Continuous probability distribution
Logistic distribution
Probability density function ![Standard logistic PDF](//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Logisticpdfunction.svg/320px-Logisticpdfunction.svg.png) |
Cumulative distribution function ![Standard logistic CDF](//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Logistic_cdf.svg/320px-Logistic_cdf.svg.png) |
Parameters |
location (real)
scale (real) |
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Support |
![{\displaystyle x\in (-\infty ,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7aea9be5e96822459afc5c7d9f911a586290dc5) |
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PDF |
![{\displaystyle {\frac {e^{-(x-\mu )/s}}{s\left(1+e^{-(x-\mu )/s}\right)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d42088b22c0c7577eb7a68d827061870368d054) |
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CDF |
![{\displaystyle {\frac {1}{1+e^{-(x-\mu )/s}}}={\frac {1+\tanh {\frac {x-\mu }{2s}}}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/841f3507b4810bd25d4211115d9d1720edfa96eb) |
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Quantile |
![{\displaystyle \mu +s\log \left({\frac {p}{1-p}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30fdc436e07663fa4f425651b2d506f86c740dc2) |
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Mean |
![{\displaystyle \mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161) |
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Median |
![{\displaystyle \mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161) |
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Mode |
![{\displaystyle \mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161) |
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Variance |
![{\displaystyle {\frac {s^{2}\pi ^{2}}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9e1141f49e6711939a3757111a9dc5576990db0) |
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Skewness |
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) |
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Excess kurtosis |
![{\displaystyle 6/5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/927a3fc853d2c544bcb48b4404bb7c0fbbbc6b4b) |
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Entropy |
![{\displaystyle \ln s+2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42bb631b6efc5eb57f1280f1233666e4f8f384b3) |
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MGF |
![{\displaystyle e^{\mu t}\mathrm {B} (1-st,1+st)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f1c97db3e7de686752b306a8d175911fd896d7f) for ![{\displaystyle t\in (-1/s,1/s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1962886fa620e6be1089ca78bc0cffca50febc75) and is the Beta function |
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CF |
![{\displaystyle e^{it\mu }{\frac {\pi st}{\sinh(\pi st)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bba837b5b7bb4a914d84aceb291c5ae49b13bf7a) |
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Expected shortfall |
where is the binary entropy function[1] ![{\displaystyle H(p)=-p\ln(p)-(1-p)\ln(1-p)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9993cb2dad1fab4405a10b508dc1cbb9861ca816) |
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In probability theory and statistics, the logistic distribution is a continuous probability distribution. Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks. It resembles the normal distribution in shape but has heavier tails (higher kurtosis). The logistic distribution is a special case of the Tukey lambda distribution.