SciTech-Mathmatics - LaTex: 數學專業 常用公式的輸入方式

abaelhe發表於2024-10-04
公式

Calculus and Mathmatics Analysis

Example 1

$\large \begin{array}{rl} Newton-Leibniz\ formula \\
      \int_{a}^{b}{f'(x) dx} =& f(b) -  f(a) \\
         =& \underset{n \rightarrow \infty}{\lim} \overset{ n }{\underset{k=1}{\sum}}  { ( f'(x_k) \cdot \Delta{x_k} ) }, 黎曼和形式\\
         =& \underset{n \rightarrow \infty}{\lim} \overset{ n }{\underset{k=1}{\sum}}  { \Delta{f(x_k)} },\ 無窮微分劃分形 \\
   \end{array}$

\(\large \begin{array}{rl} Newton-Leibniz\ formula \\ \int_{a}^{b}{f'(x) dx} =& f(b) - f(a) \\ =& \underset{n \rightarrow \infty}{\lim} \overset{ n }{\underset{k=1}{\sum}} { ( f'(x_k) \cdot \Delta{x_k} ) }, 黎曼和形式\\ =& \underset{n \rightarrow \infty}{\lim} \overset{ n }{\underset{k=1}{\sum}} { \Delta{f(x_k)} },\ 無窮微分劃分形 \\ \end{array}\)

Example 2

$$\large e = \lim_{\Delta X \to 0} \frac{ (\frac{\Delta Y}{Y}) } { (\frac{\Delta X}{X}) } = \frac{ (\frac{dY}{Y}) }{ (\frac{dX}{X}) } = (\frac{dY}{dX}) * (\frac{X}{Y})$$

\[\large {彈性係數}=\dfrac{因變數的變動比例}{自變數的變動比例} \]

hypothesis $ \large Y = f(X) $:

  • 弧彈性係數: 兩點\((X_0, Y_0)\) 與 \((X_1, Y_1)\) 之間的彈性係數:

    \[\large e = \frac{ (\frac{\Delta Y}{Y}) } { (\frac{\Delta X}{X}) } = \frac{ (\frac{\Delta Y}{\Delta X}) }{ (\frac{Y}{X}) } \]

    通常\(\large (X, Y)\)取\(\large (X=\frac{X_0+X_1}{2}, Y=\frac{Y_0+Y_1}{2})\)
  • 點彈性係數, 點\((X, Y)\)處的彈性係數, 即當 \(\Delta X \to 0\) 並且 \(\Delta Y \to 0\)時:

    \[\large e = \lim_{\Delta X \to 0} \frac{ (\frac{\Delta Y}{Y}) } { (\frac{\Delta X}{X}) } = \frac{ (\frac{dY}{Y}) }{ (\frac{dX}{X}) } = (\frac{dY}{dX}) * (\frac{X}{Y}) \]



Matrix Analysis and Advanved Linear Algebra

矩陣可以用bmatrix Environment和pmatrix Environment，分別為方括號和圓括號，例如

$$
\begin{bmatrix}
	a & b \\
	c & d
\end{bmatrix}
$$

效果是：

\[\begin{bmatrix} a & b \\ c & d \end{bmatrix} \]

如果要輸入行列式的話，可以使用vmatrix Environment，用法同上。

Probability and Statistics

  • \sim: \(\large \sim\)
    $\large X_{\theta_1} \sim {\mathcal {N}}(\mu_1 ,\sigma_1 ^{2})$, \(\large X_{\theta_1} \sim {\mathcal {N}}(\mu_1 ,\sigma_1 ^{2})\)
    $\large X_{\theta_2} \sim {\mathcal {N}}(\mu_2 ,\sigma_2 ^{2})$, \(\large X_{\theta_2} \sim {\mathcal {N}}(\mu_2 ,\sigma_2 ^{2})\)
    parameter vector is $\large \theta = [\mu, \sigma^{2}]$, \(\large \theta = [\mu, \sigma^{2}]\)

Example Population Parameters

設總體 $\large X$ 容量為 $\large N$, 個體取值為$\large \{x _i \},\ i \in [1, N]$, 定義：
 $\large \begin{array}{ll} \\
\text{ population }mean : & \mu =  \dfrac{1}{N} \overset{N}{\underset{i =1}{\sum}}{ x_i } \\
\text{ population }total : & \tau = \dfrac{1}{N} \overset{N}{\underset{i =1}{\sum}}{ x_i } = N \cdot \mu \\
\text{ population }variance : & \sigma^2 = \dfrac{1}{N} \overset{N}{\underset{i =1}{\sum}}{ (x_i - \mu)^2 } \\
\text{ population }standard\ deviation : & s = \sqrt{ \sigma^2 } \\
\end{array}$

設總體 \(\large X\) 容量為 \(\large N\), 個體取值為\(\large \{x _i \},\ i \in [1, N]\), 定義：
\(\large \begin{array}{ll} \\ \text{ population }mean : & \mu = \dfrac{1}{N} \overset{N}{\underset{i =1}{\sum}}{ x_i } \\ \text{ population }total : & \tau = \dfrac{1}{N} \overset{N}{\underset{i =1}{\sum}}{ x_i } = N \cdot \mu \\ \text{ population }variance : & \sigma^2 = \dfrac{1}{N} \overset{N}{\underset{i =1}{\sum}}{ (x_i - \mu)^2 } \\ \text{ population }standard\ deviation : & s = \sqrt{ \sigma^2 } \\ \end{array}\)

Example 1

$\large \begin{array}{rrll} \\
\bm{ Population } & \bm{ Parameters } & \bm{ Statistics } &  \bm{ Sample } \\
quantity(count)\ =& \bm{ N }  & \bm{ n } &=\ quantity (count) \\
mean\ =& \bm{ \mu } & \bm{ \overset{-}{x} } &=\ mean \\
variance \ =& \bm{ \sigma^2 } & \bm{ s^2 } &=\ variance \\
standard\ deviation \ =& \bm{ \sigma } & \bm{ s } &=\ standard\ deviation \\
\end{array}$

$\large \begin{array}{rrll} \\ \bm{ Population } & \bm{ Parameters } & \bm{ Statistics } & \bm{ Sample } \\ quantity(count)\ =& \bm{ N } & \bm{ n } &=\ quantity (count) \\ mean\ =& \bm{ \mu } & \bm{ \overset{-}{x} } &=\ mean \\ variance \ =& \bm{ \sigma^2 } & \bm{ s^2 } &=\ variance \\ standard\ deviation \ =& \bm{ \sigma } & \bm{ s } &=\ standard\ deviation \\ \end{array}$

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