https://file.aiquickdraw.com/mathbot/file/1723800159734i2iply3a.jpg

Here is the LaTeX code corresponding to the text and equations from the uploaded image:

```latex
\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{bm}

\begin{document}

\section*{腾讯 2021 年 8 月概率大书卷（偏密交流-分析精选）}

\textbf{学员已做错的数字题：}

已知某入射强度符合高斯分布：
\[
x \sim N(\mu, \sigma^2)
\]
设其光子流为 $F(x) = \frac{1}{2}\left[1+\text{erf}\left(\frac{x-\mu}{\sqrt{2}\sigma}\right)\right] \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$.

求光子流密度：
\[
G(x) = \frac{\partial F(x)}{\partial x}
\]

\textbf{条件：}
\[
(1)\; \mu = 0,\quad \sigma^2 = 2
\]
\[
(2)\; F(0) = \frac{1}{2},\quad F(1) = \frac{3}{4},\quad F(-1) = \frac{1}{4}
\]

求：
\[
F(x) = \frac{1}{2}\left[1+\text{erf}\left(\frac{x-\mu}{\sqrt{2}\sigma}\right)\right]
\]
\[
(1)\; F(x = 0)
\]
\[
(2)\; G(x = 0)
\]
\[
(3)\; x 的取值范围是多少?
\]
\[
(4)\; 结合书上公式，试解释PDF 函数 f(x) 和 CDF 函数 F(x) 之间的关系，并简要说明？
\]

提示：
\begin{itemize}
    \item PDF: 概率密度函数，它在每个点的值表示在该点的 PDF。
    \item CDF: 累积分布函数，表示变量取小于或等于某一值的概率。
    \item 使用 Python 计算 $F(x)$ 以及 $G(x)$ 并作图，实验验证公式。
\end{itemize}

\end{document}
```

This code will reproduce the mathematical expressions, questions, and explanations provided in the image.

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Here are 8 related questions to consider:

1. What is the Gaussian distribution and how is it related to the normal distribution?
2. How do you derive the probability density function (PDF) from a Gaussian distribution?
3. What is the error function (erf) and how is it used in probability and statistics?
4. Can you explain the significance of the cumulative distribution function (CDF)?
5. How do you differentiate a cumulative distribution function (CDF) to obtain the probability density function (PDF)?
6. What are the properties of the Gaussian distribution with mean $\mu = 0$ and variance $\sigma^2 = 2$?
7. How would you use Python to compute and plot $F(x)$ and $G(x)$?
8. What is the relationship between the CDF and the PDF for a Gaussian distribution?

**Tip:** When working with probability distributions, it's essential to understand the relationship between the PDF and CDF, as they provide different but complementary information about the distribution.

Explore the Gaussian distribution, probability density function (PDF), cumulative distribution function (CDF), and error function (erf) through a detailed analysis. Includes Python computations and plotting examples. Ideal for advanced learners and practitioners.

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