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Binomial Expansion 二項展式
$$\begin{aligned}
(a+b)^{n}& =C_{0}^{n}a^{n}+C_{1}^{n}a^{n-1}b+\cdots+C_{r}^{n}a^{n-r}b^{r}+\cdots+C_{n-1}^{n}ab^{n-1}+C_{n}^{n}b^{n} \\
&=\sum_{r=0}^{n}C_{r}^{n}a^{n-r}b^{r}
\end{aligned}$$
Exponential Function 指數函數
$$\begin{aligned}
e^{x}& =1+\frac{x}{1!}+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots \\
&=\sum_{r=0}^\infty\frac{x^r}{r!}
\end{aligned}$$
Logarithmic Function 對數函數
- $$\mathrm{}lnMN=lnM+lnN$$
- $$\mathrm{}ln\frac{M}{N}=lnM-lnN$$
- $$\mathrm{}lnM^{n}=n\mathrm{ln}M$$
- $$\mathrm{}\log_{a}M=\frac{\ln M}{\ln a}$$
Limits 極限
- $$\lim_{x\to a}k=k\text{,where }k\text{ is a constant.}$$
- $$\lim_{x\rightarrow a}x=a$$
- $$\lim_{x\to a}kf(x)=k\lim_{x\to a}f(x),\mathrm{where}k\mathrm{is~a~constant}.$$
- $$\lim_{x\rightarrow a}[f(x)+g(x)]=\lim_{x\rightarrow a}f\left(x\right)+\lim_{x\rightarrow a}g\left(x\right)$$
- $$\lim_{x\to a}[f(x)-g(x)]=\lim_{x\to a}f\left(x\right)-\lim_{x\to a}g\left(x\right)$$
- $$\lim_{x\rightarrow a}[f(x)\cdot g(x)]=\lim_{x\rightarrow a}f\left(x\right)\cdot\lim_{x\rightarrow a}g\left(x\right)$$
- $$\lim_{x\to a}\frac{f(x)}{g(x)}=\frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)},\mathrm{where}\lim_{x\to a}g\left(x\right)\neq0.$$
- $$\lim_{x\to a}[f(x)]^{n}=\left[\lim_{x\to a}f\left(x\right)\right]^{n}\text{,where }n\text{ is a positive integer.}$$
- $$\lim_{x\to a}\sqrt[n]{f\left(x\right)}=\sqrt[n]{\lim_{x\to a}f\left(x\right)}\text{ if}$$
$$\mathrm{(i)}\lim_{x\to a}f\left(x\right)\text{is real for positive odd number}n\mathrm{or}$$
$$\mathrm{(ii)}\lim_{x\to a}f\left(x\right)\geq0\text{ for positive even number }n.$$ - $$\text{If }\lim_{x\to a}f\left(x\right)= l\text{ and }g(x)\text{ is continuous at }x=l,$$
$$\mathrm{then}\lim_{x\to a}g\left[f\left(x\right)\right]=g\Big[\lim_{x\to a}f\left(x\right)\Big]$$ - $$\lim_{x\to+\infty}\frac{1}{x}=0\mathrm{~and}\lim_{x\to-\infty}\frac{1}{x}=0$$
- $$\lim_{x\to\pm\infty}\bigl[f(x)\pm g(x)\bigr]=\lim_{x\to\pm\infty}f\left(x\right)\pm\lim_{x\to\pm\infty}g\left(x\right)$$
- $$\lim_{x\to\pm\infty}[f(x)\cdot g(x)]=\lim_{x\to\pm\infty}f(x)\cdot\lim_{x\to\pm\infty}g(x)$$
- $$\lim_{x\to\pm\infty}kf(x)=k\lim_{x\to\pm\infty}f(x),\text{where }k\text{ is a constant}$$
- $$\lim_{x\to\pm\infty}\frac{f(x)}{g(x)}=\frac{\lim_{x\to\pm\infty}f(x)}{\lim_{x\to\pm\infty}g(x)},\mathrm{where}\lim_{x\to\pm\infty}g\left(x\right)\neq0$$
Differentiation 微分
First Principle 微分第一原理
$$\frac{dy}{dx}=\lim_{\Delta x\to0}\frac{\Delta y}{\Delta x}=\lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$
Basic Differentiation 基本微分
- $$\frac{d}{dx}\left(k\right)=0\text{,where }k\text{ is a constant.}$$
- $$\frac{d}{dx}\left(x^{n}\right)=nx^{n-1}$$
- $$\frac{d}{dx}\left(ku\right)=k\frac{du}{dx}\text{ ,where }k\text{ is a constant}.$$
- $$\frac{d}{dx}\left(u\pm v\right)=\frac{du}{dx}\pm\frac{dv}{dx}$$
- $$\frac{d}{dx}\left(uv\right)=u\frac{dv}{dx}+v\frac{du}{dx}$$
- $$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^{2}},\mathrm{where}v(x)\neq0$$
- $$\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$$
