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 density 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
Support
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)
E(X) = , Var(X) =
The shape of the distribution is very similar to 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 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 Xi - max Yi + 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 Xi - max Yi + 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.