# Logistic分布

Logistic分布的定义：设$$X$$是连续随机变量，$$X$$服从Logistic分布是指具有下列分布函数和密度函数：

$$
F(x)=P(X\leqslant x)=    \dfrac{1}{1+e^{-(x-\mu)/\gamma}}
$$

$$
f(x)=F'(X\leqslant x)=    \dfrac{e^{-(x-\mu)/\gamma}}{\gamma(1+e^{-(x-\mu)/\gamma})^2}
$$

其中，$$\mu$$为位置参数，$$\gamma \gt0$$为形状参数。

概率分布函数如下（$$\mu$$是位置函数，改变它可以平移图形）：

![](/files/-LAHC8G1JiwmkSkux9hw)

分布函数属于Logistic函数，是一条S形曲线（sigmoid curve）。该曲线以点$$(\mu, \dfrac{1}{2})$$为中心对称，即满足

$$
F(-x+\mu)-    \dfrac{1}{2} = -F(x+\mu) +    \dfrac{1}{2}
$$

曲线在中心附近增长速度比较快，两端增长速度比较慢。形状参数$$\gamma$$的值越小，曲线在中心附近增长的越快。

概率密度函数：

![](/files/-LAHC8GMnR1CZ5g8saFR)


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