# Gumbel (Type II) distribution

## Where do you meet this distribution?

- Meteorology, Seismology : Model of extreme event (Extreme value theory)
- Risk management – Operational risk

## Shape of Distribution

### Basic Properties

- Two parameters are required.
- Continuous distribution defined on semi-infinite range
- This distribution is always asymmetric.

### Probability

- Cumulative distribution function
- Probability density function
- 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 A 2 Value of parameter B Formula Description (Result) =EXP(-A4*A2^(-A3)) Cumulative distribution function for the terms above =A3*A4*A2^(-A3-1)*EXP(-A4*A2^(-A3)) Probability density function for the terms above

### Quantile

- Inverse function of cumulative distribution function
- 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 A 0.9 Value of parameter B Formula Description (Result) =POWER(-LN(A2)/A4,-1/A3) Inverse of the cumulative distribution function for the terms above

## Random Numbers

- Random number x is generated by inverse function method, which is for uniform random U,
- How to generate random numbers on Excel.

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A B Data Description 0.5 Value of parameter A 2 Value of parameter B Formula Description (Result) =POWER(-A2*LN(-NTRAND(100)/A3),1/A2) 100 Gumbel (Type II) 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

Not supported yet