- $$\frac{d}{dx}(e^{kx})=ke^{kx}$$
- $$\frac d{dx}(\ln x)=\frac1x$$
- $$\frac{d}{dx}(\log_{a}x)=\frac{1}{x\ln a}$$
- $$\frac{d}{dx}(a^{x})=a^{x}\mathrm{ln}a$$
- $$\frac{d^{2}y}{dx^{2}}=\frac{d}{dx}\left(\frac{dy}{dx}\right)$$
- $$\text{tangent equation at x =} x_{1}:\\y-y_{1}=f^{\prime}\left(x_{1}\right)\left(x-x_{1}\right)$$
Indefinite Integration 不定積分
- $$\frac d{dx}F(x)=f(x)$$
- $$\int f(x)dx=F(x)+C$$
- $$\int kf(x)dx=k\int f(x)dx,\text{where }k\text{ is a constant}$$
- $$\int[f(x)\pm g(x)]dx=\int f(x)dx\pm\int g(x)dx$$
- $$\int kdx=kx+C\text{,where }k\text{ is a constant}$$
- $$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C, n\neq-1$$
- $$\int\frac{1}{x}dx=\ln\lvert x\rvert+C,\text{where }x\neq0$$
- $$\int e^{x}dx=e^{x}+C$$
- $$\int f[g(x)]g^{\prime}(x)dx=\int f(u)du=F(u)+C$$
Definite Integration 定積分
- $$\int_{a}^{b}f(x)dx=F(b)-F(a)$$
- $$\int_{a}^{b}kf(x)dx=k\int_{a}^{b}f(x)dx,\text{where }k\mathrm{~is~a~constant}$$
- $$\int_{a}^{b}[f(x)\pm g(x)]dx=\int_{a}^{b}f(x)dx\pm\int_{a}^{b}g(x)dx$$
- $$\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx$$
- $$\int_{a}^{a}f(x)dx=0$$
- $$\int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx\text{ ,where }a<c<b$$
Area
Area between a curve and the x-axis from x=a t0 x=b
$$\int_a^bf(x)dx$$
Trapezoidal Rule
$$\int_{a}^{b}f(x)dx\approx\frac{\Delta x}{2}\left[f(x_{0})+2f(x_{1})+\cdots+2f(x_{n-1})+f(x_{n})\right]$$
Probability 概率
- $$P(B\mid A)=\frac{P(A\cap B)}{P(A)}$$
- $$P(A)=\sum_{i=1}^{n}P(E_{i})P(A\mid E_{i})$$
- $$P(E_{k}\mid A)=\frac{P(E_{k})P(A\mid E_{k})}{\sum_{i=1}^{n}P(E_{i})P(A\mid E_{i})}$$
- $$E(X)=\mu=\sum_{i=1}^{n}x_{i}f(x_{i})$$
- $$E[g(X)]=\sum_{i=1}^{n}g(x_{i})f(x_{i})$$
- $$E(aX+b)=aE(X)+b$$
- $$\mathrm{Var}\left(X\right)=\sum_{i=1}^{n}(x_{i}-\mu)^{2}f(x_{i})=E[(X-\mu)^{2}]$$
- $$\mathrm{Var}\left(X\right)=\sum_{i=1}^{n}x_{i}^{2}f(x_{i})-\mu^{2}=E(X^{2})-\mu^{2}$$
- $$\mathrm{Var}\left(aX+b\right)=a^{2}\mathrm{Var}\left(X\right)$$
Random Variable 隨機變數
Bernoulli Distribution 伯努利分佈
$$P(X=x)=p^{x}q^{1-x}$$
- $$E\left(X\right)=p$$
- $$\mathrm{Var}\left(X\right)=pq$$
Binomial Distribution 二項分佈
$$P(X=x)=C_{x}^{n}p^{x}q^{n-x}$$
- $$E\left(X\right)=np$$
- $$\mathrm{Var}\left(X\right)=npq$$
Poisson Distribution 泊松分佈
$$P(X=x)=\frac{\lambda^xe^{-\lambda}}{x!}$$
- $$E(X)=\lambda $$
- $$\mathrm{Var}\left(X\right)=\lambda $$
Normal Distribution 正態分佈
$$P(X=x)=\frac1{\sigma\sqrt{2\pi}}e^{-\frac12\left(\frac{x-\mu}\sigma\right)^2}$$
- $$E(X)=\mu $$
- $$\mathrm{Var}\left(X\right)=\sigma^{2}$$
Transforming to Normal Distribution
正態分佈變換
$$\begin{gathered}
X\sim\mathbf{N}(\mu,\sigma^{2}) \\
\operatorname{to}Z\sim\operatorname{N}(0,1) \\
E(\bar{X})=\mu \\
\mathrm{Var}(\bar{X})=\frac{\sigma^{2}}{n}
\end{gathered}$$
Confidence Interval 置信區間
$$C.l.\text{ for } \mu \text{ with known } \sigma^2$$
$$\left(\bar{X}-z_{\frac{a}{2}}\left(\frac{\sigma}{\sqrt{n}}\right),\bar{X}+z_{\frac{a}{2}}\left(\frac{\sigma}{\sqrt{n}}\right)\right)$$
$$C.l.\text{ for } \mu \text{ with unknown } \sigma^2$$
$$\left(X-z_{\frac{g}{2}}\left(\frac{s}{\sqrt{n}}\right),\bar{X}+z_{\frac{a}{2}}\left(\frac{s}{\sqrt{n}}\right)\right)$$