# Logistic distribution

**logisticbell-shaped curvelogistical**

In probability theory and statistics, the logistic distribution is a continuous probability distribution.wikipedia

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### Normal distribution

**normally distributednormalGaussian**

It resembles the normal distribution in shape but has heavier tails (higher kurtosis). This phrasing is common in the theory of discrete choice models, where the logistic distribution plays the same role in logistic regression as the normal distribution does in probit regression.

The normal distribution is sometimes informally called the bell curve. However, many other distributions are bell-shaped (such as the Cauchy, Student's t, and logistic distributions).

### Logistic regression

**logit modellogisticbinary logit model**

Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks.

Logistic regression measures the relationship between the categorical dependent variable and one or more independent variables by estimating probabilities using a logistic function, which is the cumulative logistic distribution.

### Logistic function

**logisticlogistic growthlogistic curve**

Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks.

which ties the logistic function into the logistic distribution.

### Tukey lambda distribution

**Tukey lambdaTukey's lambda distribution**

The logistic distribution is a special case of the Tukey lambda distribution.

The Tukey lambda distribution has a simple, closed form for the CDF and/or PDF only for a few exceptional values of the shape parameter, for example: λ = 2, 1, ½, 0 (see uniform distribution and the logistic distribution).

### Kurtosis

**excess kurtosisleptokurticplatykurtic**

It resembles the normal distribution in shape but has heavier tails (higher kurtosis).

Examples of leptokurtic distributions include the Student's t-distribution, Rayleigh distribution, Laplace distribution, exponential distribution, Poisson distribution and the logistic distribution.

### Quantile function

**quantileinverse distribution functionnormal quantile function**

The inverse cumulative distribution function (quantile function) of the logistic distribution is a generalization of the logit function.

The evaluation of quantile functions often involves numerical methods, as the example of the exponential distribution above is one of the few distributions where a closed-form expression can be found (others include the uniform, the Weibull, the Tukey lambda (which includes the logistic) and the log-logistic).

### Discrete choice

**nested logitdiscrete choice analysisdiscrete choice model**

This phrasing is common in the theory of discrete choice models, where the logistic distribution plays the same role in logistic regression as the normal distribution does in probit regression.

The unobserved term, ε n, is assumed to have a logistic distribution.

### Logit

**log-oddslogit functionlogit transformation**

The inverse cumulative distribution function (quantile function) of the logistic distribution is a generalization of the logit function.

In fact, the logit is the quantile function of the logistic distribution, while the probit is the quantile function of the normal distribution.

### Cumulative frequency analysis

**cumulative frequencyCumFreqfrequency distributions**

The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

*the normal distribution, the lognormal distribution, the logistic distribution, the loglogistic distribution, the exponential distribution, the Fréchet distribution, the Gumbel distribution, the Pareto distribution, the Weibull distribution and other

### Elo rating system

**ratingElo ratingElo**

Τhe United States Chess Federation and FIDE have switched its formula for calculating chess ratings from the normal distribution to the logistic distribution; see the article on Elo rating system (itself based on the normal distribution).

Therefore, the USCF and some chess sites use a formula based on the logistic distribution.

### Log-logistic distribution

**log logistic distributionlog-logisticLogLogistic**

If X ~ Logistic(μ, s) then exp(X) ~ LogLogistic, and exp(X) + γ ~ shifted log-logistic

The log-logistic distribution is the probability distribution of a random variable whose logarithm has a logistic distribution.

### Sigmoid function

**sigmoidalsigmoidsigmoid curve**

Sigmoid function

Sigmoid curves are also common in statistics as cumulative distribution functions (which go from 0 to 1), such as the integrals of the logistic distribution, the normal distribution, and Student's t probability density functions.

### Exponential distribution

**exponentialexponentially distributedexponentially**

* If X ~ Exponential(1) then

If X ~ Exp(1) then (logistic distribution):

### Shifted log-logistic distribution

**shifted log-logistic**

If X ~ Logistic(μ, s) then exp(X) ~ LogLogistic, and exp(X) + γ ~ shifted log-logistic

When \xi → 0, the shifted log-logistic reduces to the logistic distribution.

### Hyperbolic secant distribution

**GHS distributionhyperbolic secantsech distribution**

Logistic distribution mimics the sech distribution.

Johnson et al. (1995, p147) places this distribution in the context of a class of generalised forms of the logistic distribution, but use a different parameterisation of the standard distribution compared to that here.

### Probability theory

**theory of probabilityprobabilityprobability theorist**

In probability theory and statistics, the logistic distribution is a continuous probability distribution.

### Statistics

**statisticalstatistical analysisstatistician**

In probability theory and statistics, the logistic distribution is a continuous probability distribution.

### Cumulative distribution function

**distribution functionCDFcumulative probability distribution function**

Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks.

### Feedforward neural network

**feedforwardfeedforward neural networksfeedforward networks**

### Probability density function

**probability densitydensity functiondensity**

The probability density function (pdf) of the logistic distribution is given by:

### Hyperbolic function

**hyperbolic tangenthyperbolichyperbolic cosine**

Because the pdf can be expressed in terms of the square of the hyperbolic secant function "sech", it is sometimes referred to as the sech-square(d) distribution.

### Random variable

**random variablesrandom variationrandom**

In this equation, x is the random variable, μ is the mean, and s is a scale parameter proportional to the standard deviation.

### Mean

**mean valuepopulation meanaverage**

In this equation, x is the random variable, μ is the mean, and s is a scale parameter proportional to the standard deviation.

### Standard deviation

**standard deviationssample standard deviationsigma**

In this equation, x is the random variable, μ is the mean, and s is a scale parameter proportional to the standard deviation.

### Inverse function

**inverseinvertibleinvertible function**

The inverse cumulative distribution function (quantile function) of the logistic distribution is a generalization of the logit function.