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 P { X = k } = − θ k k ln ( 1 − θ ) P\{X=k\}=-\frac{\theta^k}{k\ln(1-\theta)} P { X = k } = − k l n ( 1 − θ ) θ k k = 1 , 2 , ⋯ k=1,2,\cdots k = 1 , 2 , ⋯ θ ( 0 < θ < 1 ) \theta(0<\theta<1) θ ( 0 < θ < 1 ) μ m = 1 n ∑ i = 1 n X i m \mu_m=\frac{1}{n}\sum_{i=1}^{n}X_i^m μ m = n 1 ∑ i = 1 n X i m m = 1 , 2 , 3 m=1,2,3 m = 1 , 2 , 3 θ \theta θ 1 + μ 1 μ 2 1+\frac{\mu_1}{\mu_2} 1 + μ 2 μ 1 1 − μ 1 μ 2 1-\frac{\mu_1}{\mu_2} 1 − μ 2 μ 1 1 + μ 2 μ 3 1+\frac{\mu_2}{\mu_3} 1 + μ 3 μ 2 1 − μ 2 μ 3 1-\frac{\mu_2}{\mu_3} 1 − μ 3 μ 2
设袋中红球数与黑球数之比为r r r n n n k k k r r r n k \frac{n}{k} k n n − k k \frac{n-k}{k} k n − k k n \frac{k}{n} n k k n − k \frac{k}{n-k} n − k k
设( 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 ) ( X , Y ) (X,Y) ( X , Y ) ( X , Y ) ∼ f ( x , y ) = { 1 2 θ 2 e − 2 x + y 2 θ , x > 0 , y > 0 0 , 其他 (X,Y) \sim f(x,y)=\begin{cases}\frac{1}{2\theta^2}e^{-\frac{2x+y}{2\theta}},&x>0,y>0\\0,&\text{其他}\end{cases} ( X , Y ) ∼ f ( x , y ) = { 2 θ 2 1 e − 2 θ 2 x + y , 0 , x > 0 , y > 0 其他 θ \theta θ 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 Y ‾ = 1 n ∑ j = 1 n Y j \overline{Y}=\frac{1}{n}\sum_{j=1}^{n}Y_j Y = n 1 ∑ j = 1 n Y j θ \theta θ θ ^ \hat{\theta} θ ^ D ( θ ^ ) D(\hat{\theta}) D ( θ ^ ) X ‾ + Y ‾ 2 , θ 2 4 n \overline{X}+\frac{\overline{Y}}{2},\frac{\theta^2}{4n} X + 2 Y , 4 n θ 2 X ‾ 2 + Y ‾ 4 , θ 2 2 n \frac{\overline{X}}{2}+\frac{\overline{Y}}{4},\frac{\theta^2}{2n} 2 X + 4 Y , 2 n θ 2 X ‾ + Y ‾ 2 , θ 2 2 n \overline{X}+\frac{\overline{Y}}{2},\frac{\theta^2}{2n} X + 2 Y , 2 n θ 2 X ‾ 2 + Y ‾ 4 , θ 2 4 n \frac{\overline{X}}{2}+\frac{\overline{Y}}{4},\frac{\theta^2}{4n} 2 X + 4 Y , 4 n θ 2
设总体X X X λ ( λ > 0 \lambda(\lambda>0 λ ( λ > 0 ) ) ) X 1 , X 2 , ⋯ , X n X_1,X_2,\cdots,X_n X 1 , X 2 , ⋯ , X n X X X P { X = 0 } P\{X=0\} P { X = 0 }
设总体X X X f ( x ) = { β x α 2 , 0 < x < α 0 , 其他 f(x)=\begin{cases}\frac{\beta x}{\alpha^2},&0<x<\alpha\\0,&\text{其他}\end{cases} f ( x ) = { α 2 β x , 0 , 0 < x < α 其他 α ( α > 1 ) \alpha(\alpha>1) α ( α > 1 ) X 1 , X 2 , ⋯ , X n X_1,X_2,\cdots,X_n X 1 , X 2 , ⋯ , X n X X X p = P { 0 < X < α } p=P\{0<X<\sqrt{\alpha}\} p = P { 0 < X < α } X ( 1 ) = min { X 1 , X 2 , ⋯ , X n } X_{(1)}=\min\{X_1,X_2,\cdots,X_n\} X ( 1 ) = min { X 1 , X 2 , ⋯ , X n } X ( n ) = max { X 1 , X 2 , ⋯ , X n } X_{(n)}=\max\{X_1,X_2,\cdots,X_n\} X ( n ) = max { X 1 , X 2 , ⋯ , X n } p p p p ^ \hat{p} p ^
