T-H Exponential

The pdf for the Temperature Humidity relationship and the exponential distribution is given next.

By setting m = L(U,V) in Eqn. ( 1) the exponential pdf becomes:

9.5.3.gif (3)

T-H Exponential Statistical Properties Summary

Mean or MTTF

The mean, , or mean time to failure (MTTF) for the T-H exponential model is given by:

9.511.1.gif

Substituting Eqn. ( 3) yields:

9.511.2.gif

Median

The median, for the T-H exponential model is given by:

Mode

The mode, for the T-H exponential model is given by:

Standard Deviation

The standard deviation, , for the T-H exponential model is given by:

T-H Exponential Reliability Function

The T-H exponential reliability function is given by:

9.515.1.gif

This function is the complement of the T-H exponential cumulative distribution function or:

9.515.2.gif

and:

9.515.3.gif

Conditional Reliability

The conditional reliability function for the T-H exponential model is given by:

9.516.1.gif

Reliable Life

For the T-H exponential model, the reliable life, or the mission duration for a desired reliability goal tR is given by:

9.517.1.gif

or:

Parameter Estimation

Maximum Likelihood Estimation Method

Substituting the T-H model into the exponential log-likelihood equation yields:

9.521.1.gif

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.
A is the T-H parameter (unknown, the first of three parameters to be estimated).
is the second T-H parameter (unknown, the second of three parameters to be estimated).
b is the third T-H parameter (unknown, the third of three parameters to be estimated).
Vi is the temperature level of the ith group.
Ui is the relative humidity 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 A, and b so that = 0, = 0 and = 0.