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Laplace Distribution

Where will you meet this distribution?​

Shape of Distribution​

Basic Properties​

  • Two parameters μ,ϕ\mu, \phi are required.

    ϕ>0\phi>0

  • Continuous distribution defined on entire range

  • This distribution is always symmetric.

Probability​

  • Cumulative distribution function

    F(x)={12exp⁡(x−μϕ)  (x<μ)1−12exp⁡(−x−μϕ)  (x≥μ)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}

  • Probability density function

    f(x)=12ϕexp⁡(−∣x−μ∣ϕ)f(x)=\frac{1}{2\phi}\exp\left(-\frac{|x-\mu|}{\phi}\right)

  • How to compute these on Excel.

AB
1DataDescription
20.5Value for which you want the distribution
38Value of parameter Mu
42Value of parameter Phi
5=(A2-A3)/A4Standardized variable z
6FormulaDescription (Result)
7=IF(A2<A3,0.5*EXP(A5),1-0.5*EXP(-A5))Cumulative distribution function for the terms above
8=0.5*EXP(-ABS(A5))/A4Probability density function for the terms above

Quantile​

  • Inverse function of cumulative distribution function

    F−1(P)={ϕln⁡2P+μ(P<0.5)−(ϕln⁡2(1−P)+μ)(P≥0.5)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.

AB
1DataDescription
20.7Probability associated with the distribution
31.7Value of parameter Mu
40.9Value of parameter Phi
5FormulaDescription (Result)
6=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 μ\mu.

Standard Deviation -- How wide does the distribution spread? (Definition)​

AB
1DataDescription
22Value of parameter Phi
3FormulaDescription (Result)
4=SQRT(2)*A2Standard deviation of the distribution for the terms above

Skewness -- Which side is the distribution distorted into? (Definition)​

  • Skewness of the distribution is 00.

Kurtosis -- Sharp or Dull, consequently Fat Tail or Thin Tail (Definition)​

  • Kurtosis of the distribution is 33.

Random Numbers​

  • Random number x is generated by inverse function method, which is for uniform random U,

    x={ϕln⁡2U+μ(U<0.5)−(ϕln⁡2(1−U)+μ)(U≥0.5)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.

AB
1DataDescription
20.5Value of parameter Mu
30.5Value of parameter Phi
4FormulaDescription (Result)
5=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​