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基础部分
设X 1 , X 2 X_1,X_2 X 1 , X 2 X ∼ N ( 1 , 1 ) X \sim N(1,1) X ∼ N ( 1 , 1 ) X 1 − 1 ∣ 1 − X 2 ∣ \frac{X_1-1}{|1-X_2|} ∣1 − X 2 ∣ X 1 − 1 t ( 1 ) t(1) t ( 1 ) F ( 1 , 1 ) F(1,1) F ( 1 , 1 ) χ 2 ( 1 ) \chi^2(1) χ 2 ( 1 ) N ( 1 , 1 ) N(1,1) N ( 1 , 1 )
设X 1 , X 2 , ⋯ , X 10 X_1,X_2,\cdots,X_{10} X 1 , X 2 , ⋯ , X 10 X ∼ N ( 0 , σ 2 ) ( σ > 0 ) X \sim N(0,\sigma^2)(\sigma>0) X ∼ N ( 0 , σ 2 ) ( σ > 0 ) Y 2 = 1 9 ∑ i = 2 10 X i 2 Y^2=\frac{1}{9}\sum_{i=2}^{10}X_i^2 Y 2 = 9 1 ∑ i = 2 10 X i 2 X 1 2 ∼ χ 2 ( 1 ) X_1^2 \sim \chi^2(1) X 1 2 ∼ χ 2 ( 1 ) Y 2 ∼ χ 2 ( 9 ) Y^2 \sim \chi^2(9) Y 2 ∼ χ 2 ( 9 ) X 1 ∣ Y ∣ ∼ t ( 9 ) \frac{X_1}{|Y|} \sim t(9) ∣ Y ∣ X 1 ∼ t ( 9 ) X 1 2 Y 2 ∼ F ( 9 , 1 ) \frac{X_1^2}{Y^2} \sim F(9,1) Y 2 X 1 2 ∼ F ( 9 , 1 )
设总体X X X Y Y Y N ( 0 , σ 2 ) N(0,\sigma^2) N ( 0 , σ 2 ) X 1 , X 2 , ⋯ , X n X_1,X_2,\cdots,X_n X 1 , X 2 , ⋯ , X n Y 1 , Y 2 , ⋯ , Y n Y_1,Y_2,\cdots,Y_n Y 1 , Y 2 , ⋯ , Y n X X X Y Y Y n n n X ‾ , S X 2 \overline{X},S_X^2 X , S X 2 Y ‾ , S Y 2 \overline{Y},S_Y^2 Y , S Y 2 X ‾ − Y ‾ ∼ N ( 0 , σ 2 ) \overline{X}-\overline{Y} \sim N(0,\sigma^2) X − Y ∼ N ( 0 , σ 2 ) S X 2 + S Y 2 ∼ χ 2 ( 2 n − 2 ) S_X^2+S_Y^2 \sim \chi^2(2n-2) S X 2 + S Y 2 ∼ χ 2 ( 2 n − 2 ) X ‾ − Y ‾ S X 2 + S Y 2 ∼ t ( 2 n − 2 ) \frac{\overline{X}-\overline{Y}}{\sqrt{S_X^2+S_Y^2}} \sim t(2n-2) S X 2 + S Y 2 X − Y ∼ t ( 2 n − 2 ) S X 2 S Y 2 ∼ F ( n − 1 , n − 1 ) \frac{S_X^2}{S_Y^2} \sim F(n-1,n-1) S Y 2 S X 2 ∼ F ( n − 1 , n − 1 )
设随机变量X ∼ N ( 0 , 1 ) X \sim N(0,1) X ∼ N ( 0 , 1 ) Y ∼ N ( 0 , 1 ) Y \sim N(0,1) Y ∼ N ( 0 , 1 ) X + Y X+Y X + Y X 2 + Y 2 X^2+Y^2 X 2 + Y 2 χ 2 \chi^2 χ 2 X 2 / Y 2 X^2/Y^2 X 2 / Y 2 F F F X 2 X^2 X 2 Y 2 Y^2 Y 2 χ 2 \chi^2 χ 2
设n n n X ∼ t ( n ) X \sim t(n) X ∼ t ( n ) Y ∼ F ( 1 , n ) Y \sim F(1,n) Y ∼ F ( 1 , n ) c c c P { X > c } = 2 5 P\{X>c\}=\frac{2}{5} P { X > c } = 5 2 P { Y ≤ c 2 } = P\{Y \leq c^2\}= P { Y ≤ c 2 } = 1 5 \frac{1}{5} 5 1 2 5 \frac{2}{5} 5 2 3 5 \frac{3}{5} 5 3 4 5 \frac{4}{5} 5 4
