// Home / Documentation / Gallery of Distributions / Pareto distribution

Pareto distribution

Where do you meet this distribution?

Shape of Distribution

Basic Properties

  • Two parameters a, b are required.
    a>0, b>0
  • Continuous distribution defined on semi-infinite range x>b
  • This distribution is always asymmetric.

Probability

  • Cumulative distribution function
    F(x)=1-\left(\frac{b}{x}\right)^a
  • Probability density function
    f(x)=\frac{ab^a}{x^{a+1}}
  • 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
    F^{-1}(P)=\frac{b}{(1-P)^{1/a}}
  • 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
    \frac{ab}{a-1}
  • 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
    \frac{ab^2}{(a-1)^2(a-2)}

    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
    \sqrt{\frac{a-2}{a}}\frac{2(a+1)}{a-3}
  • 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
    \frac{6(a^3+a^2-6a-2)}{a(a-3)(a-4)}
  • 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,
    x=\frac{b}{(1-U)^{1/a}}
  • 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

 

Comments are closed.