拐点与正态分布 CDF 的拐点

拐点（inflection point）指函数曲率发生改变的点，典型判别是二阶导数在该点附近变号（或等价地凸凹性改变）。

对 $X\sim N(\mu,\sigma^2)$，CDF 为

$$F(x)=\Phi\left(\frac{x-\mu}{\sigma}\right).$$

有

$$F'(x)=\frac{1}{\sigma}\varphi\left(\frac{x-\mu}{\sigma}\right),$$

其中 $\varphi$ 为标准正态密度。再求导：

$$F''(x)=\frac{1}{\sigma^2}\varphi'\left(\frac{x-\mu}{\sigma}\right) =\frac{1}{\sigma^2}\left(-\frac{x-\mu}{\sigma}\right)\varphi\left(\frac{x-\mu}{\sigma}\right).$$

令 $F''(x)=0$ 得 $x=\mu$。因此拐点坐标为

$$\boxed{\bigl(\mu,\tfrac12\bigr)}.$$

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英文解析

An inflection point refers to a point where the curvature of a function changes; a typical criterion is that the second derivative changes sign in the neighborhood of this point (or equivalently, the concavity changes).

For $X\sim N(\mu,\sigma^2)$, the CDF is

$$F(x)=\Phi\left(\frac{x-\mu}{\sigma}\right).$$

We have

$$F'(x)=\frac{1}{\sigma}\varphi\left(\frac{x-\mu}{\sigma}\right),$$

where $\varphi$ is the standard normal density. Differentiating again:

$$F''(x)=\frac{1}{\sigma^2}\varphi'\left(\frac{x-\mu}{\sigma}\right) =\frac{1}{\sigma^2}\left(-\frac{x-\mu}{\sigma}\right)\varphi\left(\frac{x-\mu}{\sigma}\right).$$

Setting $F''(x)=0$ yields $x=\mu$. Thus, the coordinates of the inflection point are

$$\boxed{\bigl(\mu,\tfrac12\bigr)}.$$