设X 1 , X 2 , ⋯ , X n ( n ≥ 2 ) X_1,X_2,\cdots,X_n(n \geq 2) X 1 , X 2 , ⋯ , X n ( n ≥ 2 ) X X X X ‾ = 1 n ∑ i = 1 n X i \overline{X}=\frac{1}{n}\sum_{i=1}^{n}X_i X = n 1 ∑ i = 1 n X i 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 S 2 = n − 1 1 ∑ i = 1 n ( X i − X ) 2 Y = X ‾ − S Y=\overline{X}-S Y = X − S E ( Y 2 ) = E(Y^2)= E ( Y 2 ) = 1 − 1 n 1-\frac{1}{n} 1 − n 1 1 + 1 n 1+\frac{1}{n} 1 + n 1 1 − 1 n − 1 1-\frac{1}{n-1} 1 − n − 1 1 1 + 1 n − 1 1+\frac{1}{n-1} 1 + n − 1 1
设X , Y X,Y X , Y N ( 0 , σ 2 ) N(0,\sigma^2) N ( 0 , σ 2 ) X 1 , ⋯ , X 9 X_1,\cdots,X_9 X 1 , ⋯ , X 9 Y 1 , ⋯ , Y 11 Y_1,\cdots,Y_{11} Y 1 , ⋯ , Y 11 X X X Y Y Y S X 2 S_X^2 S X 2 S Y 2 S_Y^2 S Y 2 S 1 2 = 1 2 ( S X 2 + S Y 2 ) S_1^2=\frac{1}{2}(S_X^2+S_Y^2) S 1 2 = 2 1 ( S X 2 + S Y 2 ) S 2 2 = 1 9 ( 4 S X 2 + 5 S Y 2 ) S_2^2=\frac{1}{9}(4S_X^2+5S_Y^2) S 2 2 = 9 1 ( 4 S X 2 + 5 S Y 2 ) S X 2 S_X^2 S X 2 S Y 2 S_Y^2 S Y 2 S 1 2 S_1^2 S 1 2 S 2 2 S_2^2 S 2 2
设X X X f ( x ; σ ) = { 2 x σ e − x 2 σ , x > 0 0 , x ≤ 0 f(x;\sigma)=\begin{cases}\frac{2x}{\sigma}e^{-\frac{x^2}{\sigma}},&x>0\\0,&x \leq 0\end{cases} f ( x ; σ ) = { σ 2 x e − σ x 2 , 0 , x > 0 x ≤ 0 σ \sigma σ X 1 , X 2 , ⋯ , X n X_1,X_2,\cdots,X_n X 1 , X 2 , ⋯ , X n X X X σ \sigma σ σ ^ = 1 n − 1 ∑ i = 1 n X i \hat{\sigma}=\frac{1}{n-1}\sum_{i=1}^{n}X_i σ ^ = n − 1 1 ∑ i = 1 n X i σ ^ = 1 n − 1 ∑ i = 1 n X i 2 \hat{\sigma}=\frac{1}{n-1}\sum_{i=1}^{n}X_i^2 σ ^ = n − 1 1 ∑ i = 1 n X i 2 σ ^ = 1 n ∑ i = 1 n X i \hat{\sigma}=\frac{1}{n}\sum_{i=1}^{n}X_i σ ^ = n 1 ∑ i = 1 n X i σ ^ = 1 n ∑ i = 1 n X i 2 \hat{\sigma}=\frac{1}{n}\sum_{i=1}^{n}X_i^2 σ ^ = n 1 ∑ i = 1 n X i 2
设某个试验有三种可能结果，其发生的概率分别为p 1 = λ 2 p_1=\lambda^2 p 1 = λ 2 p 2 = ( 1 − λ ) 2 p_2=(1-\lambda)^2 p 2 = ( 1 − λ ) 2 p 3 = 2 λ ( 1 − λ ) p_3=2\lambda(1-\lambda) p 3 = 2 λ ( 1 − λ ) λ \lambda λ 0 < λ < 1 0<\lambda<1 0 < λ < 1 n n n n 1 , n 2 , n 3 ( n 1 + n 2 + n 3 = n ) n_1,n_2,n_3(n_1+n_2+n_3=n) n 1 , n 2 , n 3 ( n 1 + n 2 + n 3 = n ) λ \lambda λ
