Page 251 - DMTH404_STATISTICS

Unit 17: System Reliability

 F is an increasing failure rate (IFR) distribution if l(t) is an increasing function of t. Notes
This is analogous to “wearing out”.
 F is a decreasing failure rate (DFR) distribution if l(t) is a decreasing function of t.
This is analogous to “burning in”.

17.6 Distribution Functions for Modeling Component Lifetimes

 Exponential Distribution
 Weibull Distribution
 Gamma Distribution
 Log-Normal Distribution

The exponential distribution with parameters l > 0 has distribution function


G(t) 1 e  ( t) , t  0.

Its failure rate function is given by
 e  t
 (t)   . 
e  t
It is considered both IFR and DFR.
The Weibull distribution with parameters  > 0,  > 0 has distribution function


( t)




G(t) 1 e , t  0.
Its failure rate function is given by

 (t)   ( t)  1
It is IFR if a  1 and DFR if 0 < a  1.
The gamma distribution with parameters  > 0,  > 0 has density function

 e  t ( t)  1  e  t ( t)  1

g(t)   , t  0.

 ( )   x  1
 e x dx
0
Its failure rate function is given by
1  x  x   1
  e   1   dx.
 (t)  t 
0
It is IFR if a  1 and DFR if 0 < a  1.

LOVELY PROFESSIONAL UNIVERSITY 243