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Weibull Distribution

Where do you meet this distribution?​

Shape of Distribution​

Basic Properties​

  • Two parameters α,β\alpha, \beta are required (How can you get these).

    α>0,β>0\alpha>0,\beta>0

  • Continuous distribution defined on semi-bounded range x≥0x \geq 0

  • This distribution is always asymmetric.

Probability​

  • How to compute these on Excel.
AB
1DataDescription
20.5Value for which you want the distribution
38Value of parameter Alpha
42Value of parameter Beta
5FormulaDescription (Result)
6=NTWEIBULLDIST(A2,A3,A4,TRUE)Cumulative distribution function for the terms above
7=NTWEIBULLDIST(A2,A3,A4,FALSE)Probability density function for the terms above

  • Cumulative distribution function

    F(x)=1−exp⁡[−(xβ)α]F(x)=1-\exp\left[-\left(\frac{x}{\beta}\right)^\alpha\right]

  • Probability density function

    f(x)=αβ(xβ)α−1exp⁡[−(xβ)α]f(x)=\frac{\alpha}{\beta}\left(\frac{x}{\beta}\right)^{\alpha-1}\exp\left[-\left(\frac{x}{\beta}\right)^{\alpha}\right]

  • Function reference : NTWEIBULLDIST

Sample distribution

Quantile​

  • Inverse function of cumulative distribution function

    F−1(P)=β(ln⁡11−P)1/αF^{-1}(P)=\beta\left(\ln\frac{1}{1-P}\right)^{1/\alpha}

  • How to compute this on Excel.

AB
1DataDescription
20.7Probability associated with the distribution
31.7Value of parameter Alpha
40.9Value of parameter Beta
5FormulaDescription (Result)
6=WEIBULLINV(A2,A3,A4)Inverse of the cumulative distribution function for the terms above

Characteristics​

Mean -- Where is the "center" of the distribution? (Definition)​

  • Mean of the distribution is given as

    βΓ(1+1α)\beta\Gamma\left(1+\frac{1}{\alpha}\right)

    where Γ(⋅)\Gamma(\cdot) is gamma function.

  • How to compute this on Excel

AB
1DataDescription
28Value of parameter Alpha
32Value of parameter Beta
4FormulaDescription (Result)
5=NTWEIBULLMEAN(A2,A3)Mean of the distribution for the terms above

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

  • Variance of the distribution is given as

    μ′(2)−m2\mu^\prime(2)-m^2

    where

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

    , mm is mean of the distribution and Γ(⋅)\Gamma(\cdot) is gamma function

    Standard Deviation is a positive square root of Variance.

  • How to compute this on Excel

AB
1DataDescription
28Value of parameter Alpha
32Value of parameter Beta
4FormulaDescription (Result)
5=NTWEIBULLSTDEV(A2,A3)Standard deviation of the distribution for the terms above

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

  • Skewness of the distribution is given as

    1σ3[μ′(3)−3mσ2−m3]\frac{1}{\sigma^3}\left[\mu^\prime(3)-3m\sigma^2-m^3\right]

    where

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

    , mm is mean of the distribution, σ2\sigma^2 is variance of the distribution, Γ(⋅)\Gamma(\cdot) is gamma function and σ\sigma is standard deviation of the distribution.

  • How to compute this on Excel

AB
1DataDescription
28Value of parameter Alpha
32Value of parameter Beta
4FormulaDescription (Result)
5=NTWEIBULLSKEW(A2,A3)Skewness of the distribution for the terms above

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

  • Kurtosis of the distribution is given as

    μ′(4)−4γ1σ3m−6m2σ2−m4σ4−3\frac{{\mu^\prime}(4)-4\gamma_1\sigma^3m-6m^2\sigma^2-m^4}{\sigma^4}-3

    where

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

    , Γ(⋅)\Gamma(\cdot) is gamma function, mm is mean of the distribution, σ\sigma is standard deviation of the distribution and γ1\gamma_1 is skewness of the distribution.

  • How to compute this on Excel

AB
1DataDescription
28Value of parameter Alpha
32Value of parameter Beta
4FormulaDescription (Result)
5=NTWEIBULLKURT(A2,A3)Kurtosis of the distribution for the terms above

Random Numbers​

  • Random number x is generated by inverse function method, which is for uniform random U,

    x=β(ln⁡11−U)1/αx=\beta\left(\ln\frac{1}{1-U}\right)^{1/\alpha}

  • How to generate random numbers on Excel.

AB
1DataDescription
20.5Value of parameter Alpha
30.5Value of parameter Beta
4FormulaDescription (Result)
5=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​