55202

مالحظات على توزيع ويبلNotes 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.

جملة العلوم جملة العلوم االقتصادية واإلداريةاالقتصادية واإلدارية

8181اجمللد اجمللد 6677العدد العدد

070070 --062062الصفحات الصفحات

162 8176

مالحظات على توزيع ويبل

8-

α

4

life time

life data analysis

38598Wallodi Weibull

:α,α > 0

> 0p.d.f.

c. d. f.

Hazard function

rth

α

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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

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23Weibull Distribution -Bi

93Weibull Distribution -Five parameter Biλ>0

θ>0≥0α>0>0

Xh(x)Xλθθ=8λ

X>

α, ,

S(x)S(x)X

(+2α)uniform random

variable

x

S(x)=R

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2p. d. f.

c. d. f.

bath tub

θ

α=18α=1

4=1α=2=23

=3α=4=4θ=0.7λ=0.1

80840447

85%78405369933287

840

82233738α834056=0.99

82%

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162 8176


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9

Si3N4

SiCZnO

Gaussian

F()

mp. d. f.

p. d. f.

(maximum likelihood)

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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=3K=2

122 8176


(AIC)

Si3N4 SiC ZnO

iNp. d. f.k

N Si3N4 SiCZnO

An

An

630072436173636673936199Si3N4

833666537166531366133869SiC

-9.76 768393714350718345805ZnO

AIC

Si3N4ZnOSiC

AIC9

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7

8

43

82

497

6degeneratesα

sharp peak15

80β=1

88α=2chi (n,)n=2 = α

Ray(α)Rayleigh Distribution

84=2

chi(n, )n=2=

Ray (

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83Xp. d. f.

extreme value(Gumbel Distribution)

7- 1- Burgher E, Reymen D, Raymen O, Wessa P, "Facilities Development and

Design" Resa Corporation, (2004)

2- Chunsheng Lu, Robert Danzer, Franz Dieter Fischer, "Fracture Statistics

of Brittle Materials: Weibull or Normal Distribution, Physical Review, Vol.

65, 067/02, (2002)

3- Devroye L, "Non- Uniform Random Variate Generation", springer-

Verlag, New York, (1986).

4- Hanagel D, "A Multi- Variate Weibull Distribution", Dept. of Statistics,

University of Pune, India, (2004)

5- Kapur J. N., Saxena H. C., "Mathematical Statistics" S. Chand &

Company Ltd., Ram-Nager, New Delhi 110055, (1978).

6- Merran Evans, Nicolas Hasings, Brian Peacock, "Statistical Distributions",

John Wiley & Sons, Inc. USA, (2000)

7- Patel J. K., Kapadia C. H., Owen D. B. "Hand Book Of Statistical

Distributions", Marcel-Dekker, (1976)

8- Vijay K., Rohatge, Ehsane Saleh A. K. Md. "An Introduction to probability

and Statistics", 2nd

Edition, John Wiley & Sons Inc., Canada, (2000).

55202

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