V (X) = 500 W = (X 1 , X 2 , , X n ) X 1 , X 2 , , X 100 E(X i ) = E(X) = 1065, V (X i ) = V (X) = 500 2 , ∀i = 1, 100 X = 1 100 (X 1 + X 2 + + X 100 ) E(X) = 1 100 (E(X 1 ) + E(X 2 ) + + E(X 100 )) = 1 100 .100.1065 = 1065 V ( X) = 1 100 2 (V (X 1 ) + V (X 1 ) + + V (X 100 )) = 1 100 2 .100.500 2 = 2500 W = (X 1 , X 2 , , X n ) X i V (X) = V X 1 + + X n n = 1 n 2 [V (X 1 ) + + V (X n )] = 1 n 2 nV (X) = V (X) n = 1000000 n Se(X) 25 ⇔ V (X) 25 ⇔ 1000000 n 25 ⇔ n 1000000 25 2 = 1600 Se(f) = V (f) = p(1 − p) √ n = √ 0, 5.0, 5 √ 120 = 0, 5 10, 9445 = 0, 0456 W = (X 1 , X 2 , X 3 , X 4 , X 5 ) X x = x 1 + + x 5 5 = 10 + 12 + 16 + 18 + 19 5 = 15 f = 3 50 = 0, 06 ˆ θ ∧θ BS( ˆ θ) = 1/n, BS(∧θ) = 0, 01 BS( ˆ θ) < BS(∧θ) ⇔ 1 n < 0, 01 ⇔ n > 100 V (X) = σ 2 X d V (X d ) = 1 4n[f(x d )] 2 = 1 4.500[f(x d )] 2 = 1 2000[f(x d )] 2 X ′ V (X ′ ) = V (X) 320 = σ 2 320 EF = V ( X ′ ) V (X d ) = σ 2 320 1 2000[f (x d )] 2 = 2000 320 [f(x d )] 2 σ 2 = 6, 25[f(x d )] 2 σ 2 = 6, 25[f(µ)] 2 σ 2 = 6, 25 1 σ 2 .2π σ 2 = 6, 25 2π < 1 X ∼ N(µ, σ 2 ) W = (X 1 , X 2 , , X n ) X d V (X d ) = 1 4n[f(x d )] 2 = 2π 4 V ( X) X V (X) X d X EF = V ( X) V (X d ) = V ( X) 2π 4 V ( X) = 4 2π ≈ 0, 64 E(X) = 8. 1 4 + 10. 1 4 + 11. 1 2 = 10 E(Y ) = 4. 1 2 + 6. 1 2 = 5 E(Z) = 32 1 8 + 40 1 8 + 44 1 4 + 48 1 8 + 60 1 8 + 66 1 8 = 50 X ∼ A(p) W 1 = (X 11 , X 12 , X 13 ) n 1 = 3 W 2 = (X 21 , X 22 , X 23 , X 24 ) n 2 = 4 f 1 , f 2 E(f 1 ) = E(f 2 ) = p V (f 1 ) = p(1 − p) n 1 = p(1 − p) 3 ; V (f 2 ) = p(1 − p) n 2 = p(1 − p) 4 V (f 2 ) < V (f 1 ) f 2 f 1 f 2 f 1 EF = V (f 1 ) V (f 2 ) = p(1−p) 3 p(1−p) 4 = 4 3 ≈ 1, 333 W 1 W 2 θ = αf 1 + (1 − α)f 2 V (θ) = V (αf 1 + (1 − α)f 2 ) = α 2 V (f 1 ) + (1 − α) 2 V (f 2 ) = α 2 p(1 − p) 3 + (1 − α) 2 p(1 − p) 4 = (7α 2 − 6α + 3) p(1 − p) 12 g(α) = 7α 2 − 6α + 3 g(α) g(α) α = 6/14 = 3/7 V (θ) 3 7 f 1 + 4 7 f 2 σ(V ) = 3σ(U) := 3. σ (i) W 1 = 1 2 U + 1 2 V ⇒ E(W 1 ) = 1 2 E(U) + 1 2 E(V ) = X (ii) W 2 = 3 4 U + 1 4 V ⇒ E(W 2 ) = 3 4 E(U) + 1 4 E(V ) = X (iii) W 3 = 1.U + 0.V ⇒ E(W 3 ) = E(U) = X W 1 , W 2 , W 3 W 1 = 1 2 U + 1 2 V V (W 1 ) = 1 4 V (U) + 1 4 V (V ) = σ 2 4 + 9σ 2 4 = 10 4 σ 2 W 2 = 3 4 U + 1 4 V V (W 2 ) = 9 16 V (U) + 1 16 V (V ) = 9σ 2 16 + 9σ 2 16 = 18 16 σ 2 W 3 = 1.U + 0.V V (W 3 ) = V (U) = σ 2 V (W 3 ) < V (W 2 ) < V (W 1 ) W 3 µ X x = 17 + 28 + 92 + 41 4 = 44, 5 f = 8 20 = 0, 4 X ∼ P (λ) f(x i , λ) = e −λ λ x i x i ! , i = 1, 3 a. L(x 1 , x 2 , x 3 , λ) = f(x 1 , λ)f(x 2 , λ)f(x 3 , λ) = e −3λ λ x 1 +x 2 +x 3 x 1 !x 2 !x 3 ! x 1 = 15, x 2 = 8, x 3 = 13 L(λ) = e −3λ λ 15+8+13 15!8!13! = e −3λ λ 36 15!8!13! b. L(λ = 5) = e −15 5 36 15!8!13! ≈ 1, 356.10 −8 L(λ = 10) = e −30 10 36 15!8!13! ≈ 2, 8501.10 −4 L(λ = 12) = e −36 