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# Laplace distribution

## Shape of Distribution

### Basic Properties

• Two parameters $\mu, \phi$ are required.
$\phi>0$
• Continuous distribution defined on entire range
• This distribution is always symmetric.

### Probability

• Cumulative distribution function
$F(x)=\begin{cases}\frac{1}{2}\exp\left(\frac{x-\mu}{\phi}\right)\;&(x<\mu)\\1-\frac{1}{2}\exp\left(-\frac{x-\mu}{\phi}\right)\;&(x\geq \mu)\end{cases}[/latex]
• Probability density function
[latex]f(x)=\frac{1}{2\phi}\exp\left(-\frac{|x-\mu|}{\phi}\right)$
• 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 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
$F^{-1}(P)=\begin{cases}\phi\ln 2P+\mu\;&(P<0.5)\\-(\phi\ln 2(1-P)+\mu)\;&(P\geq 0.5)\end{cases}$
• 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 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 $\mu$.

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

• Variance of the distribution is given as
$2\phi^2$

Standard Deviation is a positive square root of Variance.

• How to compute this on Excel

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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 $0$.

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

• Kurtosis of the distribution is $3$.

## Random Numbers

• Random number x is generated by inverse function method, which is for uniform random U,
$x=\begin{cases}\phi\ln 2U+\mu\;&(U<0.5)\\-(\phi\ln 2(1-U)+\mu)\;&(U\geq 0.5)\end{cases}$
• How to generate random numbers on Excel.

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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