Basic Analysis and Graphing - SAS

Chapter 2 Performing Univariate Analysis 77
Statistical Details for the Distribution Platform
Weibull, Weibull with Threshold, and Extreme Value
Exponential
The Weibull distribution has different shapes depending on the values of α (scale) and β (shape). It often
provides a good model for estimating the length of life, especially for mechanical devices and in biology. The
Weibull option is the same as the Weibull with threshold option, with a threshold (θ) parameter of zero. For
the Weibull with threshold option, JMP estimates the threshold as the minimum value. If you know what
the threshold should be, set it by using the Fix Parameters option. See “Fit Distribution Options” on
page 58.
The pdf for the Weibull with threshold is as follows:
β
pdf: ------ for α,β > 0;
α β ( x – θ) β – 1 x – θ
exp – ----------
α
β
θ < x
1
E(x) = θ + αΓ1
+ --
β
Var(x) = α 2 2
Γ1
+ -- Γ
β
2 1
1 + --
–
β
where
Γ ( . )
is the Gamma function.
The Extreme Value distribution is a two parameter Weibull (α, β) distribution with the transformed
parameters δ =1/β and λ =ln(α).
The exponential distribution is especially useful for describing events that randomly occur over time, such as
survival data. The exponential distribution might also be useful for modeling elapsed time between the
occurrence of non-overlapping events, such as the time between a user’s computer query and response of the
server, the arrival of customers at a service desk, or calls coming in at a switchboard.
The Exponential distribution is a special case of the two-parameter Weibull when β = 1 and α = σ, and also
a special case of the Gamma distribution when α = 1.
1
pdf: -- exp( – x ⁄ σ)
for 0 < σ; 0 ≤ x
σ
E(x) = σ
Var(x) = σ 2
Devore (1995) notes that an exponential distribution is memoryless. Memoryless means that if you check a
component after t hours and it is still working, the distribution of additional lifetime (the conditional
probability of additional life given that the component has lived until t) is the same as the original
distribution.