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  • 数理统计知识点总结

    2020-12-13 21:05:31
    数理统计知识点总结 文章目录数理统计知识点总结第六章:数理统计的基本概念 第六章:数理统计的基本概念 1、称研究对象的全体为总体,它是一个随机变量X;称组成总体的元素为个体。 2、称相互独立且与总体X同分布的...

    数理统计知识点总结

    第六章:数理统计的基本概念

    1、称研究对象的全体为总体,它是一个随机变量X;称组成总体的元素为个体。
    2、称相互独立且与总体X同分布的随机变量 X 1 , X 2 … … X n X_1,X_2……X_n X1,X2Xn为来自总体一个容量为n的简单随机样本,简称样本。称样本的一组取值 x 1 , x 2 … … x n x_1,x_2……x_n x1,x2xn为该样本的一组样本值
    3、样本的联合分布函数,分布律,概率密度都为乘积形式
    4、设 X 1 , X 2 … … X n X_1,X_2……X_n X1,X2Xn是来自总体X的一个样本, x 1 , x 2 … … x n x_1,x_2……x_n x1,x2xn是样本的一系列样本值,将其从小到大排序,并编上序号,则称函数 F n ( x ) = x 1 , x 2 … … x n 中 小 于 等 于 x 的 样 本 值 的 个 数 n = 0 ( x < x ( 1 ) ) ; k n ( x ( k ) < = x < x ( k + 1 ) ) ; 1 ( x > = x ( n ) ) F_n(x) = \frac{x_1,x_2……x_n中小于等于x的样本值的个数}{n} = 0(x<x_{(1)}) ;\frac{k}{n}(x_{(k)}<=x<x{(k+1)});1(x>=x{(n)}) Fn(x)=nx1,x2xnx=0x<x(1);nkx(k)<=x<x(k+1);1x>=x(n)为总体X的经验分布函数
    5、对于任意实数x,当n趋于 ∞ \infty 时,经验分布函数 F n ( x ) F_n(x) Fn(x)以概率1一致收敛于总体X的分布函数F(x),即 P ( l i m n − ∞ s u p − ∞ < x < ∞ ∣ F n ( x ) − F ( x ) ∣ = 0 ) = 1 P(lim_{n-\infty}sup_{-\infty<x<\infty}|F_n(x)-F(x)| = 0) = 1 P(limnsup<x<Fn(x)F(x)=0)=1
    该定理说明当样本充分大时,可以将经验分布函数当作当作分布函数使用
    6、统计量:对于不含总体X的未知参数函数 g ( x 1 , x 2 … … x n ) g(x_1,x_2……x_n) g(x1,x2xn),称 g ( X 1 , X 2 … … X n ) g(X_1,X_2……X_n) g(X1,X2Xn)为总体的一个统计量, g ( x 1 , x 2 … … x n ) g(x_1,x_2……x_n) g(x1,x2xn)为统计量的样本值。 g ( X 1 , X 2 … … X n ) g(X_1,X_2……X_n) g(X1,X2Xn)统计量也是一个随机函数。
    [重要]7、一些统计量的总结:
    样本均值:
    X ‾ = 1 n Σ i = 1 n X i \overline X = \frac{1}{n}\Sigma_{i=1}^nX_i X=n1Σi=1nXi
    样本均值的样本值:
    x ‾ = 1 n Σ i = 1 n x i \overline x = \frac{1}{n}\Sigma_{i=1}^nx_i x=n1Σi=1nxi
    样本方差:
    S 2 = 1 n − 1 Σ i = 1 n ( X i − X ‾ ) 2 = 1 n − 1 ( Σ i = 1 n X i 2 − n X ‾ 2 ) S^2 = \frac{1}{n-1}\Sigma_{i=1}^{n}(X_i-\overline X)^2 = \frac{1}{n-1}(\Sigma_{i=1}^{n}X_i^2-n\overline X^2) S2=n11Σi=1n(XiX)2=n11(Σi=1nXi2nX2)
    样本方差的样本值:
    s 2 = 1 n − 1 Σ i = 1 n ( x i − x ‾ ) 2 = 1 n − 1 ( Σ i = 1 n x i 2 − n x ‾ 2 ) s^2 = \frac{1}{n-1}\Sigma_{i=1}^{n}(x_i-\overline x)^2 = \frac{1}{n-1}(\Sigma_{i=1}^{n}x_i^2-n\overline x^2) s2=n11Σi=1n(xix)2=n11(Σi=1nxi2nx2)
    样本标准差:
    S = 1 n − 1 Σ i = 1 n ( X i − X ‾ ) 2 S = \sqrt{\frac{1}{n-1}\Sigma_{i=1}^{n}(X_i-\overline X)^2} S=n11Σi=1n(XiX)2
    样本标准差的样本值:
    s = 1 n − 1 Σ i = 1 n ( x i − x ‾ ) 2 s = \sqrt{\frac{1}{n-1}\Sigma_{i=1}^{n}(x_i-\overline x)^2} s=n11Σi=1n(xix)2
    样本K阶原点矩:
    A k = 1 n X i k A_k = \frac{1}{n}X_i^k Ak=n1Xik
    样本K阶原点矩的样本值:
    a k = 1 n x i k a_k = \frac{1}{n}x_i^k ak=n1xik
    样本K阶中心矩:
    B k = 1 n ( X i − X ‾ ) k B_k = \frac{1}{n}(X_i-\overline X)^k Bk=n1(XiX)k
    样本K阶中心距的样本值:
    b k = 1 n ( x i − x ‾ ) k b_k = \frac{1}{n}(x_i-\overline x)^k bk=n1(xix)k
    [重要]8、设X是总体,其均值 E X = μ EX=\mu EX=μ,方差 D X = σ 2 DX = \sigma^2 DX=σ2 X ‾ \overline X X S 2 S^2 S2分别为样本均值与严格不能方差,则 E X ‾ = μ , D X ‾ = σ 2 n , E S 2 = σ 2 E\overline X = \mu,D\overline X=\frac{\sigma^2}{n},ES^2 = \sigma^2 EX=μ,DX=nσ2,ES2=σ2
    8、抽样分布(统计量的分布)
    四大分布:
    标准正态分布: X − N ( 0 , 1 ) X-N(0,1) XN(0,1)
    α 分 位 点 \alpha分位点 α P ( X > z α ) = α P(X>z_\alpha) = \alpha P(X>zα)=α 1 > α > 0 1>\alpha>0 1>α>0
    χ 2 分 布 \chi^2分布 χ2 χ 2 − χ 2 ( n ) \chi^2-\chi^2(n) χ2χ2(n)
    设随机变量 X 1 , X 2 … … X n X_1,X_2……X_n X1,X2Xn相互独立且服从标准正态分布N(0,1),则称
    χ 2 = X 1 2 + X 2 2 + … … + X n 2 \chi^2 = X_1^2+X_2^2+……+X_n^2 χ2=X12+X22++Xn2
    的分布服从自由度为n的卡方分布。
    在这里插入图片描述
    卡方分布也满足上述分位点定义: P ( χ 2 > χ α 2 ( n ) ) = α P(\chi^2>\chi_\alpha^2(n)) = \alpha P(χ2>χα2(n))=α
    卡方分布有以下性质:
    ①两个相互独立的卡方分布之和为自由度之和的卡方分布
    ②卡方分布的均值为自由度n,方差为2n
    t 分 布 t分布 t T − t ( n ) T-t(n) Tt(n)
    设随机变量 X − N ( 0 , 1 ) , Y − χ 2 ( n ) X-N(0,1),Y-\chi^2(n) XN(0,1),Yχ2(n),且X,Y相互独立,则
    T = X Y / n T = \frac{X}{\sqrt{Y/n}} T=Y/n X
    的分布为服从参数为n的t分布.
    在这里插入图片描述
    t分布也满足上述分位点定义: P ( T > t α ( n ) ) = α P(T>t_\alpha(n)) = \alpha P(T>tα(n))=α
    F 分 布 F分布 F F − F ( n 1 , n 2 ) F-F(n_1,n_2) FF(n1,n2)
    设随机变量 X − χ 2 ( n 1 ) , Y − χ 2 ( n 2 ) X-\chi^2(n_1),Y-\chi^2(n_2) Xχ2(n1),Yχ2(n2),且X,Y相互独立,则
    F = X / n 1 Y / n 2 F = \frac{X/n_1}{\sqrt{Y/n_2}} F=Y/n2 X/n1
    的分布为服从参数为 ( n 1 , n 2 ) (n_1,n_2) (n1,n2)的t分布.
    在这里插入图片描述
    F分布也满足上述分位点定义: P ( F > F α ( n 1 , n 2 ) ) = α P(F>F_\alpha(n_1,n_2)) = \alpha P(F>Fα(n1,n2))=α
    F分布具有以下性质:
    1 F − F ( n 2 , n 1 ) \frac{1}{F}-F(n_2,n_1) F1Fn2,n1
    F 1 − α ( n 1 , n 2 ) = 1 F α ( n 2 , n 1 ) F_{1-\alpha}(n_1,n_2) = \frac{1}{F_\alpha(n_2,n_1)} F1α(n1,n2)=Fα(n2,n1)1
    9、八大分布
    对于单正态分布总体 X − N ( μ , σ 2 ) X-N(\mu,\sigma^2) XN(μ,σ2)
    X ‾ − μ σ / n − N ( 0 , 1 ) \frac{\overline X-\mu}{\sigma/\sqrt{n}}-N(0,1) σ/n XμN(0,1)
    X ‾ − N ( μ , σ 2 n ) \overline X -N(\mu,\frac{\sigma^2}{n}) XN(μ,nσ2)
    Σ i = 1 n ( X i − μ ) 2 σ 2 − χ 2 ( n ) \frac{\Sigma_{i=1}^n(X_i-\mu)^2}{\sigma^2}-\chi^2(n) σ2Σi=1n(Xiμ)2χ2(n)
    ( n − 1 ) S 2 σ 2 − χ 2 ( n − 1 ) \frac{(n-1)S^2}{\sigma^2}-\chi^2(n-1) σ2n1S2χ2(n1)
    Σ i = 1 n ( X i − X ‾ ) 2 σ 2 − χ 2 ( n − 1 ) \frac{\Sigma_{i=1}^n(X_i-\overline X)^2}{\sigma^2}-\chi^2(n-1) σ2Σi=1n(XiX)2χ2(n1)
    X ‾ − μ S / n − t ( n − 1 ) \frac{\overline X-\mu}{S/\sqrt{n}}-t(n-1) S/n Xμt(n1)
    对于双正态分布
    X ‾ − Y ‾ − ( μ 1 − μ 2 ) σ 2 n 1 + σ 2 n 2 − N ( 0 , 1 ) \frac{\overline X-\overline Y - (\mu_1-\mu_2)}{\sqrt{\frac{\sigma^2}{n_1}+\frac{\sigma^2}{n_2}}}-N(0,1) n1σ2+n2σ2 XY(μ1μ2)N(0,1)
    ⑥当方差未知时
    X ‾ − Y ‾ − ( μ 1 − μ 2 ) σ 2 n 1 + σ 2 n 2 − t ( n 1 + n 2 − 2 ) \frac{\overline X-\overline Y - (\mu_1-\mu_2)}{\sqrt{\frac{\sigma^2}{n_1}+\frac{\sigma^2}{n_2}}}-t(n_1+n_2-2) n1σ2+n2σ2 XY(μ1μ2)t(n1+n22)
    S w = ( n 1 − 1 ) S 1 2 + ( n 2 − 1 ) S 2 2 n 1 + n 2 − 2 S_w = \sqrt{\frac{(n_1-1)S_1^2+(n_2-1)S_2^2}{n_1+n_2-2}} Sw=n1+n22n11S12+(n21)S22
    n 2 σ 2 2 Σ i = 1 n 1 ( X i − μ 1 ) 2 n 1 σ 1 2 Σ i = 1 n 2 ( Y i − μ 2 ) 2 − F ( n 1 , n 2 ) \frac{n_2\sigma_2^2\Sigma_{i=1}^{n_1}(X_i-\mu_1)^2}{n_1\sigma_1^2\Sigma_{i=1}^{n_2}(Y_i-\mu_2)^2}-F(n_1,n_2) n1σ12Σi=1n2(Yiμ2)2n2σ22Σi=1n1(Xiμ1)2F(n1,n2)
    σ 2 2 S 1 2 σ 2 2 S 1 2 − F ( n 1 − 1 , n 2 − 1 ) \frac{\sigma_2^2S_1^2}{\sigma_2^2S_1^2}-F(n_1-1,n_2-1) σ22S12σ22S12F(n11,n21)

