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Exponential distribution

Where do you meet this distribution?

  • The lengths of the inter-arrival times in a homogeneous Poisson process
  • Nuclear physics : The time until a radioactive particle decays
  • Statistical mechanics : Molecular distribution in uniform gravitational field
  • Risk management : The time until default in reduced form credit risk modeling

Shape of Distribution

Basic Properties

  • A parameter \beta is required.
    \beta>0

    This parameter is Mean of the distribution.

  • Continuous distribution defined on semi-infinite range x \geq 0
  • This distribution is always asymmetric.

Probability

  • Cumulative distribution function
    F(x)=1-\exp\left(-\frac{x}{\beta}\right)
  • Probability density function
    f(x)=\frac{1}{\beta}\exp\left(-\frac{x}{\beta}\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 Beta
    Formula Description (Result)
    =1-EXP(-A2/A3) Cumulative distribution function for the terms above
    =EXP(-A2/A3)/A3 Probability density function for the terms above

Quantile

  • Inverse function of cumulative distribution function
    F^{-1}(P)=-\beta\ln(1-P)
  • How to compute this on Excel.
     
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    A B
    Data Description
    0.5 Probability associated with the distribution
    1.7 Value of parameter Beta
    Formula Description (Result)
    =-A3*LN(1-A2) 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 \beta.

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

  • Standard deviation of the distribution is given as \beta.

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

  • Skewness is 2.

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

  • Kurtosis is 6.

Random Numbers

  • Random number x is generated by inverse function method, which is for uniform random U,
    x=-\beta\ln(1-U)
  • How to generate random numbers on Excel.
     
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    Data Description
    0.5 Value of parameter Beta
    Formula Description (Result)
    =-A2*LN(1-NTRAND(100)) 100 exponential 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 A4:A103 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

NtRand Functions

Not supported yet

Reference

 

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