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[course] 日常
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YDX-2147483647 committed Nov 14, 2024
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2 changes: 1 addition & 1 deletion docs/course/computational-methods.md
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- 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
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30 changes: 30 additions & 0 deletions docs/course/stochastic-geometry.md
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# Stochastic Geometry

$$
\def\CC{\mathbb{C}}
\def\RR{\mathbb{R}}
\def\NN{\mathbb{N}}
\DeclareMathOperator\expect{\mathbb{E}}
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- 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
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