Applied Survival Data Analysis Assignment-
Q1. The exponential distribution is said to posses a "memoryless property". This memoryless property implies that a used unit is just as reliable as one that is new - that there is no wear out. Probabilistically this memoryless property can be stated as
Pr(T > s) = Pr(T > t + s|T > t),
for any s, t > 0. Show that for a continuous random variable, this memoryless property holds if and only if T ∼ EXP(θ).
Exponential ⇒ Memoryless:
Memoryless ⇒ Exponential:
Q2. Suppose T follows Weibull distribution with cdf
Pr(T ≤ t; η, β) = 1 - exp [-(t/η)β], t > 0.
Show that the distribution of Y = ln(T) is smallest extreme value distribution with parameters µ = ln η and σ = 1/β.
Q3. Show that if Y has a SEV(µ, σ) distribution, then -Y has a LEV(-µ, σ) distribution.
Q4. The coefficient of variation, γ2, is a useful scale-free measure of relative variability for a random variable.
a. Derive an expression for the coefficient of variation for the Weibull distribution.
b. Compute γ2 for all combinations of β = 0.5, 1, 3, 5 and η = 50, 100.
Table 1: Coefficient of variation γ2 for Weibull distribution
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β = 0.5
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β = 1
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β = 3
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β = 5
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η = 50
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η = 100
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c. Explain the effect that changes in η and β have on the γ2.