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