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  • 概率论与数理统计知识点总结
  • 概率论与数理统计知识点总结(详细版).docx
  • 概率论和数理统计知识点总结

    千次阅读 2020-12-21 16:08:37
    概率论与数理统计 随机事件和概率 1.事件的关系与运算 2.运算律 3.德摩根律 4.完全事件组 5.概率的基本公式 6.事件的独立性 7.独立重复试验 8.重要公式与结论 随机变量及其概率分布 1.随机变量及概率分布 2.分布函数...

    随机事件和概率

    1.事件的关系与运算

    (1) 子事件: A ⊂ B A \subset B AB,若 A A A发生,则 B B B发生。

    (2) 相等事件: A = B A = B A=B,即 A ⊂ B A \subset B AB,且 B ⊂ A B \subset A BA

    (3) 和事件: A ⋃ B A\bigcup B AB(或 A + B A + B A+B), A A A B B B中至少有一个发生。

    (4) 差事件: A − B A - B AB A A A发生但 B B B不发生。

    (5) 积事件: A ⋂ B A\bigcap B AB(或 A B {AB} AB), A A A B B B同时发生。

    (6) 互斥事件(互不相容): A ⋂ B A\bigcap B AB= ∅ \varnothing

    (7) 互逆事件(对立事件):
    A ⋂ B = ∅ , A ⋃ B = Ω , A = B ˉ , B = A ˉ A\bigcap B=\varnothing ,A\bigcup B=\Omega ,A=\bar{B},B=\bar{A} AB=,AB=Ω,A=Bˉ,B=Aˉ

    2.运算律

    (1) 交换律: A ⋃ B = B ⋃ A , A ⋂ B = B ⋂ A A\bigcup B=B\bigcup A,A\bigcap B=B\bigcap A AB=BA,AB=BA
    (2) 结合律: ( A ⋃ B ) ⋃ C = A ⋃ ( B ⋃ C ) (A\bigcup B)\bigcup C=A\bigcup (B\bigcup C) (AB)C=A(BC)
    (3) 分配律: ( A ⋂ B ) ⋂ C = A ⋂ ( B ⋂ C ) (A\bigcap B)\bigcap C=A\bigcap (B\bigcap C) (AB)C=A(BC)

    3.德摩根律

    A ⋃ B ‾ = A ˉ ⋂ B ˉ \overline{A\bigcup B}=\bar{A}\bigcap \bar{B} AB=AˉBˉ A ⋂ B ‾ = A ˉ ⋃ B ˉ \overline{A\bigcap B}=\bar{A}\bigcup \bar{B} AB=AˉBˉ

