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.

	A	B
1	Data	Description
2	0.5	Value for which you want the distribution
3	8	Value of parameter M
4	2	Value of parameter B
5	Formula	Description (Result)
6	=NTLOGISTICDIST(A2,A3,A4,TRUE)	Cumulative distribution function for the terms above
7	=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.

	A	B
1	Data	Description
2	0.7	Probability associated with the distribution
3	1.7	Value of parameter M
4	0.9	Value of parameter B
5	Formula	Description (Result)
6	=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

	A	B
1	Data	Description
2	2	Value of parameter B
3	Formula	Description (Result)
4	=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.

	A	B
1	Data	Description
2	0.5	Value of parameter M
3	0.5	Value of parameter B
4	Formula	Description (Result)
5	=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

• If you already have parameters of the distribution
  • Generating random numbers based on Mersenne Twister algorithm: NTRANDLOGISTIC
  • Computing probability : NTLOGISTICDIST
  • Computing mean : NTLOGISTICMEAN
  • Computing standard deviation : NTLOGISTICSTDEV
  • Computing skewness : NTLOGISTICSKEW
  • Computing kurtosis : NTLOGISTICKURT
  • Computing moments above at once : NTLOGISTICMOM
• If you know mean and standard deviation of the distribution
  • Estimating parameters of the distribution:NTLOGISTICPARAM

Reference

• Wolfram Mathworld -- Logistic Distribution
• Wikipedia -- Logistic distribution
• Statistics Online Computational Resource