Mathematics in machine learning -- common probability distribution (XIII): Logistic Distribution

Catalogues ：《 Mathematics in machine learning 》 General catalogue
Related articles ：
· Common probability distribution （ One ）： Bernoulli distribution （Bernoulli Distribution ）
· Common probability distribution （ Two ）： Category distribution （Multinoulli Distribution ）
· Common probability distribution （ 3、 ... and ）： The binomial distribution （Binomial Distribution ）
· Common probability distribution （ Four ）： Uniform distribution （Uniform Distribution ）
· Common probability distribution （ 5、 ... and ）： Gaussian distribution （Gaussian Distribution ）/ Normal distribution （Normal Distribution ）
· Common probability distribution （ 6、 ... and ）： An index distribution （Exponential Distribution ）
· Common probability distribution （ 7、 ... and ）： Laplacian distribution （Laplace Distribution ）
· Common probability distribution （ 8、 ... and ）： Dirac distribution （Dirac Distribution ）
· Common probability distribution （ Nine ）： Empirical distribution （Empirical Distribution ）
· Common probability distribution （ Ten ）： Beta distribution （Beta Distribution ）
· Common probability distribution （ 11、 ... and ）： Dirichlet distribution （Dirichlet Distribution ）
· Common probability distribution （ Twelve ）： Structured probability model / Graph model
· Common probability distribution （ 13、 ... and ）： Logistic distribution （Logistic Distribution ）

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set up $X$ It's a continuous random variable ,$X$ Obey the logistic distribution （Logistic Distribution ） Refer to $X$ It has the following distribution function and density function ：
$$F(x)=P(X\le x)=\frac{1}{1+e^{-\frac{(x-\mu )}{\gamma }}}$$

$$f(x)=F^{\prime }(x)=\frac{e^{-\frac{(x-\mu )}{\gamma }}}{\gamma (1+e^{-\frac{(x-\mu )}{\gamma }}{)}^2}$$

among ,$\mu$ For position parameters ,$\gamma >0$ For shape parameters .

Density function of logistic distribution $f(x)$ And distribution function $F(x)$ The figure of is shown in the figure below . The distribution function belongs to the logical tearing function , The graph is a S Shape curve （sigmoid curve）. The curve is in point $(\mu ,\frac{1}{2})$ It's central symmetry , The meet
$$F(-x+\mu )-\frac{1}{2}=-F(x-\mu )+\frac{1}{2}$$

The curve grows faster near the center , Slow growth at both ends . shape parameter $\gamma$ The smaller the value of , The faster the curve grows near the center .