设二维总体( X , Y ) (X,Y) ( X , Y ) f ( x , y ; λ ) = { 1 λ 2 e − x + y λ , x > 0 , y > 0 0 , 其他 f(x,y;\lambda)=\begin{cases}\frac{1}{\lambda^2}e^{-\frac{x+y}{\lambda}},&x>0,y>0\\0,&\text{其他}\end{cases} f ( x , y ; λ ) = { λ 2 1 e − λ x + y , 0 , x > 0 , y > 0 其他 λ \lambda λ ( X 1 , Y 1 ) , ( X 2 , Y 2 ) , ⋯ , ( X n , Y n ) (X_1,Y_1),(X_2,Y_2),\cdots,(X_n,Y_n) ( X 1 , Y 1 ) , ( X 2 , Y 2 ) , ⋯ , ( X n , Y n ) λ \lambda λ
设总体X X X f ( x ; θ ) = { e − ( x − θ ) , x ≥ θ 0 , 其他 f(x;\theta)=\begin{cases}e^{-(x-\theta)},&x \geq \theta\\0,&\text{其他}\end{cases} f ( x ; θ ) = { e − ( x − θ ) , 0 , x ≥ θ 其他 X 1 , X 2 , ⋯ , X n X_1,X_2,\cdots,X_n X 1 , X 2 , ⋯ , X n X X X θ \theta θ θ ^ = \hat{\theta}= θ ^ =
设X 1 , X 2 , ⋯ , X n X_1,X_2,\cdots,X_n X 1 , X 2 , ⋯ , X n X X X X X X f ( x ) = 1 2 λ e − ∣ x ∣ λ , − ∞ < x < + ∞ , λ > 0 f(x)=\frac{1}{2\lambda}e^{-\frac{|x|}{\lambda}},-\infty<x<+\infty,\lambda>0 f ( x ) = 2 λ 1 e − λ ∣ x ∣ , − ∞ < x < + ∞ , λ > 0 λ \lambda λ λ \lambda λ
设连续型总体X X X F ( x ; θ ) = { 0 , x ≤ 0 x θ , 0 < x < 1 1 , x ≥ 1 F(x;\theta)=\begin{cases}0,&x \leq 0\\x^{\sqrt{\theta}},&0<x<1\\1,&x \geq 1\end{cases} F ( x ; θ ) = ⎩ ⎨ ⎧ 0 , x θ , 1 , x ≤ 0 0 < x < 1 x ≥ 1 θ \theta θ θ > 0 \theta>0 θ > 0 X 1 , X 2 , ⋯ , X n X_1,X_2,\cdots,X_n X 1 , X 2 , ⋯ , X n X X X θ \theta θ
在数集Ω = { 0 , 1 , 2 , ⋯ , N } \Omega=\{0,1,2,\cdots,N\} Ω = { 0 , 1 , 2 , ⋯ , N } n n n X 1 , X 2 , ⋯ , X n X_1,X_2,\cdots,X_n X 1 , X 2 , ⋯ , X n N N N max { X 1 , X 2 , ⋯ , X n } \max\{X_1,X_2,\cdots,X_n\} max { X 1 , X 2 , ⋯ , X n } min { X 1 , X 2 , ⋯ , X n } \min\{X_1,X_2,\cdots,X_n\} min { X 1 , X 2 , ⋯ , X n } 1 n ∑ i = 1 n X i \frac{1}{n}\sum_{i=1}^{n}X_i n 1 ∑ i = 1 n X i n n n
设某元件的使用寿命T T T F ( t ) F(t) F ( t ) F ′ ( t ) + 2 t θ 2 [ F ( t ) − 1 ] = 0 , t ≥ 0 F'(t)+\frac{2t}{\theta^2}[F(t)-1]=0,t \geq 0 F ′ ( t ) + θ 2 2 t [ F ( t ) − 1 ] = 0 , t ≥ 0 θ \theta θ F ( 0 ) = 0 F(0)=0 F ( 0 ) = 0 Q ( θ ) = θ 2 ( ln θ 2 − 3 4 ) + θ Q(\theta)=\theta^2(\frac{\ln\theta}{2}-\frac{3}{4})+\theta Q ( θ ) = θ 2 ( 2 l n θ − 4 3 ) + θ n n n t 1 , t 2 , ⋯ , t n t_1,t_2,\cdots,t_n t 1 , t 2 , ⋯ , t n θ \theta θ θ ^ \hat{\theta} θ ^ Q Q Q Q ^ \hat{Q} Q ^
