Notes on Weibull Distribution

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Notes on Weibull Distribution
α
4002
Abstract
Weibull Distribution is one of most important distribution and it is mainly used in
reliability and in distribution of life time. The study handled two parameter and three-
parameter Weibull Distribution in addition to five –parameter Bi-Weibull distribution.
The latter being very new and was not mentioned before in many of the previous
references. This distribution depends on both the two parameter and the three –
parameter Weibull distributions by using the scale parameter (α) and the shape
parameter () in the first and adding the location parameter ()to the second and then
joining them together to produce a distribution with five parameters.
The paper also handled the relationship between Weibull Distribution and the other
known distributions such as the exponential distribution, Chi distribution, Gamma
distribution and Gumbel (extreme value) distribution.
The paper considered a number of important notes including the importance and
the application of Weibull distribution in the field of reliability.
The study depended on the most recent research papers on this subject and until
May 2004.
‫العلوم‬ ‫جملة‬‫العلوم‬ ‫جملة‬
‫واإلدارية‬ ‫االقتصادية‬‫واإلدارية‬ ‫االقتصادية‬
‫اجمللد‬‫اجمللد‬8181
‫العدد‬‫العدد‬6677
‫الصفحات‬‫الصفحات‬062062--070070

2. 162
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8-
α
4
life time
life data analysis
3
8598Wallodi Weibull
:α,α > 0
 > 0p.d.f.
c. d. f.
Hazard function
rth
α

3. 161
8176
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23
2=1Exp: αα
1Rayleigh Variate=2
3
(standard Extreme Value Variate)v: 0,1
13Parameter Estimation
M. L. E.
33
p. d.
f.X < 
Minimum Life
 > 0
α > 0 > 0
p. d. f.
c. d. f.
Hazard function

4. 163
8176
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23Weibull Distribution-Bi
93Weibull Distribution-Five parameter Bi
λ>0
θ>0≥0α>0>0
Xh(x)Xλ
θθ=8λ
X>
α, ,
S(x)S(x)X
(+2α)uniform random
variable
x
S(x)=R

5. 162
8176
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2
p. d. f.
c. d. f.
bath tub
θ
α=1
8α=1
4=1α=2=23
=3α=4=4θ=0.7λ=0.1
80
840447
85%7
8405369933287
840
82233738α834056=0.99
82%

6. 162
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7. 166
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8. 162
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9. 162
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9
Si3N4
SiCZnO
Gaussian

F()
m
p. d. f.
p. d. f.

(maximum likelihood)

10. 162
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M. L.p.d.f.
M. L.
M. L.mN
m
M. L.
(Akaike information Criterion, AIC)M. L.
M. L.4
K=3
K=2

11. 122
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(AIC)
Si3N4SiC
ZnO
iNp. d. f.k
NSi3N4SiCZnO
An
An
630072436173636673936199Si3N4
833666537166531366133869SiC
-9.76768393714350718345805ZnO
AIC
Si3N4ZnO
SiC
AIC9

12. 122
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7
8
4
3
8
2
4
9
7
6degeneratesα
sharp peak
1
5

80β=1
88α=2chi (n,)n=2
= α
Ray(α)Rayleigh Distribution
84=2
chi(n, )n=2=
Ray (

13. 121
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83Xp. d. f.
extreme value(Gumbel Distribution)
7-
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Design" Resa Corporation, (2004)
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65, 067/02, (2002)
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Company Ltd., Ram-Nager, New Delhi 110055, (1978).
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John Wiley & Sons, Inc. USA, (2000)
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Distributions", Marcel-Dekker, (1976)
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and Statistics", 2nd
Edition, John Wiley & Sons Inc., Canada, (2000).