Eyring-Exponential

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

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

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:

(3)

thus:

(4)

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

and substituting for m in Eqn. (4):

(5)

Eyring Exponential Statistical Properties Summary

Mean or MTTF

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

Median

The median, for the Eyring-exponential relationship is given by:

Mode

The mode, for the Eyring-exponential model is = 0.

Standard Deviation

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

Eyring-Exponential Reliability Function

The Eyring-exponential reliability function is given by:

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

and:

Conditional Reliability

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

Reliable Life

For the Eyring-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

The complete exponential log-likelihood function of the Eyring model is composed of two summation portions:

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.

• Vi is the stress level of the ith group.

• A is the Eyring parameter (unknown, the first of two parameters to be estimated).

• B is the second Eyring parameter (unknown, the second of two parameters to be estimated).

• 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 and so that = 0 and = 0 where:

See Also:
Eyring Relationship

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