-
韋伯分佈
韋伯分佈
(Weibull distribution)
以
指數分佈
為一特例。其
< br>p.d.f.
為
其中
α,β>0。以
得
分佈
,
以
表此分佈
,
有二參數
α,β, α
為尺度參數,
β
為形狀參數。若取
β=1,
則
表之。底下給出一些韋伯分佈
p.d.f.
之圖形。
韋伯分佈是瑞典物理學家
Waloddi Weibull,
為發展強化材料的理論
,
於西元
1939
年所引進
,
是一較新
的分佈。在可靠度理論及有關壽命檢定問題裡
,
常少不了韋伯分佈的影子。
分佈的分佈函數為
期望值與變異數分別為
Characteristic Effects of
the Shape Parameter,
β
, for
the Weibull Distribution
The Weibull
shape parameter,
β
, is also
known as the slope. This is because the value of
β
is equal to the slope of
the
regressed line in a probability
plot. Different values of the shape parameter can
have marked effects on the behavior of
the distribution. In fact, some values
of the shape parameter will cause the distribution
equations to reduce to those of
other
distributions. For example, when
β
= 1, the
pdf
of the three-parameter
Weibull reduces to that of the
two-
parameter exponential distribution or:
where
failure
rate.
The parameter
β
is a pure number,
i.e
. it is dimensionless.
The Effect of
β
on the
pdf
Figure
6-1 shows the effect of different values of the
shape parameter,
β
, on the
shape of the
pdf
. One can
see that the
shape of the
pdf
can take on a variety of
forms based on the value of
β
.
Figure 6-1: The effect of the Weibull
shape parameter on the
pdf
.
For 0 <
β
?
?
?
?
As
As
1:
(or
γ
),
,
.
f
(
T
)
decreases monotonically and is convex as
T
increases beyond the value
of
γ
.
The mode is
non-existent.
For
β
> 1:
?
?
f
(
T
)
= 0 at
T
= 0 (or
γ
).
f
(
T
)
increases as
(the mode) and decreases
thereafter.
?
For
β
< 2.6 the
Weibull
pdf
is positively
skewed (has a right tail), for 2.6 <
β
< 3.7 its coefficient of
skewness
approaches zero (no tail).
Consequently, it may approximate the normal
pdf
, and for
β
> 3.7 it is negatively
skewed (left tail).
The way the value of
β
relates to the physical
behavior of the items being modeled becomes more
apparent when we
observe how its
different values affect the reliability and
failure rate functions. Note that for
β
= 0.999,
f
(0) =
but for
β
= 1.001,
f
(
0
)
= 0.
This abrupt shift is what complicates MLE
estimation when
β
is close
to one.
The Effect
of
β
on the
cdf
and Reliability Function
,
Figure 6-2:
Effect of
β
on the
cdf
on a Weibull probability
plot with a fixed value of
η
.
Figure 6-2 shows the effect of the
value of
β
on the
cdf
, as manifested in the
Weibull
probability plot
. It
is easy to see
why this parameter is
sometimes referred to as the slope. Note that the
models represented by the three lines all have
the same value of
η
. Figure 6-3 shows the
effects of these varied values of
β
on the reliability plot,
which is a linear
analog of the
probability plot.