Arrhenius Exponential

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

The pdf of the 1-parameter exponential distribution is given by:

(3)

$images\6_3_3.gif$

It can be easily shown that the mean life for the 1-parameter exponential distribution (presented in detail in the Life Distributions chapter) is given by:

(4)

$images\6_3_4.gif$

thus,

(5)

$images\6_3_5.gif$

The Arrhenius-exponential model pdf can then be obtained by setting m = L(V) in Eqn. (1). Therefore,

$images\6_3_6.gif$

Substituting for m in Eqn. (5) yields a pdf that is both a function of time and stress or,

$images\6_3_7.gif$

Arrhenius Exponential Statistical Properties Summary

Mean or MTTF

The mean, $images\tline.gif$ , or mean time to failure (MTTF) of the Arrhenius-exponential is given by:

(6)

$images\6_311_6.gif$

Median

The median, $images\tu.gif$ , of the Arrhenius-exponential is given by:

$images\6_312_1.gif$

Mode

The mode, $images\twave.gif$ , of the Arrhenius-exponential is given by:

$images\6_313_1.gif$

Standard Deviation

The standard deviation, $images\ot.gif$ , of the Arrhenius-exponential is given by:

$images\6_314_1.gif$

Arrhenius-Exponential Reliability Function

The Arrhenius-exponential reliability function is given by:

$images\6_315_0.gif$

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

$images\6_315_1.gif$

and

$images\6_315_2.gif$

Conditional Reliability

The Arrhenius-exponential conditional reliability function is given by:

$images\6_316_0.gif$

Reliable Life

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

$images\6_317_1.gif$

$images\6_317_2.gif$

Parameter Estimation

Maximum Likelihood Estimation Method

The log-likelihood function for the exponential distribution is composed of two summation portions shown next.

$images\6_321_1.gif$

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 in the $images\ith.gif$ time-to-failure data group.

· $images\lambdal.gif$ is the failure rate parameter (unknown).

· $images\tsubi.gif$ 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.

Substituting the Arrhenius-exponential model into the log-likelihood function yields,

(7)

$images\6_321_7.gif$

The solution (parameter estimates) will be found by solving for the parameters $images\bhat.gif$ ,

$images\chat.gif$ so that $images\vb2.gif$ = 0 and $images\vc.gif$ = 0, where,

$images\6_321_2.gif$