    4.完全事件组

    A 1 A 2 ⋯ A n {{A}_{1}}{{A}_{2}}\cdots {{A}_{n}} A1A2An两两互斥,且和事件为必然事件,即 A i ∩ A j = ∅ , i ≠ j , ⋃ i = 1 n = Ω A_{i} \cap A_{j}=\varnothing, i \neq j, \bigcup_{i=1}^{n}=\Omega AiAj=,i=j,i=1n=Ω

    5.概率的基本公式

    (1)条件概率:
    P ( B ∣ A ) = P ( A B ) P ( A ) P(B|A)=\frac{P(AB)}{P(A)} P(BA)=P(A)P(AB),表示 A A A发生的条件下, B B B发生的概率。
    (2)全概率公式:
    P ( A ) = ∑ i = 1 n P ( A ∣ B i ) P ( B i ) , B i B j = ∅ , i ≠ j , ⋃ n i = 1   B i = Ω P(A)=\sum\limits_{i=1}^{n}{P(A|{{B}_{i}})P({{B}_{i}}),{{B}_{i}}{{B}_{j}}}=\varnothing ,i\ne j,\underset{i=1}{\overset{n}{\mathop{\bigcup }}}\,{{B}_{i}}=\Omega P(A)=i=1nP(ABi)P(Bi),BiBj=,i=j,i=1nBi=Ω
    (3) Bayes公式:

    P ( B j ∣ A ) = P ( A ∣ B j ) P ( B j ) ∑ i = 1 n P ( A ∣ B i ) P ( B i ) , j = 1 , 2 , ⋯   , n P({{B}_{j}}|A)=\frac{P(A|{{B}_{j}})P({{B}_{j}})}{\sum\limits_{i=1}^{n}{P(A|{{B}_{i}})P({{B}_{i}})}},j=1,2,\cdots ,n P(BjA)=i=1nP(ABi)P(Bi)P(ABj)P(Bj),j=1,2,,n
    注:上述公式中事件 B i {{B}_{i}} Bi的个数可为可列个。
    (4)乘法公式:
    P ( A 1 A 2 ) = P ( A 1 ) P ( A 2 ∣ A 1 ) = P ( A 2 ) P ( A 1 ∣ A 2 ) P({{A}_{1}}{{A}_{2}})=P({{A}_{1}})P({{A}_{2}}|{{A}_{1}})=P({{A}_{2}})P({{A}_{1}}|{{A}_{2}}) P(A1A2)=P(A1)P(A2A1)=P(A2)P(A1A2)
    P ( A 1 A 2 ⋯ A n ) = P ( A 1 ) P ( A 2 ∣ A 1 ) P ( A 3 ∣ A 1 A 2 ) ⋯ P ( A n ∣ A 1 A 2 ⋯ A n − 1 ) P({{A}_{1}}{{A}_{2}}\cdots {{A}_{n}})=P({{A}_{1}})P({{A}_{2}}|{{A}_{1}})P({{A}_{3}}|{{A}_{1}}{{A}_{2}})\cdots P({{A}_{n}}|{{A}_{1}}{{A}_{2}}\cdots {{A}_{n-1}}) P(A1A2An)=P(A1)P(A2A1)P(A3A1A2)P(AnA1A2An1)

    6.事件的独立性

    (1) A A A B B B相互独立 ⇔ P ( A B ) = P ( A ) P ( B ) \Leftrightarrow P(AB)=P(A)P(B) P(AB)=P(A)P(B)
    (2) A A A B B B C C C两两独立
    ⇔ P ( A B ) = P ( A ) P ( B ) \Leftrightarrow P(AB)=P(A)P(B) P(AB)=P(A)P(B); P ( B C ) = P ( B ) P ( C ) P(BC)=P(B)P(C) P(BC)=P(B)P(C) ; P ( A C ) = P ( A ) P ( C ) P(AC)=P(A)P(C) P(AC)=P(A)P(C);
    (3) A A A B B B C C C相互独立
    ⇔ P ( A B ) = P ( A ) P ( B ) \Leftrightarrow P(AB)=P(A)P(B) P(AB)=P(A)P(B); P ( B C ) = P ( B ) P ( C ) P(BC)=P(B)P(C) P(BC)=P(B)P(C) ;
    P ( A C ) = P ( A ) P ( C ) P(AC)=P(A)P(C) P(AC)=P(A)P(C) ; P ( A B C ) = P ( A ) P ( B ) P ( C ) P(ABC)=P(A)P(B)P(C) P(ABC)=P(A)P(B)P(C)

    7.独立重复试验

    将某试验独立重复 n n n次,若每次实验中事件A发生的概率为 p p p,则 n n n次试验中 A A A发生 k k k次的概率为:
    P ( X = k ) = C n k p k ( 1 − p ) n − k P(X=k)=C_{n}^{k}{{p}^{k}}{{(1-p)}^{n-k}} P(X=k)=Cnkpk(1p)nk

    8.重要公式与结论

    ( 1 ) P ( A ˉ ) = 1 − P ( A ) (1)P(\bar{A})=1-P(A) (1)P(Aˉ)=1P(A)
    ( 2 ) P ( A ⋃ B ) = P ( A ) + P ( B ) − P ( A B ) (2)P(A\bigcup B)=P(A)+P(B)-P(AB) (2)P(AB)=P(A)+P(B)P(AB)
    P ( A ⋃ B ⋃ C ) = P ( A ) + P ( B ) + P ( C ) − P ( A B ) − P ( B C ) − P ( A C ) + P ( A B C ) P(A\bigcup B\bigcup C)=P(A)+P(B)+P(C)-P(AB)-P(BC)-P(AC)+P(ABC) P(ABC)=P(A)+P(B)+P(C)P(AB)P(BC)P(AC)+P(ABC)
    ( 3 ) P ( A − B ) = P ( A ) − P ( A B ) (3)P(A-B)=P(A)-P(AB) (3)P(AB)=P(A)P(AB)
    ( 4 ) P ( A B ˉ ) = P ( A ) − P ( A B ) , P ( A ) = P ( A B ) + P ( A B ˉ ) , (4)P(A\bar{B})=P(A)-P(AB),P(A)=P(AB)+P(A\bar{B}), (4)P(ABˉ)=P(A)P(AB),P(A)=P(AB)+P(ABˉ),
    P ( A ⋃ B ) = P ( A ) + P ( A ˉ B ) = P ( A B ) + P ( A B ˉ ) + P ( A ˉ B ) P(A\bigcup B)=P(A)+P(\bar{A}B)=P(AB)+P(A\bar{B})+P(\bar{A}B) P(AB)=P(A)+P(AˉB)=P(AB)+P(ABˉ)+P(AˉB)
    (5)条件概率 P ( ⋅ ∣ B ) P(\centerdot |B) P(B)满足概率的所有性质,
    例如:. P ( A ˉ 1 ∣ B ) = 1 − P ( A 1 ∣ B ) P({{\bar{A}}_{1}}|B)=1-P({{A}_{1}}|B) P(Aˉ1B)=1P(A1B)
    P ( A 1 ⋃ A 2 ∣ B ) = P ( A 1 ∣ B ) + P ( A 2 ∣ B ) − P ( A 1 A 2 ∣ B ) P({{A}_{1}}\bigcup {{A}_{2}}|B)=P({{A}_{1}}|B)+P({{A}_{2}}|B)-P({{A}_{1}}{{A}_{2}}|B) P(A1A2B)=P(A1B)+P(A2B)P(A1A2B)
    P ( A 1 A 2 ∣ B ) = P ( A 1 ∣ B ) P ( A 2 ∣ A 1 B ) P({{A}_{1}}{{A}_{2}}|B)=P({{A}_{1}}|B)P({{A}_{2}}|{{A}_{1}}B) P(A1A2B)=P(A1B)P(A2A1B)
    (6)若 A 1 , A 2 , ⋯   , A n {{A}_{1}},{{A}_{2}},\cdots ,{{A}_{n}} A1,A2,,An相互独立,则 P ( ⋂ i = 1 n A i ) = ∏ i = 1 n P ( A i ) , P(\bigcap\limits_{i=1}^{n}{{{A}_{i}}})=\prod\limits_{i=1}^{n}{P({{A}_{i}})}, P(i=1nAi)=i=1nP(Ai),
    P ( ⋃ i = 1 n A i ) = ∏ i = 1 n ( 1 − P ( A i ) ) P(\bigcup\limits_{i=1}^{n}{{{A}_{i}}})=\prod\limits_{i=1}^{n}{(1-P({{A}_{i}}))} P(i=1nAi)=i=1n(1P(Ai))
    (7)互斥、互逆与独立性之间的关系:
    A A A B B B互逆 ⇒ \Rightarrow A A A B B B互斥,但反之不成立, A A A B B B互斥(或互逆)且均非零概率事件$\Rightarrow $ A A A B B B不独立.
    (8)若 A 1 , A 2 , ⋯   , A m , B 1 , B 2 , ⋯   , B n {{A}_{1}},{{A}_{2}},\cdots ,{{A}_{m}},{{B}_{1}},{{B}_{2}},\cdots ,{{B}_{n}} A1,A2,,Am,B1,B2,,Bn相互独立,则 f ( A 1 , A 2 , ⋯   , A m ) f({{A}_{1}},{{A}_{2}},\cdots ,{{A}_{m}}) f(A1,A2,,Am) g ( B 1 , B 2 , ⋯   , B n ) g({{B}_{1}},{{B}_{2}},\cdots ,{{B}_{n}}) g(B1,B2,,Bn)也相互独立,其中 f ( ⋅ ) , g ( ⋅ ) f(\centerdot ),g(\centerdot ) f(),g()分别表示对相应事件做任意事件运算后所得的事件,另外,概率为1(或0)的事件与任何事件相互独立.

