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Where do you meet this distribution?

• Economic

“Job Insecurity isnâ€™t Always Efficient” by David J. Balan and Dan Hanner

Shape of Distribution

Basic Properties

• Two parameters $a, b$ are required.
$a

These parameters are minimum and maximum value of variable respectively.

• Continuous distribution defined on bounded range $a\leq x \leq b$
• This distribution is always symmetric.

Probability

• Cumulative distribution function
$F(x)=\frac{a}{3}\left[(x-b)^3+(b-a)^3\right]$

where

$\alpha=\frac{12}{(b-a)^3},\;\beta=\frac{a+b}{2}$
• Probability density function
$f(x)=a(x-b)^2$
• How to compute these on Excel.

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A B
Data Description
2 Value for which you want the distribution
1 Value of parameter A
5 Value of parameter B
=12/((A4-A3)^3) Vertical scale
=(A3+A4)/2 Mean of the distribution
Formula Description (Result)
=A5*((A2-A6)^3+(A6-A5)^3)/3 Cumulative distribution function for the terms above
=A5*(A2-A6)^2 Probability density function for the terms above

Quantile

• Inverse of cumulative distribution function
$F^{-1}(P)=\left[\frac{3P}{\alpha}-(\beta-\alpha)^3\right]^{1/3}+\beta$

where

$\alpha=\frac{12}{(b-a)^3},\;\beta=\frac{a+b}{2}$
• How to compute this on Excel.

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A B
Data Description
0.5 Probability associated with the distribution
1 Value of parameter A
5 Value of parameter B
=12/((A4-A3)^3) Vertical scale
=(A3+A4)/2 Mean of the distribution
Formula Description (Result)
=(3*A2/A5-(A6-A5)^3)^(1/3)+A6 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
$\frac{a+b}{2}$
• 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)
=(A2+A3)/2 Mean of the distribution for the terms above

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

• Variance of the distribution is given as
$\frac{3}{20}(b-a)^2$

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)
=SQRT(3)*(A3-A2)/(2*SQRT(5)) 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 given as
$\frac{3}{112}(a-b)^4$
• 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)
=3*(A3-A2)^4/112 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,
$x=\left[\frac{3U}{\alpha}-(\beta-\alpha)^3\right]^{1/3}+\beta$

where

$\alpha=\frac{12}{(b-a)^3},\;\beta=\frac{a+b}{2}$
• How to generate random numbers on Excel.

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A B
Data Description
1 Value of parameter A
5 Value of parameter B
=12/((A3-A2)^3) Vertical scale
=(A2+A3)/2 Mean of the distribution
Formula Description (Result)
=(3*NTRAND(100)/A2-(A3-A2)^3)^(1/3)+A3 100 U-quadratic 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 A7:A106 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

NtRand Functions

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