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# Logistic distribution

## Where do you meet this distribution?

• Biology : how species populations grow in competition
• Energy : the diffusion and substitution of primary energy so
• Marketing : the diffusion of new-product sales
• Psychology : learning curve
• Technology : to describe how new technologies diffuse and substitute for each other

## Shape of Distribution

### Basic Properties

• Two parameters $m, b$ are required (How can you get these).
$b>0$
• Continuous distribution defined on entire range
• This distribution is always symmetric.

### Probability

• Cumulative distribution function
$F(x)=\frac{1}{2}\left[1+\tanh\left(\frac{x-m}{2b}\right)\right]$
• Probability density function
$f(x)=\frac{1}{4 b}\text{sech}^2\left(\frac{x-m}{2b}\right)$
• 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 M
2 Value of parameter B
Formula Description (Result)
=NTLOGISTICDIST(A2,A3,A4,TRUE) Cumulative distribution function for the terms above
=NTLOGISTICDIST(A2,A3,A4,FALSE) Probability density function for the terms above

• Function reference : NTLOGISTICDIST

### Quantile

• Inverse function of cumulative distribution function
$F^{-1}(P)=2b\tanh^{-1}(2P-1)+m$
• 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 M
0.9 Value of parameter B
Formula Description (Result)
=NTLOGISTICINV(A2,A3,A4) Inverse of the cumulative distribution function for the terms above
• Function reference : NTLOGISTICINV

## Characteristics

### Mean – Where is the “center” of the distribution? (Definition)

• Mean of the distribution is given as $m$

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

• Variance of the distribution is given as
$\frac{\pi^2b^2}{3}$

Standard Deviation is a positive square root of Variance.

• How to compute this on Excel

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A B
Data Description
2 Value of parameter B
Formula Description (Result)
=NTLOGISTICSTDEV(A2) Standard deviation of the distribution for the terms above
• Function reference : NTLOGISTICSTDEV

### 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 $1.2$.

## Random Numbers

• Random number x is generated by inverse function method, which is for uniform random U,
$x=2b\tanh^{-1}(2U-1)+m$
• How to generate random numbers on Excel.

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A B
Data Description
0.5 Value of parameter M
0.5 Value of parameter B
Formula Description (Result)
=NTRANDLOGISTIC(100,A2,A3) 100 Logistic 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.