设总体X X X E ( X ) = 0 E(X)=0 E ( X ) = 0 D ( X ) = σ 2 D(X)=\sigma^2 D ( X ) = σ 2 X 1 , X 2 , ⋯ , X n ( n > 2 ) X_1,X_2,\cdots,X_n(n>2) X 1 , X 2 , ⋯ , X n ( n > 2 ) X X X X ‾ = 1 n ∑ i = 1 n X i \overline{X}=\frac{1}{n}\sum_{i=1}^{n}X_i X = n 1 ∑ i = 1 n X i 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 S 2 = n − 1 1 ∑ i = 1 n ( X i − X ) 2 σ 2 \sigma^2 σ 2 n X ‾ 2 + S 2 n\overline{X}^2+S^2 n X 2 + S 2 1 2 ( n X ‾ 2 + S 2 ) \frac{1}{2}(n\overline{X}^2+S^2) 2 1 ( n X 2 + S 2 ) 1 3 ( n X ‾ 2 + S 2 ) \frac{1}{3}(n\overline{X}^2+S^2) 3 1 ( n X 2 + S 2 ) 1 4 ( n X ‾ 2 + S 2 ) \frac{1}{4}(n\overline{X}^2+S^2) 4 1 ( n X 2 + S 2 )
设μ \mu μ X X X σ \sigma σ X X X X 1 , X 2 , ⋯ , X n X_1,X_2,\cdots,X_n X 1 , X 2 , ⋯ , X n X X X σ 2 \sigma^2 σ 2 1 n − 1 ∑ i = 1 n ( X i − μ ) 2 \frac{1}{n-1}\sum_{i=1}^{n}(X_i-\mu)^2 n − 1 1 ∑ i = 1 n ( X i − μ ) 2 μ \mu μ 1 n ∑ i = 1 n ( X i − μ ) 2 \frac{1}{n}\sum_{i=1}^{n}(X_i-\mu)^2 n 1 ∑ i = 1 n ( X i − μ ) 2 μ \mu μ 1 n − 1 ∑ i = 1 n ( X i − μ ) 2 \frac{1}{n-1}\sum_{i=1}^{n}(X_i-\mu)^2 n − 1 1 ∑ i = 1 n ( X i − μ ) 2 μ \mu μ 1 n ∑ i = 1 n ( X i − μ ) 2 \frac{1}{n}\sum_{i=1}^{n}(X_i-\mu)^2 n 1 ∑ i = 1 n ( X i − μ ) 2 μ \mu μ
设σ \sigma σ X X X X 1 , X 2 , ⋯ , X n X_1,X_2,\cdots,X_n X 1 , X 2 , ⋯ , X n X X X S S S σ \sigma σ
设X 1 X_1 X 1 X ∼ N ( 0 , σ 2 ) ( σ > 0 ) X \sim N(0,\sigma^2)(\sigma>0) X ∼ N ( 0 , σ 2 ) ( σ > 0 ) x 1 x_1 x 1 σ 2 \sigma^2 σ 2
设总体X X X X X X P P P θ 3 \theta^3 θ 3 3 θ 2 ( 1 − θ ) 3\theta^2(1-\theta) 3 θ 2 ( 1 − θ ) 3 θ ( 1 − θ ) 2 3\theta(1-\theta)^2 3 θ ( 1 − θ ) 2 ( 1 − θ ) 3 (1-\theta)^3 ( 1 − θ ) 3 0 < θ < 1 0<\theta<1 0 < θ < 1 X 1 , X 2 , ⋯ , X n X_1,X_2,\cdots,X_n X 1 , X 2 , ⋯ , X n X X X θ \theta θ θ \theta θ
