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)

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)

thus:

(5)

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

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

Arrhenius Exponential Statistical Properties Summary

Mean or MTTF

The mean, , or mean time to failure (MTTF) of the Arrhenius-exponential model is given by:

(6)

Median

The median, , of the Arrhenius-exponential model is given by:

Mode

The mode, , of the Arrhenius-exponential model is given by:

Standard Deviation

The standard deviation, , of the Arrhenius-exponential model is given by:

Arrhenius-Exponential Reliability Function

The Arrhenius-exponential reliability function is given by:

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

and:

Conditional Reliability

The Arrhenius-exponential conditional reliability function is given by:

Reliable Life

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

or

Parameter Estimation

Maximum Likelihood Estimation Method

The log-likelihood function for the exponential distribution is:

where:

and:

• Fe is the number of groups of exact times-to-failure data points.

• Ni is the number of times-to-failure in the ith time-to-failure data group.

• λ is the failure rate parameter (unknown).

• 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.

• FI is 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.

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

(7)

where:

The solution (parameter estimates) will be found by solving for the parameters , so that = 0 and = 0, where:

See Also:
Arrhenius Relationship

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