Weibull Distribution

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

$$ f(x) = \frac{β}{α} x^{β−1}e^{−\frac{1}{α}x^β} $$

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.

• Another generalization of the exponential distribution
• Popular in reliability and survival analysis
• Probability density function

f(x) = \frac{κ}{λ} \left(\frac{x}{λ}\right)^{κ-1} e^{-(x/λ)^κ} x>0

• Positive scale parameter \lambda
• Positive shape parameter \kappa
• Exponential distribution is a special case when \kappa = 1
• Rayleigh distribution is a special case when \kappa = 2
• When 3 < \kappa < 4, the pdf is nearly symmetric

• The mode and median are equal when \kappa ~ 3.26 • Closed-form cumulative distribution function

The cumulative distribution function on the support of X is

$$ F(x) = P(X ≤ x) = 1 − e^{ − \frac{1}{α} x^β} $$

The survivor function on the support of X is

$$ S(x) = P(X ≥ x) = e^{ − \frac{1}{α} x^β} $$

The hazard function on the support of X is h(x) = \frac{f(x)}{S(x)} = \frac{β}{α} x^{β−1}

The cumulative hazard function on the support of X is H(x) = − ln S(x) = \frac{1}{α} x^β

The median of X is (α ln 2)^{\frac{1}{β}}

http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Weibull.pdf

How do we use weibull probabilistic failure statistics to determine what stresses will cause arbitrary failure/success rates when subjected to a given stress condition

Variability

Which is more useful?

• What is the yield strength of this alloy?
• What is the probability of yielding for this alloy under these conditions?

How to report Average value - Always do more than a few measurements! Use the average

Standard deviation- Shows the spread of data