设一批零件的长度服从正态分布N ( μ , σ 2 ) N(\mu,\sigma^2) N ( μ , σ 2 ) σ 2 \sigma^2 σ 2 μ \mu μ n n n x ‾ \overline{x} x μ \mu μ μ 0 \mu_0 μ 0 x ‾ > μ 0 − σ n z 0.10 \overline{x}>\mu_0-\frac{\sigma}{\sqrt{n}}z_{0.10} x > μ 0 − n σ z 0.10 x ‾ > μ 0 + σ n z 0.05 \overline{x}>\mu_0+\frac{\sigma}{\sqrt{n}}z_{0.05} x > μ 0 + n σ z 0.05 x ‾ > μ 0 + σ n z 0.10 \overline{x}>\mu_0+\frac{\sigma}{\sqrt{n}}z_{0.10} x > μ 0 + n σ z 0.10 x ‾ > μ 0 − σ n z 0.05 \overline{x}>\mu_0-\frac{\sigma}{\sqrt{n}}z_{0.05} x > μ 0 − n σ z 0.05
设X 1 , X 2 , ⋯ , X n X_1,X_2,\cdots,X_n X 1 , X 2 , ⋯ , X n X X X X ‾ \overline{X} X E ( X ) = θ E(X)=\theta E ( X ) = θ H 0 : θ = 0 ; H 1 : θ ≠ 0 H_0:\theta=0;H_1:\theta \neq 0 H 0 : θ = 0 ; H 1 : θ = 0 W 1 = { ∣ X ‾ ∣ > 1 } W_1=\{|\overline{X}|>1\} W 1 = { ∣ X ∣ > 1 } W 2 = { ∣ X ‾ ∣ > 2 } W_2=\{|\overline{X}|>2\} W 2 = { ∣ X ∣ > 2 } α 1 \alpha_1 α 1 α 2 \alpha_2 α 2 α 1 = α 2 \alpha_1=\alpha_2 α 1 = α 2 α 1 > α 2 \alpha_1>\alpha_2 α 1 > α 2 α 1 < α 2 \alpha_1<\alpha_2 α 1 < α 2 α 1 \alpha_1 α 1 α 2 \alpha_2 α 2
设X 1 , x 2 X_1,x_2 X 1 , x 2 N ( μ , 1 ) N(\mu,1) N ( μ , 1 ) H 0 : μ = 2 H_0:\mu=2 H 0 : μ = 2 H 1 : μ = 4 H_1:\mu=4 H 1 : μ = 4 W = { X ‾ > 3 } W=\{\overline{X}>3\} W = { X > 3 } X ‾ = 1 2 ∑ i = 1 2 X i \overline{X}=\frac{1}{2}\sum_{i=1}^{2}X_i X = 2 1 ∑ i = 1 2 X i α , β \alpha,\beta α , β α = β = 1 − Φ ( 2 ) \alpha=\beta=1-\Phi(\sqrt{2}) α = β = 1 − Φ ( 2 ) α = 1 − Φ ( 2 ) , β = Φ ( 2 ) \alpha=1-\Phi(\sqrt{2}),\beta=\Phi(\sqrt{2}) α = 1 − Φ ( 2 ) , β = Φ ( 2 ) α = Φ ( 2 ) , β = 1 − Φ ( 2 ) \alpha=\Phi(\sqrt{2}),\beta=1-\Phi(\sqrt{2}) α = Φ ( 2 ) , β = 1 − Φ ( 2 ) α = β = Φ ( 2 ) \alpha=\beta=\Phi(\sqrt{2}) α = β = Φ ( 2 )
设总体X ∼ ( 1 2 3 θ 2 2 θ ( 1 − θ ) ( 1 − θ ) 2 ) X \sim \begin{pmatrix}1&2&3\\\theta^2&2\theta(1-\theta)&(1-\theta)^2\end{pmatrix} X ∼ ( 1 θ 2 2 2 θ ( 1 − θ ) 3 ( 1 − θ ) 2 ) H 0 : θ = 0.1 ; H 1 : θ = 0.9 H_0:\theta=0.1;H_1:\theta=0.9 H 0 : θ = 0.1 ; H 1 : θ = 0.9 W W W { X 1 = 1 , X 2 = 1 , X 3 = 1 } \{X_1=1,X_2=1,X_3=1\} { X 1 = 1 , X 2 = 1 , X 3 = 1 }