The logistic distribution is a continuous probability distribution with probability density function given by:
subject to the following conditions:
The logistic variate Logistic(a,b) is related to the standardized variate Logistic(0,1) by Logistic(0,1) ~ (Logistic(a,b)-a)/b.
The standard logistic variate is related to the standard Exponential variate by Logistic(0,1) -log(exp(-Exponential(1))/(1+exp(-Exponential(1)))).
The logistic variate with location parameter 0 and scale parameter b is related to two independent Gumbel variates G1 and G2 by Logistic(0,b) ~ G1 - G2.
The standardized logistic variate is related to the Pareto variate with location parameter a and shape parameter c by Logistic(0,1) −log⁡Pareto⁡a,cac−1.
The standardized logistic variate is related to the Power variate with scale parameter 1 and shape parameter c by Logistic(0,1) ~ -log(Power(1,c)^(-c)-1).
Note that the Logistic command is inert and should be used in combination with the RandomVariable command.
X ≔ RandomVariable⁡Logistic⁡a,b:
Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.
Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.
Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.
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