Weibull distribution (from The shorthand X Weibull Xleemis/chart/UDR/PDFs/Weibull.pdfWeibull...
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Weibull distribution (from http://www.math.wm.edu/ ˜ leemis/chart/UDR/UDR.html) The shorthand X ∼ Weibull(α, β) is used to indicate that the random variable X has the Weibull distribution with scale parameter α > 0 and shape parameter β > 0. A Weibull random variable X has probability density function f (x)= β α x β-1 e -(1/α)x β x > 0. The Weibull distribution is used in reliability and survival analysis to model the lifetime of an object, the lifetime of a organism, or a service time. The accelerated life and Cox proportional hazards model are identical when the baseline distribution is Weibull. The probability density function is plotted below for α = 1 and β = 1/2, 1, 2, 3. 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 x f (x) β = 1 β = 2 β = 3 β = 1/2 The cumulative distribution function on the support of X is F (x)= P(X ≤ x)= 1 - e -(1/α)x β x > 0. The survivor function on the support of X is S(x)= P(X ≥ x)= e -(1/α)x β x > 0. The hazard function on the support of X is h(x)= f (x) S(x) = β α x β-1 x > 0. The cumulative hazard function on the support of X is H (x)= - ln S(x)= 1 α x β x > 0. The inverse distribution function of X is F -1 (u)=[-α ln(1 - u)] 1/β 0 < u < 1. 1
Transcript of Weibull distribution (from The shorthand X Weibull Xleemis/chart/UDR/PDFs/Weibull.pdfWeibull...
Weibull distribution (from http://www.math.wm.edu/ ˜ leemis/chart/UDR/UDR.html )The shorthand X ∼ Weibull(α, β) is used to indicate that the random variable X has the Weibulldistribution with scale parameter α > 0 and shape parameter β > 0. A Weibull random variable Xhas probability density function
f (x) =βα
xβ−1e−(1/α)xβx> 0.
The Weibull distribution is used in reliability and survival analysis to model the lifetime of anobject, the lifetime of a organism, or a service time. The accelerated life and Cox proportionalhazards model are identical when the baseline distribution is Weibull. The probability densityfunction is plotted below for α = 1 and β = 1/2, 1, 2, 3.
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
x
f (x)
β=1
β=2β=3
β=1/2
The cumulative distribution function on the support of X is
F(x) = P(X ≤ x) = 1−e−(1/α)xβx> 0.
The survivor function on the support of X is
S(x) = P(X ≥ x) = e−(1/α)xβx> 0.
The hazard function on the support of X is
h(x) =f (x)S(x)
=βα
xβ−1 x> 0.
The cumulative hazard function on the support of X is
H(x) =− lnS(x) =1α
xβ x> 0.
The inverse distribution function of X is
F−1(u) = [−α ln(1−u)]1/β 0 < u< 1.
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The median of X is(α ln2)1/β.
The moment generating function of X is
M(t) = E[
etX]=∫ ∞
0etx β
αxβ−1e−(1/α)xβ β
αxβ−1e−(1/α)xβ
dx t> 0.
The characteristic function of X is
φ(t) = E[
eitX]=∫ ∞
0eitx β
αxβ−1e−(1/α)xβ
dx t> 0.
The population mean, variance, skewness, and kurtosis of X are
E[X] =αβ
Γ(
1β
)
V[X] = α2
{
2β
Γ(
2β
)
−
[
1β
Γ(
1β
)]2}
E
[
(
X−µσ
)3]
=
{
2β
Γ(
2β
)
−
[
1β
Γ(
1β
)]2}−3/2{
3β
Γ(
3β
)
−6β2 Γ
(
1β
)
Γ(
2β
)
+2
[
1β
Γ(
1β
)]3}
E
[
(
X−µσ
)4]
=
{
2β
Γ(
2β
)
−
[
1β
Γ(
1β
)]2}−2
{
4β
Γ(
4β
)
−12β2 Γ
(
1β
)
Γ(
3κ
)
+12β3
[
Γ(
1β
)]2
Γ(
2β
)
−3β4
[
Γ(
1β
)]4}
.
APPL verification: The APPL statements
X := WeibullRV(1 / alpha, beta);CDF(X);HF(X);IDF(X);Mean(X);Variance(X);Skewness(X);Kurtosis(X);MGF(X);
verify the cumulative distribution function, hazard function, population mean, variance, skewness,kurtosis, and moment generating function.
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