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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)$-\log \left({\left(\frac{\mathrm{Pareto}\left(a\,c\right)}{a}\right)}^c-1\right)$.

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.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{Logistic}\left(a\,b\right)\right)\:$

$\mathrm{PDF}\left(X\,u\right)$

$\frac{{\ⅇ}^{\frac{u-a}{b}}}{b{\left(1+{\ⅇ}^{\frac{u-a}{b}}\right)}^2}$

$\mathrm{PDF}\left(X\,0.5\right)$

$\frac{{\ⅇ}^{\frac{0.5-1.a}{b}}}{b{\left(1.+{\ⅇ}^{\frac{0.5-1.a}{b}}\right)}^2}$

$\mathrm{Mean}\left(X\right)$

$a$

$\mathrm{Variance}\left(X\right)$

$\frac{b^2{\mathrm{\pi }}^2}{3}$

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.