Weibull Distribution

The Weibull Distribution density function $f_{X} (x)$ is defined by

\begin{array}{l} f_{X} (x) & = \frac{β}{α} {(\frac{x - t}{α})}^{β - 1} \exp [- {(\frac{x - t}{α})}^{β}]; & x \geq t \\ = 0 & otherwise \end{array}

where

α > 0

is the scale parameter,

β > 0

is the shape parameter, and

t

is the location parameter (called Threshold in Isight).

The Weibull probability distribution function is

F_{X} (x) = 1 - \exp [- {(\frac{x - t}{α})}^{β}] x \geq t .

Generally, location parameter t can have values from $- \infty$ to $+ \infty$ . However, Isight allows for nonnegative values only.

The mean value and standard deviation of the random variable $X$ with the Weibull distribution are given as follows:

μ_{X} = t + α Γ (1 + \frac{1}{β})

and

σ_{X} = α {Γ (1 + \frac{2}{β}) - {[Γ (1 + \frac{1}{β})]}^{2}}^{\frac{1}{2}}

where

Γ

is the well-known gamma function

Γ (k) = \int_{0}^{\infty} e^{- u} u^{k - 1} d u

Γ (k) = (k - 1)!

when

k

is a positive integer.

The Weibull distribution can take different shapes, as shown in the following figure. For example, this distribution is often used to describe the life of capacitors and ball bearings.