设X 1 , X 2 , ⋯ , X n X_1,X_2,\cdots,X_n X 1 , X 2 , ⋯ , X n X ∼ N ( 1 , σ 2 ) X \sim N(1,\sigma^2) X ∼ N ( 1 , σ 2 ) σ > 0 \sigma>0 σ > 0 σ 2 \sigma^2 σ 2 σ ^ 2 \hat{\sigma}^2 σ ^ 2 D ( σ ^ 2 ) = D(\hat{\sigma}^2)= D ( σ ^ 2 ) =
设X X X f ( x ; θ ) = { 1 , θ − 1 2 ≤ x ≤ θ + 1 2 0 , 其他 f(x;\theta)=\begin{cases}1,&\theta-\frac{1}{2} \leq x \leq \theta+\frac{1}{2}\\0,&\text{其他}\end{cases} f ( x ; θ ) = { 1 , 0 , θ − 2 1 ≤ x ≤ θ + 2 1 其他 − ∞ < θ < + ∞ -\infty<\theta<+\infty − ∞ < θ < + ∞ X 1 , X 2 , ⋯ , X n X_1,X_2,\cdots,X_n X 1 , X 2 , ⋯ , X n X X X X ( 1 ) = min { X 1 , X 2 , ⋯ , X n } X_{(1)}=\min\{X_1,X_2,\cdots,X_n\} X ( 1 ) = min { X 1 , X 2 , ⋯ , X n } X ( n ) = max { X 1 , X 2 , ⋯ , X n } X_{(n)}=\max\{X_1,X_2,\cdots,X_n\} X ( n ) = max { X 1 , X 2 , ⋯ , X n } θ \theta θ θ ^ L \hat{\theta}_L θ ^ L
设总体X X X [ − θ , θ ] ( θ > 0 ) [-\theta,\theta](\theta>0) [ − θ , θ ] ( 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
设总体X X X ( 0 , 1 θ ] (0,\frac{1}{\theta}] ( 0 , θ 1 ] 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 θ ^ \hat{\theta} θ ^ θ ^ \hat{\theta} θ ^ P { θ < θ ^ ≤ θ + 1 } P\{\theta<\hat{\theta} \leq \theta+1\} P { θ < θ ^ ≤ θ + 1 }
设总体X ∼ U [ θ 0 , θ 0 + θ ] X \sim U[\theta_0,\theta_0+\theta] X ∼ U [ θ 0 , θ 0 + θ ] θ 0 \theta_0 θ 0 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 θ ^ 1 \hat{\theta}_1 θ ^ 1 E ( θ ^ 1 ) E(\hat{\theta}_1) E ( θ ^ 1 ) θ \theta θ θ ^ 2 \hat{\theta}_2 θ ^ 2 E ( θ ^ 2 ) E(\hat{\theta}_2) E ( θ ^ 2 )
设总体X X X ( 0 , θ ] (0,\theta] ( 0 , 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 X ( 1 ) = min { X 1 , X 2 , ⋯ , X n } X_{(1)}=\min\{X_1,X_2,\cdots,X_n\} X ( 1 ) = min { X 1 , X 2 , ⋯ , X n } X ( n ) = max { X 1 , X 2 , ⋯ , X n } X_{(n)}=\max\{X_1,X_2,\cdots,X_n\} X ( n ) = max { X 1 , X 2 , ⋯ , X n } θ \theta θ θ ^ \hat{\theta} θ ^ Z = θ ^ θ Z=\frac{\hat{\theta}}{\theta} Z = θ θ ^ P { θ ^ < θ < θ 0 } = 1 − α P\{\hat{\theta}<\theta<\theta_0\}=1-\alpha P { θ ^ < θ < θ 0 } = 1 − α 0 < α < 1 0<\alpha<1 0 < α < 1 θ 0 \theta_0 θ 0
设总体X ∼ f ( x ; θ ) = { x θ e − x 2 2 θ , x > 0 0 , x ≤ 0 X \sim f(x;\theta)=\begin{cases}\frac{x}{\theta}e^{-\frac{x^2}{2\theta}},&x>0\\0,&x \leq 0\end{cases} X ∼ f ( x ; θ ) = { θ x e − 2 θ x 2 , 0 , x > 0 x ≤ 0 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 ^ M \hat{\theta}_M θ ^ M θ ^ L \hat{\theta}_L θ ^ L θ \theta θ θ ^ M \hat{\theta}_M θ ^ M θ ^ L \hat{\theta}_L θ ^ L
