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Pareto 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.

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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.

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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

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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

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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

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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

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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.

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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

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