    随机变量及其概率分布

    1.随机变量及概率分布

    取值带有随机性的变量,严格地说是定义在样本空间上,取值于实数的函数称为随机变量,概率分布通常指分布函数或分布律

    2.分布函数的概念与性质

    定义: F ( x ) = P ( X ≤ x ) , − ∞ < x < + ∞ F(x) = P(X \leq x), - \infty < x < + \infty F(x)=P(Xx),<x<+

    性质:(1) 0 ≤ F ( x ) ≤ 1 0 \leq F(x) \leq 1 0F(x)1

    (2) F ( x ) F(x) F(x)单调不减

    (3) 右连续 F ( x + 0 ) = F ( x ) F(x + 0) = F(x) F(x+0)=F(x)

    (4) F ( − ∞ ) = 0 , F ( + ∞ ) = 1 F( - \infty) = 0,F( + \infty) = 1 F()=0,F(+)=1

    3.离散型随机变量的概率分布

    P ( X = x i ) = p i , i = 1 , 2 , ⋯   , n , ⋯ p i ≥ 0 , ∑ i = 1 ∞ p i = 1 P(X = x_{i}) = p_{i},i = 1,2,\cdots,n,\cdots\quad\quad p_{i} \geq 0,\sum_{i =1}^{\infty}p_{i} = 1 P(X=xi)=pi,i=1,2,,n,pi0,i=1pi=1

    4.连续型随机变量的概率密度

    概率密度 f ( x ) f(x) f(x);非负可积,且:

    (1) f ( x ) ≥ 0 , f(x) \geq 0, f(x)0,

    (2) ∫ − ∞ + ∞ f ( x ) d x = 1 \int_{- \infty}^{+\infty}{f(x){dx} = 1} +f(x)dx=1

    (3) x x x f ( x ) f(x) f(x)的连续点,则:

    f ( x ) = F ′ ( x ) f(x) = F'(x) f(x)=F(x)分布函数 F ( x ) = ∫ − ∞ x f ( t ) d t F(x) = \int_{- \infty}^{x}{f(t){dt}} F(x)=xf(t)dt

    5.常见分布

    (1) 0-1分布: P ( X = k ) = p k ( 1 − p ) 1 − k , k = 0 , 1 P(X = k) = p^{k}{(1 - p)}^{1 - k},k = 0,1 P(X=k)=pk(1p)1k,k=0,1

    (2) 二项分布: B ( n , p ) B(n,p) B(n,p) P ( X = k ) = C n k p k ( 1 − p ) n − k , k = 0 , 1 , ⋯   , n P(X = k) = C_{n}^{k}p^{k}{(1 - p)}^{n - k},k =0,1,\cdots,n P(X=k)=Cnkpk(1p)nk,k=0,1,,n

    (3) Poisson分布: p ( λ ) p(\lambda) p(λ) P ( X = k ) = λ k k ! e − λ , λ > 0 , k = 0 , 1 , 2 ⋯ P(X = k) = \frac{\lambda^{k}}{k!}e^{-\lambda},\lambda > 0,k = 0,1,2\cdots P(X=k)=k!λkeλ,λ>0,k=0,1,2

    (4) 均匀分布 U ( a , b ) U(a,b) U(a,b) f ( x ) = { 1 b − a , a < x < b 0 , f(x) = \{ \begin{matrix} & \frac{1}{b - a},a < x< b \\ & 0, \\ \end{matrix} f(x)={ba1,a<x<b0,

    (5) 正态分布: N ( μ , σ 2 ) : N(\mu,\sigma^{2}): N(μ,σ2): φ ( x ) = 1 2 π σ e − ( x − μ ) 2 2 σ 2 , σ > 0 , ∞ < x < + ∞ \varphi(x) =\frac{1}{\sqrt{2\pi}\sigma}e^{- \frac{{(x - \mu)}^{2}}{2\sigma^{2}}},\sigma > 0,\infty < x < + \infty φ(x)=2π σ1e2σ2(xμ)2,σ>0,<x<+

    (6)指数分布: E ( λ ) : f ( x ) = { λ e − λ x , x > 0 , λ > 0 0 , E(\lambda):f(x) =\{ \begin{matrix} & \lambda e^{-{λx}},x > 0,\lambda > 0 \\ & 0, \\ \end{matrix} E(λ):f(x)={λeλx,x>0,λ>00,

    (7)几何分布: G ( p ) : P ( X = k ) = ( 1 − p ) k − 1 p , 0 < p < 1 , k = 1 , 2 , ⋯   . G(p):P(X = k) = {(1 - p)}^{k - 1}p,0 < p < 1,k = 1,2,\cdots. G(p):P(X=k)=(1p)k1p,0<p<1,k=1,2,.

