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,

(3)

$images\9_5_3.gif$

T-H Exponential Statistical Properties Summary

Mean or MTTF

The mean, $images\tline.gif$ , or mean time to failure (MTTF) for the T-H exponential model is given by:

$images\9_511_1.gif$

Substituting Eqn. (3) yields,

$images\9_511_2.gif$

Median

The median, $images\tu.gif$ for the T-H exponential model is given by:

$images\9_512_1.gif$

Mode

The mode, $images\twave.gif$ for the T-H exponential model is given by:

$images\9_513_1.gif$

Standard Deviation

The standard deviation, $images\ot.gif$ , for the T-H exponential model is given by:

$images\9_514_1.gif$

T-H Exponential Reliability Function

The T-H exponential reliability function is given by:

$images\9_515_1.gif$

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

$images\9_515_2.gif$

and,

$images\9_515_3.gif$

Conditional Reliability

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

$images\9_516_1.gif$

Reliable Life

For the T-H exponential model, the reliable life, or the mission duration for a desired reliability goal $images\tr2.gif$ is given by:

$images\9_517_1.gif$

or,

$images\9_517_2.gif$

Parameter Estimation

Maximum Likelihood Estimation Method

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

$images\9_521_1.gif$

where:

· $images\fe.gif$ is the number of groups of exact times-to-failure data points.

· $images\ni2.gif$ is the number of times-to-failure data points in the $images\ith.gif$ time-to-failure data group.

· A is the T-H parameter (unknown, the first of three parameters to be estimated).

· $images\phi.gif$ 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).

· $images\vi.gif$ is the temperature level of the $images\ith.gif$ group.

· $images\ui.gif$ is the relative humidity level of the $images\ith.gif$ group.

· $images\tsubi.gif$ is the exact failure time of the $images\ith.gif$ group.

· S is the number of groups of suspension data points.

· $images\nlinei.gif$ is the number of suspensions in the $images\ith.gif$ group of suspension data points.

· $images\tlinei.gif$ is the running time of the $images\ith.gif$ suspension data group.

The solution (parameter estimates) will be found by solving for the parameters A, $images\phi.gif$ and b so that $images\va.gif$ = 0, $images\vlambda.gif$ = 0 and $images\vb3.gif$ = 0.