设总体X X X f ( x ; b ) = { a e − 2 ( x − b ) , x ≥ b 0 , x < b f(x;b)=\begin{cases}ae^{-2(x-b)},&x \geq b\\0,&x f ( x ; b ) = { a e − 2 ( x − b ) , 0 , x ≥ b x < b b > 0 b>0 b > 0 X 1 , X 2 , ⋯ , X n X_1,X_2,\cdots,X_n X 1 , X 2 , ⋯ , X n X X X a a a b b b b ^ M \hat{b}_M b ^ M b ^ L \hat{b}_L b ^ L b ^ L \hat{b}_L b ^ L b b b
设X 1 , X 2 X_1,X_2 X 1 , X 2 X ∼ U [ 0 , 2 θ ] X \sim U[0,2\theta] X ∼ U [ 0 , 2 θ ] Y 1 , Y 2 , Y 3 Y_1,Y_2,Y_3 Y 1 , Y 2 , Y 3 Y ∼ U [ 0 , 4 θ ] Y \sim U[0,4\theta] Y ∼ U [ 0 , 4 θ ] θ ( θ > 0 ) \theta(\theta>0) θ ( θ > 0 ) X 1 , X 2 , Y 1 , Y 2 , Y 3 X_1,X_2,Y_1,Y_2,Y_3 X 1 , X 2 , Y 1 , Y 2 , Y 3 θ \theta θ θ ^ \hat{\theta} θ ^ D ( θ ^ ) D(\hat{\theta}) D ( θ ^ )
设X X X 2 θ \frac{2}{\theta} θ 2 X = x ( x > 0 ) X=x(x>0) X = x ( x > 0 ) Y Y Y f Y ∣ X ( y ∣ x ) = { 1 θ e − y − x θ , 0 < x < y 0 , 其他 f_{Y|X}(y|x)=\begin{cases}\frac{1}{\theta}e^{-\frac{y-x}{\theta}},&0<x<y\\0,&\text{其他}\end{cases} f Y ∣ X ( y ∣ x ) = { θ 1 e − θ y − x , 0 , 0 < x < y 其他 θ \theta θ ( 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 ) ( X , Y ) (X,Y) ( X , Y ) θ \theta θ θ ^ \hat{\theta} θ ^ D ( θ ^ ) D(\hat{\theta}) D ( θ ^ )
设总体X X X f ( x ; α , β ) = { α β α x α + 1 , x ≥ β 0 , x < β f(x;\alpha,\beta)=\begin{cases}\frac{\alpha\beta^{\alpha}}{x^{\alpha+1}},&x \geq \beta\\0,&x<\beta\end{cases} f ( x ; α , β ) = { x α + 1 α β α , 0 , x ≥ β x < β α , β \alpha,\beta α , β X 1 , X 2 , ⋯ , X n X_1,X_2,\cdots,X_n X 1 , X 2 , ⋯ , X n X X X α , β \alpha,\beta α , β α ^ , β ^ \hat{\alpha},\hat{\beta} α ^ , β ^ ε > 0 \varepsilon>0 ε > 0 a a a lim n → ∞ P { ∣ β ^ − a ∣ ≥ ε } = 0 \lim\limits_{n \to \infty}P\{|\hat{\beta}-a| \geq \varepsilon\}=0 n → ∞ lim P { ∣ β ^ − a ∣ ≥ ε } = 0 E ( ln X 1 ) E(\ln X_1) E ( ln X 1 )
设总体X X X X X X P P P 1 − p 1-p 1 − p p p p 0 < p < 1 0<p<1 0 < p < 1 X 1 , X 2 , ⋯ , X n X_1,X_2,\cdots,X_n X 1 , X 2 , ⋯ , X n X X X x 1 , x 2 , ⋯ , x n x_1,x_2,\cdots,x_n x 1 , x 2 , ⋯ , x n p p p
设总体X X X F ( x ) = { 0 , x < 1 θ , 1 ≤ x < 3 2 1 , x ≥ 3 2 F(x)=\begin{cases}0,&x<1\\\theta,&1 \leq x<\frac{3}{2}\\1,&x \geq \frac{3}{2}\end{cases} F ( x ) = ⎩ ⎨ ⎧ 0 , θ , 1 , x < 1 1 ≤ x < 2 3 x ≥ 2 3 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 ^ M \hat{\theta}_M θ ^ M n n n n 1 n_1 n 1 n 2 n_2 n 2 θ \theta θ θ ^ L \hat{\theta}_L θ ^ L
