// Home / Documentation / Gallery of Distributions / Weibull distribution

Weibull distribution

Where do you meet this distribution?

Shape of Distribution

Basic Properties

  • Two parameters \alpha, \beta are required (How can you get these).
    \alpha>0,\beta>0
  • Continuous distribution defined on semi-bounded range x \geq 0
  • This distribution is always asymmetric.

Probability

  • How to compute these on Excel.

     
    1
    2
    3
    4
    5
    6
       
    7
       
    A B
    Data Description
    0.5 Value for which you want the distribution
    8 Value of parameter Alpha
    2 Value of parameter Beta
    Formula Description (Result)
    =NTWEIBULLDIST(A2,A3,A4,TRUE) Cumulative distribution function for the terms above
    =NTWEIBULLDIST(A2,A3,A4,FALSE) Probability density function for the terms above

Sample distribution

Quantile

  • Inverse function of cumulative distribution function
    F^{-1}(P)=\beta\left(\ln\frac{1}{1-P}\right)^{1/\alpha}
  • How to compute this on Excel.

     
    1
    2
    3
    4
    5
    6
       
    A B
    Data Description
    0.7 Probability associated with the distribution
    1.7 Value of parameter Alpha
    0.9 Value of parameter Beta
    Formula Description (Result)
    =WEIBULLINV(A2,A3,A4) Inverse of the cumulative distribution function for the terms above
  • Function reference : NTWEIBULLINV

Characteristics

Mean – Where is the “center” of the distribution? (Definition)

  • Mean of the distribution is given as
    \beta\Gamma\left(1+\frac{1}{\alpha}\right)

    where \Gamma(\cdot) is gamma function.

  • How to compute this on Excel

     
    1
    2
    3
    4
    5
    A B
    Data Description
    8 Value of parameter Alpha
    2 Value of parameter Beta
    Formula Description (Result)
    =NTWEIBULLMEAN(A2,A3) Mean of the distribution for the terms above
  • Function reference : NTWEIBULLMEAN

Standard Deviation – How wide does the distribution spread? (Definition)

  • Variance of the distribution is given as
    \mu^\prime(2)-{\mu^\prime}^2(1)

    where

    \mu^\prime(r)=\beta\Gamma\left(1+\frac{r}{\alpha}\right)

    and \Gamma(\cdot) is gamma function

    Standard Deviation is a positive square root of Variance.

  • How to compute this on Excel

     
    1
    2
    3
    4
    5
       
    A B
    Data Description
    8 Value of parameter Alpha
    2 Value of parameter Beta
    Formula Description (Result)
    =NTWEIBULLSTDEV(A2,A3) Standard deviation of the distribution for the terms above
  • Function reference : NTWEIBULLSTDEV

Skewness – Which side is the distribution distorted into? (Definition)

  • Skewness of the distribution is given as
    \frac{1}{\sigma^3}\left[2\mu^\prime(1)-3\mu^\prime(1)\mu^\prime(2)+\mu^\prime(3)\right]

    where

    \mu^\prime(r)=\beta\Gamma\left(1+\frac{r}{\alpha}\right)

    , \Gamma(\cdot) is gamma function and \sigma is standard deviation of the distribution.

  • How to compute this on Excel

     
    1
    2
    3
    4
    5
    A B
    Data Description
    8 Value of parameter Alpha
    2 Value of parameter Beta
    Formula Description (Result)
    =NTWEIBULLSKEW(A2,A3) Skewness of the distribution for the terms above
  • Function reference : NTWEIBULLSKEW

Kurtosis – Sharp or Dull, consequently Fat Tail or Thin Tail (Definition)

  • Kurtosis of the distribution is given as
    \frac{-6{\mu^\prime}^4(1)+12{\mu^\prime}^2(1)\mu^\prime(2)-3{\mu^\prime}^2(2)-4\mu^\prime(1)\mu^\prime(3)+\mu^\prime(4)}{\sigma^4}

    where

    \mu^\prime(r)=\beta\Gamma\left(1+\frac{r}{\alpha}\right)

    , \Gamma(\cdot) is gamma function and \sigma is standard deviation of the distribution.

  • How to compute this on Excel

     
    1
    2
    3
    4
    5
    A B
    Data Description
    8 Value of parameter Alpha
    2 Value of parameter Beta
    Formula Description (Result)
    =NTWEIBULLKURT(A2,A3) Kurtosis of the distribution for the terms above
  • Function reference : NTWEIBULLKURT

Random Numbers

  • Random number x is generated by inverse function method, which is for uniform random U,
    x=\beta\left(\ln\frac{1}{1-U}\right)^{1/\alpha}
  • How to generate random numbers on Excel.
     
    1
    2
    3
    4
    5
       
    A B
    Data Description
    0.5 Value of parameter Alpha
    0.5 Value of parameter Beta
    Formula Description (Result)
    =NTRANDWEIBULL(100,A2,A3,0) 100 Weibull deviates based on Mersenne-Twister algorithm for which the parameters above

    Note The formula in the example must be entered as an array formula. After copying the example to a blank worksheet, select the range A5:A104 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

NtRand Functions

Reference

 

Comments are closed.