Introduction to Probability and Statistics for Engineers and Scientists

14.5The Weibull Distribution in Life Testing 601

Weibull (1, 0.5)

x

012345

1.2

1.0

0.8

0.6

0.4

0.2

0.0

Weibull (1, 2)

x

012345

1.0

0.8

0.6

0.4

0.2

0.0

Weibull (1, 1)

x

012345

1.0

0.8

0.6

0.4

0.2

0.0

Weibull (1, 3)

x

012345

1.2

1.0

0.8

0.6

0.4

0.2

0.0

FIGURE 14.2 Weibull density functions.

and

∂

∂α

logf(x 1 ,...,xn)=

n

α

−

∑n

i= 1

xβi

∂

∂β

logf(x 1 ,...,xn)=

n

β

+

∑n

i= 1

logxi−α

∑n

i= 1

xiβlogxi

Equating to zero shows that the maximum likelihood estimatesαˆandβˆare the solutions of

n

αˆ

=

∑n

i= 1

x

βˆ

i

n

βˆ

+

∑n

i= 1

logxi=ˆα

∑n

i= 1

x

βˆ

i logxi