📂Probability Distribution

Weibull Distribution

Definition

A Weibull Distribution is a probability distribution with the following probability density function, given scale parameter $\lambda > 0$ and shape parameter $k > 0$. $$ f(x) = {{ k } \over { \lambda }} \left( {{ x } \over { \lambda }} \right)^{k-1} e^{-(x/\lambda)^{k}} \qquad , x \ge 0 $$

Theorems

• [1] A Generalization of the Exponential Distribution: The Weibull Distribution becomes the Exponential Distribution when $k=1$.
• [2] A Generalization of the Rayleigh Distribution: The Weibull Distribution becomes the Rayleigh Distribution when $k=2$.

Description

As evident from its mathematical representation of the probability density function, the Weibull distribution is most commonly viewed as a generalization of the exponential distribution. Its applications are exceedingly varied, but the most representative one is in survival analysis, similar to the exponential distribution. However, unlike the constant failure rate in the exponential distribution, it changes according to $k$:

• If $k<1$, the failure rate is seen to decrease over time.
  • It is said to well explain phenomena where there is a sharp drop after a certain period, such as infant mortality.
• If $k=1$, the failure rate is constant over time, exactly as in the exponential distribution.
• If $k>1$, the failure rate is seen to increase over time.
  • This is often mentioned in research papers estimating the incubation or recovery period of infectious diseases, especially viruses.

Generalization to Three Parameters ¹

A Three-parameter Weibull Distribution is a probability distribution with the following probability density function, given scale parameter $\alpha > 0$, location parameter $\beta > 0$, and shape parameter $\gamma > 0$. $$ f(x) = {{ \gamma } \over { \alpha }} \left( {{ x-\beta } \over { \alpha }} \right)^{\gamma-1} e^{- \left( (x - \beta) / \alpha \right)^{\gamma}} \qquad , x \ge \beta $$ If $X \sim \text{Weibull} (\alpha, \beta, \gamma)$, then its mean and variance are as follows. $$ \begin{align*} E(X) =& \alpha \Gamma \left( 1 + {{ 1 } \over { \gamma }} \right) + \beta \\ \text{Var} (X) =& \alpha^{2} \left[ \Gamma \left(1 + {{ 2 } \over { \gamma }} \right) - \left( \Gamma \left( 1 + {{ 1 } \over { \gamma }} \right)^{2} \right) \right] \end{align*} $$ Of course, if $X \sim \text{Weibull} (\lambda, k)$ for the two parameters, it is as follows. $$ \begin{align*} E(X) =& \lambda \Gamma \left( 1 + {{ 1 } \over { k }} \right) \\ \text{Var} (X) =& \lambda^{2} \left[ \Gamma \left(1 + {{ 2 } \over { k }} \right) - \left( \Gamma \left( 1 + {{ 1 } \over { k }} \right)^{2} \right) \right] \end{align*} $$

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• Here, $\Gamma$ is the Gamma function.