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Statistics[Distributions]

 Logistic
 logistic distribution

 Calling Sequence Logistic(a, b) LogisticDistribution(a, b)

Parameters

 a - location parameter b - scale parameter

Description

 • The logistic distribution is a continuous probability distribution with probability density function given by:

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

 subject to the following conditions:

$a::\mathrm{real},0

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

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{Logistic}\left(a,b\right)\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\frac{{{ⅇ}}^{\frac{{u}{-}{a}}{{b}}}}{{b}{}{\left({1}{+}{{ⅇ}}^{\frac{{u}{-}{a}}{{b}}}\right)}^{{2}}}$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\frac{{{ⅇ}}^{\frac{{0.5}{-}{1.}{}{a}}{{b}}}}{{b}{}{\left({1.}{+}{{ⅇ}}^{\frac{{0.5}{-}{1.}{}{a}}{{b}}}\right)}^{{2}}}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 ${a}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 $\frac{{{b}}^{{2}}{}{{\mathrm{\pi }}}^{{2}}}{{3}}$ (4)

References

 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.

 See Also