    (8)超几何分布: H ( N , M , n ) : P ( X = k ) = C M k C N − M n − k C N n , k = 0 , 1 , ⋯   , m i n ( n , M ) H(N,M,n):P(X = k) = \frac{C_{M}^{k}C_{N - M}^{n -k}}{C_{N}^{n}},k =0,1,\cdots,min(n,M) H(N,M,n):P(X=k)=CNnCMkCNMnk,k=0,1,,min(n,M)

    6.随机变量函数的概率分布

    (1)离散型: P ( X = x 1 ) = p i , Y = g ( X ) P(X = x_{1}) = p_{i},Y = g(X) P(X=x1)=pi,Y=g(X)

    则: P ( Y = y j ) = ∑ g ( x i ) = y i P ( X = x i ) P(Y = y_{j}) = \sum_{g(x_{i}) = y_{i}}^{}{P(X = x_{i})} P(Y=yj)=g(xi)=yiP(X=xi)

    (2)连续型: X   ~ f X ( x ) , Y = g ( x ) X\tilde{\ }f_{X}(x),Y = g(x) X ~fX(x),Y=g(x)

    则: F y ( y ) = P ( Y ≤ y ) = P ( g ( X ) ≤ y ) = ∫ g ( x ) ≤ y f x ( x ) d x F_{y}(y) = P(Y \leq y) = P(g(X) \leq y) = \int_{g(x) \leq y}^{}{f_{x}(x)dx} Fy(y)=P(Yy)=P(g(X)y)=g(x)yfx(x)dx f Y ( y ) = F Y ′ ( y ) f_{Y}(y) = F'_{Y}(y) fY(y)=FY(y)

    7.重要公式与结论

    (1) X ∼ N ( 0 , 1 ) ⇒ φ ( 0 ) = 1 2 π , Φ ( 0 ) = 1 2 , X\sim N(0,1) \Rightarrow \varphi(0) = \frac{1}{\sqrt{2\pi}},\Phi(0) =\frac{1}{2}, XN(0,1)φ(0)=2π 1,Φ(0)=21, Φ ( − a ) = P ( X ≤ − a ) = 1 − Φ ( a ) \Phi( - a) = P(X \leq - a) = 1 - \Phi(a) Φ(a)=P(Xa)=1Φ(a)

    (2) X ∼ N ( μ , σ 2 ) ⇒ X − μ σ ∼ N ( 0 , 1 ) , P ( X ≤ a ) = Φ ( a − μ σ ) X\sim N\left( \mu,\sigma^{2} \right) \Rightarrow \frac{X -\mu}{\sigma}\sim N\left( 0,1 \right),P(X \leq a) = \Phi(\frac{a -\mu}{\sigma}) XN(μ,σ2)σXμN(0,1),P(Xa)=Φ(σaμ)

    (3) X ∼ E ( λ ) ⇒ P ( X > s + t ∣ X > s ) = P ( X > t ) X\sim E(\lambda) \Rightarrow P(X > s + t|X > s) = P(X > t) XE(λ)P(X>s+tX>s)=P(X>t)

    (4) X ∼ G ( p ) ⇒ P ( X = m + k ∣ X > m ) = P ( X = k ) X\sim G(p) \Rightarrow P(X = m + k|X > m) = P(X = k) XG(p)P(X=m+kX>m)=P(X=k)

    (5) 离散型随机变量的分布函数为阶梯间断函数;连续型随机变量的分布函数为连续函数,但不一定为处处可导函数。

    (6) 存在既非离散也非连续型随机变量。

    多维随机变量及其分布

    1.二维随机变量及其联合分布

    由两个随机变量构成的随机向量 ( X , Y ) (X,Y) (X,Y), 联合分布为 F ( x , y ) = P ( X ≤ x , Y ≤ y ) F(x,y) = P(X \leq x,Y \leq y) F(x,y)=P(Xx,Yy)

    2.二维离散型随机变量的分布

    (1) 联合概率分布律 P { X = x i , Y = y j } = p i j ; i , j = 1 , 2 , ⋯ P\{ X = x_{i},Y = y_{j}\} = p_{{ij}};i,j =1,2,\cdots P{X=xi,Y=yj}=pij;i,j=1,2,

    (2) 边缘分布律 p i ⋅ = ∑ j = 1 ∞ p i j , i = 1 , 2 , ⋯ p_{i \cdot} = \sum_{j = 1}^{\infty}p_{{ij}},i =1,2,\cdots pi=j=1pij,i=1,2, p ⋅ j = ∑ i ∞ p i j , j = 1 , 2 , ⋯ p_{\cdot j} = \sum_{i}^{\infty}p_{{ij}},j = 1,2,\cdots pj=ipij,j=1,2,

    (3) 条件分布律 P { X = x i ∣ Y = y j } = p i j p ⋅ j P\{ X = x_{i}|Y = y_{j}\} = \frac{p_{{ij}}}{p_{\cdot j}} P{X=xiY=yj}=pjpij
    P { Y = y j ∣ X = x i } = p i j p i ⋅ P\{ Y = y_{j}|X = x_{i}\} = \frac{p_{{ij}}}{p_{i \cdot}} P{Y=yjX=xi}=pipij

    3. 二维连续性随机变量的密度

    (1) 联合概率密度 f ( x , y ) : f(x,y): f(x,y):

    1. f ( x , y ) ≥ 0 f(x,y) \geq 0 f(x,y)0

    2. ∫ − ∞ + ∞ ∫ − ∞ + ∞ f ( x , y ) d x d y = 1 \int_{- \infty}^{+ \infty}{\int_{- \infty}^{+ \infty}{f(x,y)dxdy}} = 1 ++f(x,y)dxdy=1

    (2) 分布函数: F ( x , y ) = ∫ − ∞ x ∫ − ∞ y f ( u , v ) d u d v F(x,y) = \int_{- \infty}^{x}{\int_{- \infty}^{y}{f(u,v)dudv}} F(x,y)=xyf(u,v)dudv

    (3) 边缘概率密度: f X ( x ) = ∫ − ∞ + ∞ f ( x , y ) d y f_{X}\left( x \right) = \int_{- \infty}^{+ \infty}{f\left( x,y \right){dy}} fX(x)=+f(x,y)dy f Y ( y ) = ∫ − ∞ + ∞ f ( x , y ) d x f_{Y}(y) = \int_{- \infty}^{+ \infty}{f(x,y)dx} fY(y)=+f(x,y)dx

    (4) 条件概率密度: f X ∣ Y ( x | y ) = f ( x , y ) f Y ( y ) f_{X|Y}\left( x \middle| y \right) = \frac{f\left( x,y \right)}{f_{Y}\left( y \right)} fXY(xy)=fY(y)f(x,y) f Y ∣ X ( y ∣ x ) = f ( x , y ) f X ( x ) f_{Y|X}(y|x) = \frac{f(x,y)}{f_{X}(x)} fYX(yx)=fX(x)f(x,y)

    4.常见二维随机变量的联合分布

    (1) 二维均匀分布: ( x , y ) ∼ U ( D ) (x,y) \sim U(D) (x,y)U(D) , f ( x , y ) = { 1 S ( D ) , ( x , y ) ∈ D 0 , 其 他 f(x,y) = \begin{cases} \frac{1}{S(D)},(x,y) \in D \\ 0,其他 \end{cases} f(x,y)={S(D)1,(x,y)D0,

    (2) 二维正态分布: ( X , Y ) ∼ N ( μ 1 , μ 2 , σ 1 2 , σ 2 2 , ρ ) (X,Y)\sim N(\mu_{1},\mu_{2},\sigma_{1}^{2},\sigma_{2}^{2},\rho) (X,Y)N(μ1,μ2,σ12,σ22,ρ), ( X , Y ) ∼ N ( μ 1 , μ 2 , σ 1 2 , σ 2 2 , ρ ) (X,Y)\sim N(\mu_{1},\mu_{2},\sigma_{1}^{2},\sigma_{2}^{2},\rho) (X,Y)N(μ1,μ2,σ12,σ22,ρ)

    f ( x , y ) = 1 2 π σ 1 σ 2 1 − ρ 2 . exp ⁡ { − 1 2 ( 1 − ρ 2 ) [ ( x − μ 1 ) 2 σ 1 2 − 2 ρ ( x − μ 1 ) ( y − μ 2 ) σ 1 σ 2 + ( y − μ 2 ) 2 σ 2 2 ] } f(x,y) = \frac{1}{2\pi\sigma_{1}\sigma_{2}\sqrt{1 - \rho^{2}}}.\exp\left\{ \frac{- 1}{2(1 - \rho^{2})}\lbrack\frac{{(x - \mu_{1})}^{2}}{\sigma_{1}^{2}} - 2\rho\frac{(x - \mu_{1})(y - \mu_{2})}{\sigma_{1}\sigma_{2}} + \frac{{(y - \mu_{2})}^{2}}{\sigma_{2}^{2}}\rbrack \right\} f(x,y)=2πσ1σ21ρ2 1.exp{2(1ρ2)1[σ12(xμ1)22ρσ1σ2(xμ1)(yμ2)+σ22(yμ2)2]}

