Probability Playground: The Logistic Distribution

The logistic distribution is a continuous distribution that plays a role in logistic regression. This is used for modeling the probabilities of categorical variables which can only take a fixed set of values. The cumulative distribution function is the logistic function, which is often used to map the real numbers to probabilities in the interval [0, 1].

Parameter	Range	Description
μ	−∞ < μ < ∞	Location parameter
β	β > 0	Scale parameter

Probability Density Function

f (x; μ, β) = \frac{1}{β} \frac{e^{- (x - μ) / β}}{{[1 + e^{- (x - μ) / β}]}^{2}}

F (x; μ, β) = \frac{1}{1 + e^{- (x - μ) / β}}

Support

- \infty < x < \infty

Mean

Variance

Example	μ	β
Suppose a probability p is chosen randomly from the interval [0, 1]. Then the log odds given by log(p/(1 − p)) has a logistic(0, 1) distribution.	0.000	1.000
Monthly percent returns for the S&P 500 index for the period 1871 - 2022 have an approximate logistic(0.96, 2.02) distribution.	0.9600	2.020
Monthly percent changes for gold prices for the period 1993 - 2022 have an approximate logistic(0.53, 1.96) distribution.	0.5300	1.960

X ∼ Logistic(μ, β)

μ = β =

E(X) = , Var(X) =

The pdf of the logistic distribution is very similar to that of the normal distribution, unimodal and symmetric about the mean μ. The tails are, however, slightly heavier than those of a normal distribution with the same variance.

The cdf of the logistic distribution is very similar to the that of the normal distribution, with the same property that F(μ - x) = 1 - F(μ + x).

The illustration above shows two samples of n points (marked red and purple) chosen independently and at random from an exponential(β) distribution. The random variable max X_i − max Y_i + μ converges to a logistic(μ, β) distribution as n approaches infinity, where μ denotes a location parameter.

The simulation above shows two samples of n points (marked red and purple) chosen independently and at random from an exponential(β) distribution. The light blue line shows the value of max X_i − max Y_i + μ, where μ denotes a location parameter. The distribution of this value converges to a logistic(μ, β) distribution as n approaches infinity. The histogram accumulates the results of each simulation.

Y = (X − μ)/β ∼ Standard Logistic

E(Y) = , Var(Y) =

Proof

X ∼ Logistic(μ, β)

Y = (X − μ)/β ∼ Standard Logistic