概率密度部分

1. 期望和峰值 当outliers很严重
   

2. skewness and kurtosis
   $\begin{array}{l} \text { Skewness: } \frac{E\left[(x-E[x])^{3}\right]}{(D[x])^{3}} \\ \text { Kurtosis: } \frac{E\left[(x-E[x])^{4}\right]}{(D[x])^{4}}-3 \end{array}$ Skewness: (D[x])3E[(x−E[x])3]​ Kurtosis: (D[x])4E[(x−E[x])4]​−3​
   

3. Moment 生成函数
   As the limit, if the moments of all orders are specifed, the probability distribution is uniquely determined.
   $M_{x}(t)=E\left[e^{t x}\right]=\left\{\begin{array}{ll} \sum_{x} e^{t x} f(x) & \text { (Discrete) } \\ \int e^{t x} f(x) \mathrm{d} x & \text { (Continuous) } \end{array}\right.$Mx​(t)=E[etx]={∑x​etxf(x)∫etxf(x)dx​ (Discrete)  (Continuous) ​
   $e^{t x}=1+(t x)+\frac{(t x)^{2}}{2 !}+\frac{(t x)^{3}}{3 !}+\cdots$etx=1+(tx)+2!(tx)2​+3!(tx)3​+⋯
   $E\left[e^{t x}\right]=M_{x}(t)=1+t \mu_{1}+t^{2} \frac{\mu_{2}}{2 !}+t^{3} \frac{\mu_{3}}{3 !}+\cdots$E[etx]=Mx​(t)=1+tμ1​+t22!μ2​​+t33!μ3​​+⋯
   $\begin{aligned} M_{x}^{\prime}(t) &=\mu_{1}+\mu_{2} t+\frac{\mu_{3}}{2 !} t^{2}+\frac{\mu_{4}}{3 !} t^{3}+\cdots \\ M_{x}^{\prime \prime}(t) &=\mu_{2}+\mu_{3} t+\frac{\mu_{4}}{2 !} t^{2}+\frac{\mu_{5}}{3 !} t^{3}+\cdots \\ & \vdots \\ M_{x}^{(k)}(t) &=\mu_{k}+\mu_{k+1} t+\frac{\mu_{k+2}}{2 !} t^{2}+\frac{\mu_{k+3}}{3 !} t^{3}+\cdots \end{aligned}$Mx′​(t)Mx′′​(t)Mx(k)​(t)​=μ1​+μ2​t+2!μ3​​t2+3!μ4​​t3+⋯=μ2​+μ3​t+2!μ4​​t2+3!μ5​​t3+⋯⋮=μk​+μk+1​t+2!μk+2​​t2+3!μk+3​​t3+⋯​

4. 分布的特征方程
   是概率密度的傅里叶变换
   $\varphi_{x}(t)=M_{i x}(t)=M_{x}(i t)$φx​(t)=Mix​(t)=Mx​(it)

5. TRANSFORMATION OF RANDOM VARIABLES
   要乘上雅克比行列式：
   Integration of function $f(x)$f(x) over $\mathcal{X}$X can be expressed by using function $g(r)$g(r) on $\mathcal{R}$R such that
   $x=g(r) \text { and } X=g(\mathcal{R})$x=g(r) and X=g(R)
   as
   $\int_{\mathcal{X}} f(x) \mathrm{d} x=\int_{\mathcal{R}} f(g(r))\left|\frac{\mathrm{d} x}{\mathrm{d} r}\right| \mathrm{d} r$∫X​f(x)dx=∫R​f(g(r))∣∣∣∣​drdx​∣∣∣∣​dr