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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
  • Epidemiology: spreading of epidemics
  • 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

    Sample distribution

  • 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.

NtRand Functions

Reference

 


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