given by:
Median
The median, 
for the T-NT
exponential model is given by:
Mode
The mode, 
for the T-NT
exponential model is given by:
Standard Deviation
The standard deviation, 
,
for the T-NT exponential model is given by:
T-NT Exponential Reliability Function
The T-NT exponential reliability function is given by:
This function is the complement of the T-NT exponential cumulative distribution function or:
and:
Conditional Reliability
The conditional reliability function for the T-NT exponential model is given by:
Reliable Life
For the T-NT exponential model , the reliable life, or the mission duration for a desired reliability goal tR, is given by:
or:
Parameter Estimation
Maximum Likelihood Estimation Method
Substituting the T-NT relationship into the exponential log-likelihood equation yields:
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.
B is the T-NT parameter (unknown, the first of three parameters to be estimated).
C is the second T-NT parameter (unknown, the second of three parameters to be estimated).
n is the third T-NT parameter (unknown, the third of three parameters to be estimated).
Vi is the temperature level of the ith group.
Ui is the non-thermal 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 the
parameters B, C and n so that 
= 0, 
= 0 and 
= 0.
See Also:
Temperature-NonThermal Relationship
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