// Home / Documentation / Gallery of Distributions / Gumbel (Type I) distribution

# Gumbel (Type I) distribution

## Shape of Distribution

### Basic Properties

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

### Probability

• Cumulative distribution function
$F(x)=\exp\left[-\exp\left(-\frac{x-\alpha}{\beta}\right)\right]$
• Probability density function
$f(x)=\frac{1}{\beta}\exp\left(-\frac{x-\alpha}{\beta}\right)\exp\left[-\exp\left(-\frac{x-\alpha}{\beta}\right)\right]$
• 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)
=NTGUMBELDIST(A2,A3,A4,TRUE) Cumulative distribution function for the terms above
=NTGUMBELDIST(A2,A3,A4,FALSE) Probability density function for the terms above

• Function reference : NTGUMBELDIST

### Quantile

• Inverse function of cumulative distribution function
$F^{-1}(P)=\alpha-\beta\ln\ln\frac{1}{P}$
• 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)
=GUMBELINV(A2,A3,A4) Inverse of the cumulative distribution function for the terms above
• Function reference : NTGUMBELINV

## Characteristics

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

• Mean of the distribution is given as
$\alpha+\gamma\beta$

where $\gamma$ is Euler’s constant.

• 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)
=NTGUMBELMEAN(A2,A3) Mean of the distribution for the terms above
• Function reference : NTGUMBELMEAN

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

• Variance of the distribution is given as
$\beta^2\zeta(2)$

where $\zeta(\cdot)$ is Riemann zeta function.

Standard Deviation is a positive square root of Variance.

• How to compute this on Excel

1
2
3
4
A B
Data Description
2 Value of parameter B
Formula Description (Result)
=NTGUMBELSTDEV(A2) Standard deviation of the distribution for the terms above
• Function reference : NTGUMBELSTDEV

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

• Skewness of the distribution is given as
$-\frac{12\sqrt{6}\zeta(3)}{\pi^3}=-1.139547099\cdots$

where $\zeta(\cdot)$ is Riemann zeta function.

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

• Kurtosis of the distribution is $2.4$

## Random Numbers

• Random number x is generated by inverse function method, which is for uniform random U,
$x=\alpha-\beta\ln\ln\frac{1}{U}$
• 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)
=NTRANDGUMBEL(100,A2,A3,0) 100 Gumbel Type I 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.