# Pareto 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.

1 2 3 4 5 6 7 A B Data Description 5 Value for which you want the distribution 8 Value of parameter A 2 Value of parameter B Formula Description (Result) =1-POWER(A4/A2,A3) Cumulative distribution function for the terms above =A3*A4^A3/POWER(A2,A3+1) Probability density function for the terms above

### Quantile

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

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

- Variance of the distribution is given as
Standard Deviation is a positive square root of Variance.

- How to compute this on Excel

1 2 3 4 5

A B Data Description 8 Value of parameter A 2 Value of parameter B Formula Description (Result) =A3/(A2-1)*SQRT(A2/(A2-2)) 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
- How to compute this on Excel

1 2 3 4 A B Data Description 8 Value of parameter A Formula Description (Result) =SQRT((A2-2)/A2)*2*(A2+1)/(A2-3) 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
- How to compute this on Excel

1 2 3 4

A B Data Description 8 Value of parameter A Formula Description (Result) =6*(A2^3+A2^2-6A2-2)/(A2*(A2-3)*(A2-4)) 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,
- How to generate random numbers on Excel.

1 2 3 4 5

A B Data Description 0.5 Value of parameter A 2 Value of parameter B Formula Description (Result) =A3/POWER(1-NTRAND(100),1/A2) 100 Pareto 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

## Reference

- Wolfram Mathworld – Pareto distribution
- Wikipedia – Pareto distribution
- Statistics Online Computational Resource