    5.随机变量的独立性和相关性

    X X X Y Y Y的相互独立: ⇔ F ( x , y ) = F X ( x ) F Y ( y ) \Leftrightarrow F\left( x,y \right) = F_{X}\left( x \right)F_{Y}\left( y \right) F(x,y)=FX(x)FY(y):

    ⇔ p i j = p i ⋅ ⋅ p ⋅ j \Leftrightarrow p_{{ij}} = p_{i \cdot} \cdot p_{\cdot j} pij=pipj(离散型)
    ⇔ f ( x , y ) = f X ( x ) f Y ( y ) \Leftrightarrow f\left( x,y \right) = f_{X}\left( x \right)f_{Y}\left( y \right) f(x,y)=fX(x)fY(y)(连续型)

    X X X Y Y Y的相关性:

    相关系数 ρ X Y = 0 \rho_{{XY}} = 0 ρXY=0时,称 X X X Y Y Y不相关,
    否则称 X X X Y Y Y相关

    6.两个随机变量简单函数的概率分布

    离散型: P ( X = x i , Y = y i ) = p i j , Z = g ( X , Y ) P\left( X = x_{i},Y = y_{i} \right) = p_{{ij}},Z = g\left( X,Y \right) P(X=xi,Y=yi)=pij,Z=g(X,Y) 则:

    P ( Z = z k ) = P { g ( X , Y ) = z k } = ∑ g ( x i , y i ) = z k P ( X = x i , Y = y j ) P(Z = z_{k}) = P\left\{ g\left( X,Y \right) = z_{k} \right\} = \sum_{g\left( x_{i},y_{i} \right) = z_{k}}^{}{P\left( X = x_{i},Y = y_{j} \right)} P(Z=zk)=P{g(X,Y)=zk}=g(xi,yi)=zkP(X=xi,Y=yj)

    连续型: ( X , Y ) ∼ f ( x , y ) , Z = g ( X , Y ) \left( X,Y \right) \sim f\left( x,y \right),Z = g\left( X,Y \right) (X,Y)f(x,y),Z=g(X,Y)
    则:

    F z ( z ) = P { g ( X , Y ) ≤ z } = ∬ g ( x , y ) ≤ z f ( x , y ) d x d y F_{z}\left( z \right) = P\left\{ g\left( X,Y \right) \leq z \right\} = \iint_{g(x,y) \leq z}^{}{f(x,y)dxdy} Fz(z)=P{g(X,Y)z}=g(x,y)zf(x,y)dxdy f z ( z ) = F z ′ ( z ) f_{z}(z) = F'_{z}(z) fz(z)=Fz(z)

    7.重要公式与结论

    (1) 边缘密度公式: f X ( x ) = ∫ − ∞ + ∞ f ( x , y ) d y , f_{X}(x) = \int_{- \infty}^{+ \infty}{f(x,y)dy,} fX(x)=+f(x,y)dy,
    f Y ( y ) = ∫ − ∞ + ∞ f ( x , y ) d x f_{Y}(y) = \int_{- \infty}^{+ \infty}{f(x,y)dx} fY(y)=+f(x,y)dx

    (2) P { ( X , Y ) ∈ D } = ∬ D f ( x , y ) d x d y P\left\{ \left( X,Y \right) \in D \right\} = \iint_{D}^{}{f\left( x,y \right){dxdy}} P{(X,Y)D}=Df(x,y)dxdy

    (3) 若 ( X , Y ) (X,Y) (X,Y)服从二维正态分布 N ( μ 1 , μ 2 , σ 1 2 , σ 2 2 , ρ ) N(\mu_{1},\mu_{2},\sigma_{1}^{2},\sigma_{2}^{2},\rho) N(μ1,μ2,σ12,σ22,ρ)
    则有:

    1. X ∼ N ( μ 1 , σ 1 2 ) , Y ∼ N ( μ 2 , σ 2 2 ) . X\sim N\left( \mu_{1},\sigma_{1}^{2} \right),Y\sim N(\mu_{2},\sigma_{2}^{2}). XN(μ1,σ12),YN(μ2,σ22).

    2. X X X Y Y Y相互独立 ⇔ ρ = 0 \Leftrightarrow \rho = 0 ρ=0,即 X X X Y Y Y不相关。

    3. C 1 X + C 2 Y ∼ N ( C 1 μ 1 + C 2 μ 2 , C 1 2 σ 1 2 + C 2 2 σ 2 2 + 2 C 1 C 2 σ 1 σ 2 ρ ) C_{1}X + C_{2}Y\sim N(C_{1}\mu_{1} + C_{2}\mu_{2},C_{1}^{2}\sigma_{1}^{2} + C_{2}^{2}\sigma_{2}^{2} + 2C_{1}C_{2}\sigma_{1}\sigma_{2}\rho) C1X+C2YN(C1μ1+C2μ2,C12σ12+C22σ22+2C1C2σ1σ2ρ)

    4.   X {\ X}  X关于 Y = y Y=y Y=y的条件分布为: N ( μ 1 + ρ σ 1 σ 2 ( y − μ 2 ) , σ 1 2 ( 1 − ρ 2 ) ) N(\mu_{1} + \rho\frac{\sigma_{1}}{\sigma_{2}}(y - \mu_{2}),\sigma_{1}^{2}(1 - \rho^{2})) N(μ1+ρσ2σ1(yμ2),σ12(1ρ2))

    5. Y Y Y关于 X = x X = x X=x的条件分布为: N ( μ 2 + ρ σ 2 σ 1 ( x − μ 1 ) , σ 2 2 ( 1 − ρ 2 ) ) N(\mu_{2} + \rho\frac{\sigma_{2}}{\sigma_{1}}(x - \mu_{1}),\sigma_{2}^{2}(1 - \rho^{2})) N(μ2+ρσ1σ2(xμ1),σ22(1ρ2))

    (4) 若 X X X Y Y Y独立,且分别服从 N ( μ 1 , σ 1 2 ) , N ( μ 1 , σ 2 2 ) , N(\mu_{1},\sigma_{1}^{2}),N(\mu_{1},\sigma_{2}^{2}), N(μ1,σ12),N(μ1,σ22),
    则: ( X , Y ) ∼ N ( μ 1 , μ 2 , σ 1 2 , σ 2 2 , 0 ) , \left( X,Y \right)\sim N(\mu_{1},\mu_{2},\sigma_{1}^{2},\sigma_{2}^{2},0), (X,Y)N(μ1,μ2,σ12,σ22,0),

    C 1 X + C 2 Y   ~ N ( C 1 μ 1 + C 2 μ 2 , C 1 2 σ 1 2 C 2 2 σ 2 2 ) . C_{1}X + C_{2}Y\tilde{\ }N(C_{1}\mu_{1} + C_{2}\mu_{2},C_{1}^{2}\sigma_{1}^{2} C_{2}^{2}\sigma_{2}^{2}). C1X+C2Y ~N(C1μ1+C2μ2,C12σ12C22σ22).

