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# Gumbel (Type II) distribution

## Shape of Distribution

### Basic Properties

• Two parameters $a, b$ are required.
$a>0, b>0$
• Continuous distribution defined on semi-infinite range $x>0$
• This distribution is always asymmetric.

### Probability

• Cumulative distribution function
$F(x)=\exp\left(-bx^{-a}\right)$
• Probability density function
$f(x)=abx^{-a-1}\exp\left(-bx^{-a}\right)$
• 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
$F^{-1}(P)=\left(-\frac{\ln P}{b}\right)^{-1/a}$
• 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,
$x=\left(-\frac{\ln U}{b}\right)^{-1/a}$
• 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