设总体X ∼ ( 0 1 2 θ 4 N θ 2 N 4 N − 3 θ 4 N ) X \sim \begin{pmatrix}0&1&2\\\frac{\theta}{4N}&\frac{\theta}{2N}&\frac{4N-3\theta}{4N}\end{pmatrix} X ∼ ( 0 4 N θ 1 2 N θ 2 4 N 4 N − 3 θ ) N > 0 N>0 N > 0 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 n 0 n_0 n 0 n 1 n_1 n 1 n 2 n_2 n 2 n 0 + n 1 + n 2 = n n_0+n_1+n_2=n n 0 + n 1 + n 2 = n θ \theta θ θ ^ 1 \hat{\theta}_1 θ ^ 1 θ ^ 2 \hat{\theta}_2 θ ^ 2 θ ^ 1 \hat{\theta}_1 θ ^ 1 θ ^ 2 \hat{\theta}_2 θ ^ 2 θ ^ 1 \hat{\theta}_1 θ ^ 1 θ ^ 2 \hat{\theta}_2 θ ^ 2
设总体X X X 1 − θ 1-\theta 1 − θ θ − θ 2 \theta-\theta^2 θ − θ 2 θ 2 \theta^2 θ 2 θ ∈ ( 0 , 1 ) \theta \in(0,1) θ ∈ ( 0 , 1 ) N i N_i N i X X X n n n i i i a 1 , a 2 , a 3 a_1,a_2,a_3 a 1 , a 2 , a 3 T = ∑ i = 1 3 a i N i T=\sum_{i=1}^{3}a_iN_i T = ∑ i = 1 3 a i N i θ \theta θ T T T
对总体X X X k k k X = k X=k X = k X X X a a a b b b X X X
设总体X X X λ ( λ > 0 ) \lambda(\lambda>0) λ ( λ > 0 ) X 1 X_1 X 1 x 1 = 3 x_1=3 x 1 = 3 e − 2 λ e^{-2\lambda} e − 2 λ e − 2 X 1 , e − 6 e^{-2X_1},e^{-6} e − 2 X 1 , e − 6 e − X 1 , e − 3 e^{-X_1},e^{-3} e − X 1 , e − 3 ( − 1 ) X 1 , − 1 (-1)^{X_1},-1 ( − 1 ) X 1 , − 1
设总体X X X θ \theta θ θ ^ 1 \hat{\theta}_1 θ ^ 1 θ ^ 2 \hat{\theta}_2 θ ^ 2 D ( θ ^ 2 ) = 2 D ( θ ^ 1 ) D(\hat{\theta}_2)=2D(\hat{\theta}_1) D ( θ ^ 2 ) = 2 D ( θ ^ 1 ) θ ^ = a θ ^ 1 + b θ ^ 2 \hat{\theta}=a\hat{\theta}_1+b\hat{\theta}_2 θ ^ = a θ ^ 1 + b θ ^ 2 θ ^ \hat{\theta} θ ^ a = 1 3 , b = 1 3 a=\frac{1}{3},b=\frac{1}{3} a = 3 1 , b = 3 1 a = 2 3 , b = 1 3 a=\frac{2}{3},b=\frac{1}{3} a = 3 2 , b = 3 1 a = 1 3 , b = 2 3 a=\frac{1}{3},b=\frac{2}{3} a = 3 1 , b = 3 2 a = 2 3 , b = 2 3 a=\frac{2}{3},b=\frac{2}{3} a = 3 2 , b = 3 2
设总体X X X X X X a a a Φ ( 1.96 ) = 0.975 \Phi(1.96)=0.975 Φ ( 1.96 ) = 0.975 X X X ( a − 0.49 , a + 0.49 ) (a-0.49,a+0.49) ( a − 0.49 , a + 0.49 ) ( a − 0.327 , a + 0.327 ) (a-0.327,a+0.327) ( a − 0.327 , a + 0.327 ) ( a − 0.196 , a + 0.196 ) (a-0.196,a+0.196) ( a − 0.196 , a + 0.196 ) ( a − 0.025 , a + 0.025 ) (a-0.025,a+0.025) ( a − 0.025 , a + 0.025 )
设总体X ∼ N ( μ , 2 2 ) X \sim N(\mu,2^2) X ∼ N ( μ , 2 2 ) μ \mu μ X 1 , X 2 , ⋯ , X 9 X_1,X_2,\cdots,X_9 X 1 , X 2 , ⋯ , X 9 X X X μ \mu μ L L L L L L E ( L ) = E(L)= E ( L ) = 2 3 z 0.025 \frac{2}{3}z_{0.025} 3 2 z 0.025 4 3 z 0.025 \frac{4}{3}z_{0.025} 3 4 z 0.025 2 3 z 0.05 \frac{2}{3}z_{0.05} 3 2 z 0.05 4 3 z 0.05 \frac{4}{3}z_{0.05} 3 4 z 0.05