    (5) 若 X X X Y Y Y相互独立, f ( x ) f\left( x \right) f(x) g ( x ) g\left( x \right) g(x)为连续函数, 则 f ( X ) f\left( X \right) f(X) g ( Y ) g(Y) g(Y)也相互独立。

    随机变量的数字特征

    1.数学期望

    离散型: P { X = x i } = p i , E ( X ) = ∑ i x i p i P\left\{ X = x_{i} \right\} = p_{i},E(X) = \sum_{i}^{}{x_{i}p_{i}} P{X=xi}=pi,E(X)=ixipi

    连续型: X ∼ f ( x ) , E ( X ) = ∫ − ∞ + ∞ x f ( x ) d x X\sim f(x),E(X) = \int_{- \infty}^{+ \infty}{xf(x)dx} Xf(x),E(X)=+xf(x)dx

    性质:

    (1) E ( C ) = C , E [ E ( X ) ] = E ( X ) E(C) = C,E\lbrack E(X)\rbrack = E(X) E(C)=C,E[E(X)]=E(X)

    (2) E ( C 1 X + C 2 Y ) = C 1 E ( X ) + C 2 E ( Y ) E(C_{1}X + C_{2}Y) = C_{1}E(X) + C_{2}E(Y) E(C1X+C2Y)=C1E(X)+C2E(Y)

    (3) 若 X X X Y Y Y独立,则 E ( X Y ) = E ( X ) E ( Y ) E(XY) = E(X)E(Y) E(XY)=E(X)E(Y)

    (4) [ E ( X Y ) ] 2 ≤ E ( X 2 ) E ( Y 2 ) \left\lbrack E(XY) \right\rbrack^{2} \leq E(X^{2})E(Y^{2}) [E(XY)]2E(X2)E(Y2)

    2.方差

    D ( X ) = E [ X − E ( X ) ] 2 = E ( X 2 ) − [ E ( X ) ] 2 D(X) = E\left\lbrack X - E(X) \right\rbrack^{2} = E(X^{2}) - \left\lbrack E(X) \right\rbrack^{2} D(X)=E[XE(X)]2=E(X2)[E(X)]2

    3.标准差

    D ( X ) \sqrt{D(X)} D(X)

    4.离散型

    D ( X ) = ∑ i [ x i − E ( X ) ] 2 p i D(X) = \sum_{i}^{}{\left\lbrack x_{i} - E(X) \right\rbrack^{2}p_{i}} D(X)=i[xiE(X)]2pi

    5.连续型

    D ( X ) = ∫ − ∞ + ∞ [ x − E ( X ) ] 2 f ( x ) d x D(X) = {\int_{- \infty}^{+ \infty}\left\lbrack x - E(X) \right\rbrack}^{2}f(x)dx D(X)=+[xE(X)]2f(x)dx

    基本性质

    (1)   D ( C ) = 0 , D [ E ( X ) ] = 0 , D [ D ( X ) ] = 0 \ D(C) = 0,D\lbrack E(X)\rbrack = 0,D\lbrack D(X)\rbrack = 0  D(C)=0,D[E(X)]=0,D[D(X)]=0

    (2) X X X Y Y Y相互独立,则 D ( X ± Y ) = D ( X ) + D ( Y ) D(X \pm Y) = D(X) + D(Y) D(X±Y)=D(X)+D(Y)

    (3)   D ( C 1 X + C 2 ) = C 1 2 D ( X ) \ D\left( C_{1}X + C_{2} \right) = C_{1}^{2}D\left( X \right)  D(C1X+C2)=C12D(X)

    (4) 一般有 D ( X ± Y ) = D ( X ) + D ( Y ) ± 2 C o v ( X , Y ) = D ( X ) + D ( Y ) ± 2 ρ D ( X ) D ( Y ) D(X \pm Y) = D(X) + D(Y) \pm 2Cov(X,Y) = D(X) + D(Y) \pm 2\rho\sqrt{D(X)}\sqrt{D(Y)} D(X±Y)=D(X)+D(Y)±2Cov(X,Y)=D(X)+D(Y)±2ρD(X) D(Y)

    (5)   D ( X ) < E ( X − C ) 2 , C ≠ E ( X ) \ D\left( X \right) < E\left( X - C \right)^{2},C \neq E\left( X \right)  D(X)<E(XC)2,C=E(X)

    (6)   D ( X ) = 0 ⇔ P { X = C } = 1 \ D(X) = 0 \Leftrightarrow P\left\{ X = C \right\} = 1  D(X)=0P{X=C}=1

    6.随机变量函数的数学期望

    (1) 对于函数 Y = g ( x ) Y = g(x) Y=g(x)

    X X X为离散型: P { X = x i } = p i , E ( Y ) = ∑ i g ( x i ) p i P\{ X = x_{i}\} = p_{i},E(Y) = \sum_{i}^{}{g(x_{i})p_{i}} P{X=xi}=pi,E(Y)=ig(xi)pi

    X X X为连续型: X ∼ f ( x ) , E ( Y ) = ∫ − ∞ + ∞ g ( x ) f ( x ) d x X\sim f(x),E(Y) = \int_{- \infty}^{+ \infty}{g(x)f(x)dx} Xf(x),E(Y)=+g(x)f(x)dx

    (2) Z = g ( X , Y ) Z = g(X,Y) Z=g(X,Y); ( X , Y ) ∼ P { X = x i , Y = y j } = p i j \left( X,Y \right)\sim P\{ X = x_{i},Y = y_{j}\} = p_{{ij}} (X,Y)P{X=xi,Y=yj}=pij; E ( Z ) = ∑ i ∑ j g ( x i , y j ) p i j E(Z) = \sum_{i}^{}{\sum_{j}^{}{g(x_{i},y_{j})p_{{ij}}}} E(Z)=ijg(xi,yj)pij ( X , Y ) ∼ f ( x , y ) \left( X,Y \right)\sim f(x,y) (X,Y)f(x,y); E ( Z ) = ∫ − ∞ + ∞ ∫ − ∞ + ∞ g ( x , y ) f ( x , y ) d x d y E(Z) = \int_{- \infty}^{+ \infty}{\int_{- \infty}^{+ \infty}{g(x,y)f(x,y)dxdy}} E(Z)=++g(x,y)f(x,y)dxdy

    7.协方差

    C o v ( X , Y ) = E [ ( X − E ( X ) ( Y − E ( Y ) ) ] Cov(X,Y) = E\left\lbrack (X - E(X)(Y - E(Y)) \right\rbrack Cov(X,Y)=E[(XE(X)(YE(Y))]

    8.相关系数

    ρ X Y = C o v ( X , Y ) D ( X ) D ( Y ) \rho_{{XY}} = \frac{Cov(X,Y)}{\sqrt{D(X)}\sqrt{D(Y)}} ρXY=D(X) D(Y) Cov(X,Y), k k k阶原点矩 E ( X k ) E(X^{k}) E(Xk);
    k k k阶中心矩 E { [ X − E ( X ) ] k } E\left\{ {\lbrack X - E(X)\rbrack}^{k} \right\} E{[XE(X)]k}

    性质

    (1)   C o v ( X , Y ) = C o v ( Y , X ) \ Cov(X,Y) = Cov(Y,X)  Cov(X,Y)=Cov(Y,X)

    (2)   C o v ( a X , b Y ) = a b C o v ( Y , X ) \ Cov(aX,bY) = abCov(Y,X)  Cov(aX,bY)=abCov(Y,X)

    (3)   C o v ( X 1 + X 2 , Y ) = C o v ( X 1 , Y ) + C o v ( X 2 , Y ) \ Cov(X_{1} + X_{2},Y) = Cov(X_{1},Y) + Cov(X_{2},Y)  Cov(X1+X2,Y)=Cov(X1,Y)+Cov(X2,Y)

    (4)   ∣ ρ ( X , Y ) ∣ ≤ 1 \ \left| \rho\left( X,Y \right) \right| \leq 1  ρ(X,Y)1

