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# Johnson SB distribution

## Shape of Distribution

### Basic Properties

• Four parameters $\gamma, \delta,\lambda,\xi$ are required.
$\delta>0,\lambda>0$
• Continuous distribution defined on bounded range $\xi\leq x \leq \xi+\lambda$
• This distribution can be symmetric or asymmetric.

### Probability

• Cumulative distribution function
$F(x)=\Phi\left(\gamma+\delta\ln\frac{z}{1-z}\right)$

where

$z=\frac{x-\xi}{\lambda}$

and $\Phi(\cdot)$ is cumulative distribution function of standard normal distribution.

• Probability density function
$f(x)=\frac{\delta}{\lambda\sqrt{2\pi}z(1-z)}\exp\left[-\frac{1}{2}\left(\gamma+\delta\ln\frac{z}{1-z}\right)^2\right]$
• 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 Gamma
2 Value of parameter Delta
2 Value of parameter Lambda
2 Value of parameter Xi
=(A2-A5)/A4 Standardized variable z
Formula Description (Result)
=NORMSDIST(A3+A4*LN(A7/(1-A7))) Cumulative distribution function for the terms above
=A4*EXP(-0.5*(A3+A4*LN(A7/(1-A7)))^2)/(SQRT(2*PI())*A5*A7*(1-A7)) Probability density function for the terms above

### Quantile

• Inverse function of cumulative distribution function
$F^{-1}(P)=\frac{\lambda\exp\left(\frac{\Phi^{-1}(P)-\gamma}{\delta}\right)}{1+\exp\left(\frac{\Phi^{-1}(P)-\gamma}{\delta}\right)}+\xi$

where $\Phi(\cdot)$ is cumulative distribution function of standard normal distribution.

• 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 Gamma
0.9 Value of parameter Delta
0.9 Value of parameter Lambda
0.9 Value of parameter Xi
Formula Description (Result)
=A5*EXP((NORMSINV(A2)-A3)/A4)/(1+EXP((NORMSINV(A2)-A3)/A4))+A6 Inverse of the cumulative distribution function for the terms above

## Random Numbers

• Random number x is generated by inverse function method, which is for uniform random U,
$x=\frac{\lambda\exp\left(\frac{\Phi^{-1}(U)-\gamma}{\delta}\right)}{1+\exp\left(\frac{\Phi^{-1}(U)-\gamma}{\delta}\right)}+\xi$

where $\Phi(\cdot)$ is cumulative distribution function of standard normal distribution.

• How to generate random numbers on Excel.

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A B
Data Description
0.5 Value of parameter Gamma
0.5 Value of parameter Delta
0.5 Value of parameter Lambda
0.5 Value of parameter Xi
Formula Description (Result)
=A4*EXP((NORMSINV(NTRAND(100))-A2)/A3)/(1+EXP((NORMSINV(NTRAND(100))-A2)/A3))+A5 100 Johnson SB 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

Not supported yet