Weibull Distribution

Weibull distribution was first discovered by a Swedish physicist Wallodi Weibull in 1939 A.D. to study the breaking strength of metals. this distribution is appropriate model to describe the time of failure of service equipment or mechines or products manufactured by a company.

Definition:

A continious random variableX is said to have a Weibull distribution with three parameters μ,α and β, if its probability density function (pdf) is

$$f(x; \ \mu, \ \alpha, \ \beta) \ = \ \frac{\alpha}{\beta} \ { \left ( \frac{x-\mu}{\beta} \ \right )^\alpha}^{-1} \ {e^-}^{\left(\frac{x-\mu}{\beta} \ \right ) \ \alpha} \ \ \ ; \ x \ > \ \mu, \ \alpha, \ \beta \ > \ 0$$

Thisα is called the shape parameter andβ is called the scale parameter of the Weibull distribution.

This definition is denoated by W(μ,α,β).Ifμ = 0, andβ = 1, then the resulting distribution is called standard Weibull distribution which depends upon only one parameter α.

Definition:

A continiousrandom variableX is said to follow a standard Weibull distribution with parameter α if its probability density function (pdf) is given by

$$f(x) \ = \ \alpha \ {x^\alpha}^{-1} \ {e^-}^{x \ \alpha} \ \ \ ; \ x \ > \ 0, \ \alpha \ > \ 0.$$

The distribution function of the standard Weibull distribution is given by

$$F (x) \ = \ \int_0^{x} \ f(u) \ du$$

$$= \ \alpha \ \int_0^{x} \ {u^\alpha}^{-1} \ {e^-}^{u \ \alpha} \ du$$

$$= \ \int_0^{x} \ {e^-}^{y} \ dy$$
Where y = u^αso that dy =αu^α-1 du

$$= \ \left [ \ - \ {e^-}^{y} \ \right ]_0^{x}$$

$$= \ \left [ \ - \ {e^-}^{u \ \alpha} \ \right ]_0^{x}$$

$$= \ 1 \ - \ {e^-}^{x \ \alpha}$$

$$\therefore \ F (x) \ = \ 1 \ - \ {e^-}^{x^\alpha} \ \ \ ; \ x \ > \ 0$$

Otherwise 0.

It is noted that ifα = 1, the standard Weibull distribution reduces to an exponential distribution with probability density function (pdf)

$$f(x) \ = \ {e^-}^{x} \ \ \ ; \ x \ > \ 0.$$

Moments of Standard Weibull Distribution

Let a random variable X has a standard Weibull distribution with an parameter α. The r^th moment about origin of X is given by

$$\mu_r^{'} \ = \ E(X^r) \ = \ \int_0^{\infty} \ x^r \ f(x) \ dx$$

$$= \ \alpha \ \int_0^{\infty} \ x^r \ {x^\alpha}^{- 1} \ {e^-}^{x \ \alpha} \ dx$$

$$put \ y \ = \ x^\alpha \ so \ that \ dy \ = \ \alpha \ {x^\alpha}^{-1} \ dx$$

$$= \ \int_{0}^{\infty} \ \left ( \ y^{\frac{1}{\alpha}} \ \right )^r \ {e^-}^{y} \ dy$$

$$= \ \int_0^{\infty} \ y^{\frac{r}{\alpha}} \ {e^-}^{y} \ dy$$

$$\Rightarrow \ \mu_r^{'} \ = \ \Gamma \left ( \frac{r}{\alpha} \ + \ 1 \ \right )$$

$$When \ r \ = \ 1, \ \mu_1^{'} \ = \ \Gamma\left ( \frac{1}{\alpha} \ + \ 1 \ \right )$$

$$When \ r = \ 2, \ \mu_2^{'} \ = \ \Gamma\left ( \frac{r}{\alpha} \ + \ 1 \ \right )$$

and so on

$$\therefore \ Mean \ = \ \mu_1^{'} \ =\ \Gamma \left ( \frac{r}{\alpha} \ + \ 1 \ \right )$$

$$and \ Variance \ = \ \mu_2^{'} \ - \ (\mu_1^{'})^2$$

$$= \ \Gamma\left ( \frac{r}{\alpha} \ + \ 1 \ \right ) \ - \ \left [ \\ \Gamma \left ( \frac{r}{\alpha} \ + \ 1 \ \right ) \ \right ]^2$$

Hazard Function and Failure Rate

The Weibull distribution with two parametersα and β has probability density function (pdf)

$$f(x) \ = \ \frac{\alpha}{\beta^\alpha} \ {x^\alpha}^{-1} \ {e^-}^{\ \left ( \ \frac{x}{\beta} \ \right )^\alpha} \ \ ; \ x \≥ \ 0, \ \alpha, \ \beta \ > \ 0.$$

The scale parameter β characterizes the time to failure of a machine or any product(s) manufactured by a company. The function

$$g(x) \ = \ \frac{\alpha}{\beta^\alpha} \ {x^\alpha}^{-1} \ \ \ \ ; x \≥ \ 0$$

is called Weibull hazard function or simply hazard function of time x to failure of machine. This Hazard function g(X) gives the failure time of the machines or products and this failure time is called failure rate of the machines. The failure rate g(x) depends upon the value of the shape parameter α. If α = 1, then g(x) = 1/β which is a constant. Hence, if α = 1, the failure rate g(x) is constant, if α > 1, the failure rate g(x) increases and if α < 1, the failure rate g(x) decreases.

Uses of Weibull Distribution:

Weibull distribution has many applications. Some uses of this distribution are given below:

• This distribution was used to study experimentally the breaking strength and elasticity of metal such as steel, copper etc.
• This distribution was used by Koa, J.H.k in reliability theory and quality control of products.
• This distribution is used to describe the failure time of machines or products and to study the variation of length of service of radio equipments.

Bibliography

Sukubhattu N.P. (2013). Probability & Inference - II. Asmita Books Publishers & Distributors (P) Ltd., Kathmandu.

Larson H.J. Introduction to Probability Theory and Statistical Inference. WileyInternational, New York.