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ๆŠฝๆ ทๅˆ†ๅธƒ

ๆญฃๆ€ๅˆ†ๅธƒ

  • ๆญฃๆ€ๅˆ†ๅธƒ:

    • ็‰นๅพๅ‡ฝๆ•ฐ: \(\varphi_X(t) = e^{it\mu - \frac{1}{2}\sigma^2t^2}\)

  • ๆญฃๆ€ๅ˜้‡็บฟๆ€งๅ‡ฝๆ•ฐ็š„ๅˆ†ๅธƒ :

\[ X\sim N(\mu,\sigma^2),Y=aX+b \implies Y\sim N(a\mu+b,a^2\sigma^2) \]

  • \(\bar{X} = \frac{1}{n}\sum X_i \sim N(\mu,\frac{\sigma^2}{n})\)

  • ๆญฃๅคชๅ˜้‡ๆ ทๆœฌๅ‡ๅ€ผๅ’Œๆ ทๆœฌๆ–นๅทฎ็š„ๅˆ†ๅธƒๅ’Œๅฎƒไปฌ็š„็‹ฌ็ซ‹ๆ€ง

    • ่ฎพ \(X_1,X_2,\cdots,X_n\; \text{i.i.d.} \sim N(\mu,\sigma^2),\bar{X}= \frac{1}{n}\sum X_i, S^2 = \frac{1}{n-1}\sum (X_i-\bar{X})^2\), ๅˆ™

      • \(\bar{X}\sim N(\mu,\frac{\sigma^2}{n})\)

      • \((n-1)S^2/\sigma^2 \sim \chi^2(n-1)\)

      • \(\bar{X}\) ๅ’Œ \(S^2\) ็‹ฌ็ซ‹

ๆฌกๅบ็ปŸ่ฎก้‡็š„ๅˆ†ๅธƒ

่ฎพ \(X_1,X_2,\cdots,X_n\) ๆ˜ฏๆฅ่‡ชๆ€ปไฝ“ \(X\) ็š„ไธ€ไธชๆ ทๆœฌ, \(X_{(1)}\leq X_{(2)}\leq \cdots \leq X_{(n)}\) ๆ˜ฏๅฎƒ็š„ๆฌกๅบ็ปŸ่ฎก้‡

  • \(X_{(m)}\) ็š„ๅˆ†ๅธƒ

    • \(F_{X_{(m)}}(x) = P(X_{(m)}\leq x) = \sum\limits_{k=m}^n C_n^k F(x)^k(1-F(x))^{n-k}\)

    • \(f_{X_{(m)}}(x) = C_m^1 C_n^m f(x)F(x)^{m-1}(1-F(x))^{n-m}\)

  • \(X_{(1)}\) ็š„ๅˆ†ๅธƒ

    • \(F_{X_{(1)}}(x) = 1-(1-F(x))^n\)

    • \(f_{X_{(1)}}(x) = n f(x)(1-F(x))^{n-1}\)

  • \(X_{(n)}\) ็š„ๅˆ†ๅธƒ

    • \(F_{X_{(n)}}(x) = F(x)^n\)

    • \(f_{X_{(n)}}(x) = n f(x)F(x)^{n-1}\)

  • \(X_{(1)},\cdots,X_{(n)}\) ็š„่”ๅˆๅˆ†ๅธƒ

    • \(f_{X_{(1)},\cdots,X_{(n)}}(x_1,\cdots,x_n) = n! f(x_1)\cdots f(x_n),x_1\leq \cdots \leq x_n\)

\(\chi^2\) ๅˆ†ๅธƒ, \(t\) ๅˆ†ๅธƒ, \(F\) ๅˆ†ๅธƒ

\(\chi^2\) ๅˆ†ๅธƒ

่ฎพ \(X_1,X_2,\cdots,X_n\; \text{i.i.d.}\;\sim N(0,1)\), ๅˆ™็งฐ \(Y = \sum\limits_{i=1}^n X_i^2\) ไธบ่‡ช็”ฑๅบฆไธบ \(n\) ็š„ \(\chi^2\) ๅกๆ–นๅ˜้‡, ่ฎฐไธบ \(Y\sim \chi^2(n)\)

  • \(f_n(x)=\begin{cases} \frac{1}{2^{n/2}\Gamma(n/2)}x^{n/2-1}e^{-x/2} & x>0 \\ 0 & x\leq 0 \end{cases}\)

  • ไปค \(\xi \sim \chi^2(n),\alpha\in (0,1),P(\xi\geq c) = \alpha\), ๅˆ™็งฐ \(c=\chi_n^2(\alpha)\) ไธบๅกๆ–นๅˆ†ๅธƒ็š„ไธŠไพง \(\alpha\) ๅˆ†ไฝๆ•ฐ

  • \(\varphi_{\chi^2(n)}(t) = (1-2it)^{-n/2},t<1/2\)

  • \(E(\xi) = n,Var(\xi) = 2n\)

\(t\) ๅˆ†ๅธƒ

่ฎพ \(X\sim N(0,1),Y\sim \chi^2(n)\) ไธ” \(X,Y\) ็‹ฌ็ซ‹, ๅˆ™็งฐ \(T = \frac{X}{\sqrt{Y/n}}\) ไธบ่‡ช็”ฑๅบฆไธบ \(n\) ็š„ \(t\) ๅ˜้‡, ่ฎฐไธบ \(T\sim t_n\)

  • \(t_n(x)=\frac{\Gamma(\frac{n+1}{2})}{\sqrt{n\pi}\Gamma(n/2)}(1+x^2/n)^{-\frac{n+1}{2}},x\in\mathbb{R}\)

  • \(\lim\limits_{n\to\infty}t_n(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}\)

  • \(E(T) = 0,Var(T) = \frac{n}{n-2},n>2\)

\(F\) ๅˆ†ๅธƒ

่ฎพ \(U\sim \chi^2(m),V\sim \chi^2(n)\) ไธ” \(U,V\) ็‹ฌ็ซ‹, ๅˆ™็งฐ \(F = \frac{U/m}{V/n}\) ไธบ่‡ช็”ฑๅบฆไธบ \(m,n\) ็š„ \(F\) ๅ˜้‡, ่ฎฐไธบ \(F\sim F_{m,n}\)

  • \(f_{m,n}(x) = \begin{cases} \frac{\Gamma(\frac{m+n}{2})}{\Gamma(\frac{m}{2})\Gamma(\frac{n}{2})}m^{m/2}n^{n/2} x^{m/2-1}(n+mx)^{-\frac{m+n}{2}} & x>0 \\ 0 & x\leq 0 \end{cases}\)

  • ่ฎพ \(F\sim F_{m,n},\alpha\in (0,1),P(F\geq c) = \alpha\), ๅˆ™็งฐ \(c=F_{m,n}(\alpha)\) ไธบ \(F\) ๅˆ†ๅธƒ็š„ไธŠไพง \(\alpha\) ๅˆ†ไฝๆ•ฐ

  • ่‹ฅ \(F\sim F_{m,n}\), ๅˆ™ \(1/F\sim F_{n,m}\)

  • ่‹ฅ \(T\sim t_n\), ๅˆ™ \(T^2\sim F_{1,n}\)

  • \(F_{m,n}(\alpha) = \frac{1}{F_{n,m}(1-\alpha)}\)

ๅ……ๅˆ†็ปŸ่ฎก้‡