# Laplace distribution

## Where will you meet this distribution?

- Marketing
“On the Laplace Distribution of Firms Growth Rates” by Giulio Bottazzi and Angelo Secchi

- Computer graphics

## Shape of Distribution

### Basic Properties

- Two parameters are required.
- Continuous distribution defined on entire range
- This distribution is always symmetric.

### Probability

- Cumulative distribution function
- How to compute these on Excel.

1 2 3 4 5 6 7

8

A B Data Description 0.5 Value for which you want the distribution 8 Value of parameter Mu 2 Value of parameter Phi =(A2-A3)/A4 Standardized variable z Formula Description (Result) =IF(A2<A3,0.5*EXP(A5),1-0.5*EXP(-A5)) Cumulative distribution function for the terms above =0.5*EXP(-ABS(A5))/A4 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 Mu 0.9 Value of parameter Phi Formula Description (Result) =IF(P<0.5,A4*LN(2*A2)+A3,-(A4*LN(2*(1-A2))+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 .

### 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 A B Data Description 2 Value of parameter Phi Formula Description (Result) =SQRT(2)*A2 Standard deviation of the distribution for the terms above

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

- Skewness of the distribution is .

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

- Kurtosis of the distribution is .

## 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 Mu 0.5 Value of parameter Phi Formula Description (Result) =IF(NTRAND(100)<0.5,A3*LN(2*NTRAND(100))+A2,-(A3*LN(2*(1-NTRAND(100)))+A2)) 100 Laplace 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 – Laplace distribution
- Wikipedia – Laplace distribution
- Statistics Online Computational Resource