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The standard logistic distribution is a logistic distribution with location parameter 0 and scale parameter 1. It represents the log odds of a probability p chosen uniformly at random from the interval [0, 1].

Probability Density Function

f ( x ) = e − x ( 1 + e − x ) 2
F ( x ) = 1 1 + e − x

Support

− ∞ < x < ∞

Mean

Variance

Example
A probability p is chosen randomly from the interval [0, 1]. The log odds given by log(p/(1 − p)) have a standard logistic distribution.
Suppose P has a standard Pareto distribution. Then log(P − 1) has a standard logistic distribution.
Suppose Y has an exponential(1) distribution. Then log(eY − 1) has a standard logistic distribution.

X ∼ Standard Logistic

Chart of the standard logistic distribution Chart area for displaying the standard logistic pdf, cdf, and simulation

E(X) = , Var(X) =

The shape of the pdf of the standard logistic distribution is very similar to that of the standard normal distribution, unimodal and symmetric about zero. The variance is, however, greater than that of a standard normal distribution.

The shape of the cdf of the standard logistic distribution is very similar to that of the standard normal distribution, with the same property that F(-x) = 1 - F(x). The variance is, however, greater than that of a standard normal distribution.

The illustration above shows a point p chosen from a standard uniform distribution. The random variable X = log(p/(1 − p)) has a standard logistic distribution.

The simulation above shows a point p chosen from a standard uniform distribution on the y-axis. The light blue circle on the x-axis shows the log odds of p, given by the random variable X = log(p/(1 − p)). This has a standard logistic distribution. The histogram accumulates the results of each simulation.

Y = μ + βX ∼ Logistic(μ, β) eX + 1 ∼ Standard Pareto 1/(1 + e−X) ∼ Standard Uniform

Chart of the related distribution Chart area for displaying the related pdf, cdf, and simulation

E(Y) = , Var(Y) =

This follows since the logistic distribution is a location-scale family of distributions.

Note that a standard logistic random variable X represents the log odds of a standard uniform random variable. So, eX yields the odds, and adding 1 gives the reciprocal of a standard uniform random variable. Hence, a standard Pareto random variable is the reciprocal of a standard uniform random variable.

This is an example of the probability integral transformation. The transformation can also be stated as Y = FX(X), where FX is the standard logistic cdf. This type of transformation always results in a standard uniform random variable when FX is continuous.