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# Weibull 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.
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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

### 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.
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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
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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
$\mu^\prime(2)-m^2$

where

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

, $m$ is mean of the distribution and $\Gamma(\cdot)$ is gamma function

Standard Deviation is a positive square root of Variance.

• How to compute this on Excel
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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
$\frac{1}{\sigma^3}\left[\mu^\prime(3)-3m\sigma^2-m^3\right]$

where

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

, $m$ is mean of the distribution, $\sigma^2$ is variance of the distribution, $\Gamma(\cdot)$ is gamma function and $\sigma$ is standard deviation of the distribution.

• How to compute this on Excel
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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
$\frac{{\mu^\prime}(4)-4\gamma_1\sigma^3m-6m^2\sigma^2-m^4}{\sigma^4}-3$

where

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

, $\Gamma(\cdot)$ is gamma function, $m$ is mean of the distribution, $\sigma$ is standard deviation of the distribution and $\gamma_1$ is skewness of the distribution.

• How to compute this on Excel
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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.
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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.