From bd4b1c07d18d84933e21fcb2defb1e86197dd21b Mon Sep 17 00:00:00 2001
From: "Y.D.X." <73375426+YDX-2147483647@users.noreply.github.com>
Date: Thu, 14 Nov 2024 22:37:59 +0800
Subject: [PATCH] =?UTF-8?q?[course]=20=E6=97=A5=E5=B8=B8?=
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---
 docs/course/computational-methods.md |  2 +-
 docs/course/stochastic-geometry.md   | 30 ++++++++++++++++++++++++++++
 2 files changed, 31 insertions(+), 1 deletion(-)

diff --git a/docs/course/computational-methods.md b/docs/course/computational-methods.md
index 298f040..cef1803 100644
--- a/docs/course/computational-methods.md
+++ b/docs/course/computational-methods.md
@@ -167,7 +167,7 @@ flowchart LR
 - 计算量：显😀、隐（默认）。
 - 相容性：相容（默认）、条件相容👻。
 - 稳定性：绝对稳定🎯、条件稳定🎲、绝对不稳定💣。
-- 局部截断误差：$\Order(\tau + h)$ 1️⃣、$\Order(\tau + h^2)$ 1️⃣⁺、$\Order(\tau^2 + \tau h^2)$ 2️⃣。
+- 局部截断误差：$\Order(\tau + h)$ 1️⃣、$\Order(\tau + h^2)$ 1️⃣⁺、$\Order(\tau^2 + \tau h + \tau h^2)$ 2️⃣。
 
 ```mermaid
 flowchart LR
diff --git a/docs/course/stochastic-geometry.md b/docs/course/stochastic-geometry.md
index 43289c5..d64566c 100644
--- a/docs/course/stochastic-geometry.md
+++ b/docs/course/stochastic-geometry.md
@@ -6,6 +6,7 @@ relevant:
 # Stochastic Geometry
 
 $$
+\def\CC{\mathbb{C}}
 \def\RR{\mathbb{R}}
 \def\NN{\mathbb{N}}
 \DeclareMathOperator\expect{\mathbb{E}}
@@ -18,6 +19,35 @@ $$
     - 概率、随机过程和随机几何及其应用
     - Probability, Random Process and Stochastic Geometry in Engineering
 
+## Random Variables
+
+### Transforms
+
+> :material-clock-edit-outline: 2024年11月14日。
+
+- Probability generating function (PGF, $G$):
+
+  $z \mapsto \expect z^\xi$, $\xi \in \NN$.
+
+- Moment generating function (MGF, $M$):
+
+  $s \mapsto \expect e^{s \xi}$, $\xi \in \RR$. Note that it may not converge for all $s \in \CC$.
+
+- Characteristic function (CF, $\varphi$):
+
+  $\nu \mapsto \expect e^{j \nu \xi}$, $\nu \in \RR$. Note that $\abs{\expect e^{j \cdots}} \leq \expect \abs{e^{j \cdots}} = 1$.
+
+### Bounds of probabilities
+
+> :material-clock-edit-outline: 2024年11月14日。
+
+For a random variable $\xi \in \RR$, there exist the following bounds of $\Pr(\xi \geq x)$. They describe how fast $\xi$, as a sum, converges to the central limit theorem.
+
+- Markov: $\expect \xi \geq x \Pr(\xi \geq x)$, where $x \in \RR^+$.
+- Generalized Markov: $\expect \xi^r \geq x^r \Pr(\xi \geq x)$, where $x,r \in \RR^+$.
+- Чебышёв: $\expect \xi^2 \geq x^2 \Pr(\abs{\xi} \geq x)$, where $x \in \RR^+$.
+- Chernoff: $M(s) \coloneqq \expect e^{s \xi} \geq e^{s x} \Pr(\xi \geq x)$, where $x\in \RR, s \in \RR^+$.
+
 ## Filters
 
 ### Matched filter