Click or press any key to close help Help screen describing the interactive elements

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

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, visualization, and simulation

E(X) = , Var(X) =

The shape of the standard logistic distribution is very similar to the standard normal distribution, unimodal and symmetric about zero. 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, visualization, 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.