    (5)   ρ ( X , Y ) = 1 ⇔ P ( Y = a X + b ) = 1 \ \rho\left( X,Y \right) = 1 \Leftrightarrow P\left( Y = aX + b \right) = 1  ρ(X,Y)=1P(Y=aX+b)=1 ,其中 a > 0 a > 0 a>0

    ρ ( X , Y ) = − 1 ⇔ P ( Y = a X + b ) = 1 \rho\left( X,Y \right) = - 1 \Leftrightarrow P\left( Y = aX + b \right) = 1 ρ(X,Y)=1P(Y=aX+b)=1
    ,其中 a < 0 a < 0 a<0

    9.重要公式与结论

    (1)   D ( X ) = E ( X 2 ) − E 2 ( X ) \ D(X) = E(X^{2}) - E^{2}(X)  D(X)=E(X2)E2(X)

    (2)   C o v ( X , Y ) = E ( X Y ) − E ( X ) E ( Y ) \ Cov(X,Y) = E(XY) - E(X)E(Y)  Cov(X,Y)=E(XY)E(X)E(Y)

    (3) ∣ ρ ( X , Y ) ∣ ≤ 1 , \left| \rho\left( X,Y \right) \right| \leq 1, ρ(X,Y)1, ρ ( X , Y ) = 1 ⇔ P ( Y = a X + b ) = 1 \rho\left( X,Y \right) = 1 \Leftrightarrow P\left( Y = aX + b \right) = 1 ρ(X,Y)=1P(Y=aX+b)=1,其中 a > 0 a > 0 a>0

    ρ ( X , Y ) = − 1 ⇔ P ( Y = a X + b ) = 1 \rho\left( X,Y \right) = - 1 \Leftrightarrow P\left( Y = aX + b \right) = 1 ρ(X,Y)=1P(Y=aX+b)=1,其中 a < 0 a < 0 a<0

    (4) 下面5个条件互为充要条件:

    ρ ( X , Y ) = 0 \rho(X,Y) = 0 ρ(X,Y)=0 ⇔ C o v ( X , Y ) = 0 \Leftrightarrow Cov(X,Y) = 0 Cov(X,Y)=0 ⇔ E ( X , Y ) = E ( X ) E ( Y ) \Leftrightarrow E(X,Y) = E(X)E(Y) E(X,Y)=E(X)E(Y) ⇔ D ( X + Y ) = D ( X ) + D ( Y ) \Leftrightarrow D(X + Y) = D(X) + D(Y) D(X+Y)=D(X)+D(Y) ⇔ D ( X − Y ) = D ( X ) + D ( Y ) \Leftrightarrow D(X - Y) = D(X) + D(Y) D(XY)=D(X)+D(Y)

    注: X X X Y Y Y独立为上述5个条件中任何一个成立的充分条件,但非必要条件。

    数理统计的基本概念

    1.基本概念

    总体:研究对象的全体,它是一个随机变量,用 X X X表示。

    个体:组成总体的每个基本元素。

    简单随机样本:来自总体 X X X n n n个相互独立且与总体同分布的随机变量 X 1 , X 2 ⋯   , X n X_{1},X_{2}\cdots,X_{n} X1,X2,Xn,称为容量为 n n n的简单随机样本,简称样本。

    统计量:设 X 1 , X 2 ⋯   , X n , X_{1},X_{2}\cdots,X_{n}, X1,X2,Xn,是来自总体 X X X的一个样本, g ( X 1 , X 2 ⋯   , X n ) g(X_{1},X_{2}\cdots,X_{n}) g(X1,X2,Xn))是样本的连续函数,且 g ( ) g() g()中不含任何未知参数,则称 g ( X 1 , X 2 ⋯   , X n ) g(X_{1},X_{2}\cdots,X_{n}) g(X1,X2,Xn)为统计量。

    样本均值: X ‾ = 1 n ∑ i = 1 n X i \overline{X} = \frac{1}{n}\sum_{i = 1}^{n}X_{i} X=n1i=1nXi

    样本方差: S 2 = 1 n − 1 ∑ i = 1 n ( X i − X ‾ ) 2 S^{2} = \frac{1}{n - 1}\sum_{i = 1}^{n}{(X_{i} - \overline{X})}^{2} S2=n11i=1n(XiX)2

    样本矩:样本 k k k阶原点矩: A k = 1 n ∑ i = 1 n X i k , k = 1 , 2 , ⋯ A_{k} = \frac{1}{n}\sum_{i = 1}^{n}X_{i}^{k},k = 1,2,\cdots Ak=n1i=1nXik,k=1,2,

    样本 k k k阶中心矩: B k = 1 n ∑ i = 1 n ( X i − X ‾ ) k , k = 1 , 2 , ⋯ B_{k} = \frac{1}{n}\sum_{i = 1}^{n}{(X_{i} - \overline{X})}^{k},k = 1,2,\cdots Bk=n1i=1n(XiX)k,k=1,2,

    2.分布

    χ 2 \chi^{2} χ2分布: χ 2 = X 1 2 + X 2 2 + ⋯ + X n 2 ∼ χ 2 ( n ) \chi^{2} = X_{1}^{2} + X_{2}^{2} + \cdots + X_{n}^{2}\sim\chi^{2}(n) χ2=X12+X22++Xn2χ2(n),其中 X 1 , X 2 ⋯   , X n , X_{1},X_{2}\cdots,X_{n}, X1,X2,Xn,相互独立,且同服从 N ( 0 , 1 ) N(0,1) N(0,1)

    t t t分布: T = X Y / n ∼ t ( n ) T = \frac{X}{\sqrt{Y/n}}\sim t(n) T=Y/n Xt(n) ,其中 X ∼ N ( 0 , 1 ) , Y ∼ χ 2 ( n ) , X\sim N\left( 0,1 \right),Y\sim\chi^{2}(n), XN(0,1),Yχ2(n), X X X Y Y Y 相互独立。

    F F F分布: F = X / n 1 Y / n 2 ∼ F ( n 1 , n 2 ) F = \frac{X/n_{1}}{Y/n_{2}}\sim F(n_{1},n_{2}) F=Y/n2X/n1F(n1,n2),其中 X ∼ χ 2 ( n 1 ) , Y ∼ χ 2 ( n 2 ) , X\sim\chi^{2}\left( n_{1} \right),Y\sim\chi^{2}(n_{2}), Xχ2(n1),Yχ2(n2), X X X Y Y Y相互独立。

    分位数:若 P ( X ≤ x α ) = α , P(X \leq x_{\alpha}) = \alpha, P(Xxα)=α,则称 x α x_{\alpha} xα X X X α \alpha α分位数

    3.正态总体的常用样本分布

    (1) 设 X 1 , X 2 ⋯   , X n X_{1},X_{2}\cdots,X_{n} X1,X2,Xn为来自正态总体 N ( μ , σ 2 ) N(\mu,\sigma^{2}) N(μ,σ2)的样本,

    X ‾ = 1 n ∑ i = 1 n X i , S 2 = 1 n − 1 ∑ i = 1 n ( X i − X ‾ ) 2 , \overline{X} = \frac{1}{n}\sum_{i = 1}^{n}X_{i},S^{2} = \frac{1}{n - 1}\sum_{i = 1}^{n}{{(X_{i} - \overline{X})}^{2},} X=n1i=1nXi,S2=n11i=1n(XiX)2,则:

    1. X ‾ ∼ N ( μ , σ 2 n )    \overline{X}\sim N\left( \mu,\frac{\sigma^{2}}{n} \right){\ \ } XN(μ,nσ2)  或者 X ‾ − μ σ n ∼ N ( 0 , 1 ) \frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{n}}}\sim N(0,1) n σXμN(0,1)

    2. ( n − 1