1. The Exponential Distribution

2. The Exponential Distribution
Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.
Copyright (c) 2009  John Wiley & Sons, Inc.

3. The Weibull Distribution

4. The Weibull Distribution
When β = 1, the Weibull reduces to the exponential
Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.
Copyright (c) 2009  John Wiley & Sons, Inc.

5. Distributions Applicable to Reliability
Exponential distribution.
Normal distribution.
Weibull distribution.

6. Distributions Applicable to Reliability
Reliability Curves:
The reliability curves for the exponential, normal and Weibull distributions as a function of time (Shown in the next slide).

7. Frequency f(t)
Reliability Rt
Failure rate 

8. Distributions Applicable to Reliability
Failure-Rate Curve:
It is important in describing the life-history curve of a product.

9. Distributions Applicable to Reliability
The curve, sometimes referred to as the “bathtub” curve, is a comparison of failure rate with time.
It has three distinct phases:
The debugging phase.
The chance failure phase.
The wear-out phase.

10. Distributions Applicable to Reliability
Wear Out Phase
Chance Failure Phase
Debugging Phase
“Infant mortality period”

11. Distributions Applicable to Reliability
The debugging phase:
It is characterized by marginal and short-life parts that cause a rapid decrease in the failure rate.
It may be part of the testing activity prior to shipment for some products.
The Weibull distribution ß<1 is used to describe the occurrence of failures.

12. Distributions Applicable to Reliability
The chance failure phase:
Failures occur in a random manner due to the constant failure rate. The Exponential and the Weibull distributions β= 1 are best suited to describe this phase.
The wear-out phase: Is depicted by a sharp raise in failure rates. The Normal distribution and the Weibull distribution ß >1 are used to describe this phase.

13. Normal Failure Analysis
The Normal distribution for reliability in the wear out phase.
R(t): Reliability at time t
P(t): Probability of failure or area of the normal curve to the left of time t. Table A.

14. Exponential Failure Analysis
Exponential distribution for reliability in the chance failure phase :
Rt = e –t/ө
Where:
t: Time or cycles.
ө: Mean life.

15. Weibull Failure Analysis
Exponential distribution for reliability for all phases
Can be used for the debugging phase (ß<1) and the chance failure phase (ß=1).
By setting = 1, the Weibull equals the exponential.
By setting ß=3.4, the Weibull approximates the Normal.
Rt = e –(t/ө)ß
Where ß is the Weibull slope.