12 36 15!8!13! = 5, 0075.10 −4 L(λ = 15) = e −45 15 36 15!8!13! ≈ 1, 9043.10 −4 L(λ = 20) = e −60 20 36 15!8!13! ≈ 1, 8328.10 −6 L(λ = 25) = e −75 25 36 15!8!13! ≈ 1, 728.10 −9 λ = 12 = x 1 +x 2 +x 3 3 L(x 1 , x 2 , , x n , λ) = e −nλ λ n i=1 x i x 1 !x 2 ! x n ! ln L(x 1 , x 2 , , x n , λ) = −nλ + (ln λ) n i=1 x i − ln(x 1 !x 2 ! x n !) ∂ ln L(x 1 , x 2 , , x n , λ) ∂λ = −n + 1 λ n i=1 x i ∂ ln L(x 1 , x 2 , , x n , λ) ∂λ = 0 ⇔ λ = 1 n n i=1 x i = x ∂ 2 ln L(x 1 , x 2 , , x n , λ) ∂λ 2 = − 1 λ 2 n i=1 x i ⇒ ∂ 2 ln L(x 1 , x 2 , , x n , λ) ∂λ 2 λ= x = − n x x 2 < 0 x λ X ∼ E(λ) W = (X 1 , , X 5 ) f(x i , λ) = λe −λx i x i 0 i = 1, 5 0 x i < 0 x i < 0 x i 0, ∀i L(x 1 , x 2 , x 3 , x 4 , x 5 , λ) = 5 i=1 f(x i , λ) = λ 5 e −λ 5 i=1 x i x 1 = 1, 2; x 2 = 7, 5; x 3 = 1, 8; x 4 = 3, 7; x 5 = 0, 8 L(λ) = λ 5 e −15λ • λ = 0, 1 ⇒ L(λ = 0, 1) = 0, 1 5 e −15.0,1 = 0, 1 5 e −1,5 ≈ 2, 2313.10 −6 • λ = 0, 2 ⇒ L(λ = 0, 2) = 0, 2 5 .e −15.0,2 = 0, 2 5 e −3 ≈ 15, 9319.10 −6 • λ = 0, 3 ⇒ L(λ = 0, 3) = 0, 3 5 e −15.0,3 = 0, 3 5 e −4,5 ≈ 26, 9947.10 −6 • λ = 0, 4 ⇒ L(λ = 0, 4) = 0, 4 5 e −15.0,4 = 0, 4 5 .e −6 ≈ 25, 3824.10 −6 • λ = 0, 5 ⇒ L(λ = 0, 5) = 0, 5 5 .e −15.0,5 = 0, 5 5 e −7,5 ≈ 17, 2839.10 −6 L(λ = 0, 3) L(x 1 , , x n , λ) = n i=1 f(x i , λ) = λ n e −λ n i=1 x i ln L(x 1 , , x n , λ) = n ln λ − λ n i=1 x i ∂ ln L(x 1 , , x n , λ) ∂λ = n λ − n i=1 x i = n λ − n x ∂ ln L(x 1 , , x n , λ) ∂λ = 0 ⇔ n λ − n x = 0 ⇔ λ = 1 x ∂ 2 ln L(x 1 , , x n , λ) ∂λ 2 = − n λ 2 < 0, ∀λ λ 1/x L(λ = 0, 3) L(λ) λ = 1 x = 5 1,2+7,5+1,8+3,7+0,8 = 1 3 L 1 3 = 1 3 5 e −15.(1/3) = 1 3 5 e −5 ≈ 27, 7282.10 −6 X ∼ B(n, p) W = (X 1 , , X m ) X i X i ∼ A(p), ∀i = 1, n P (X i = x i ) = f(x i , p) = p x i (1 − p) 1−x i ; x i = 0, 1. X = m i=1 X i , f = X m f = 0, 6 ⇒ x = f.m = 0, 6. 5 = 3 L(x 1 , , x 5 , p) = 5 i=1 f(x i , p) = 5 i=1 p x i (1 − p) 1−x i = p 5 i=1 x i (1 − p) 5− 5 i=1 x i = p x (1 − p) 5−x = p 3 (1 − p) 5−3 = p 3 (1 − p) 2 L(p) = p 3 (1 − p) 2 L(0) = 0 3 .1 2 = 0 L(0, 1) = 0, 1 3 .0, 9 2 = 81.10 −5 L(0, 2) = 0, 2 3 .0, 8 2 = 512.10 −5 L(0, 3) = 0, 3 3 .0, 7 2 = 1323.10 −5 L(0, 4) = 0, 4 3 .0, 6 2 = 2304.10 −5 L(0, 5) = 0, 5 3 .0, 5 2 = 3125.10 −5 L(0, 6) = 0, 6 3 .0, 4 2 = 3456.10 −5 L(0, 7) = 0, 7 3 .0, 3 2 = 3087.10 −5 L(0, 8) = 0, 8 3 .0, 2 2 = 2048.10 −5 L(0, 9) = 0, 9 3 .0, 1 2 = 729.10 −5 L(1) = 1 3 .0 2 = 0 L(x 1 , , x m , p) = m i=1 f(x i , p) = m i=1 p x i (1 − p) 1−x i = p m i=1 x i (1 − p) m− m i=1 x i = p x (1 − p) m−x ln L(x 1 , , x m , p) = ln p x (1 − p) m−x = x ln p + (m − x) ln(1 − p) ∂ ln L(x 1 , , x m , p) ∂p = x 1 p + (m − x) −1 1 − p = x − mp p(1 − p) ∂ ln L(x 1 , , x m , p) ∂p = 0 ⇔ x − mp) p(1 − p) = 0 ⇔ p = x m = f ∂ 2 ln L(x 1 , , x m , p) ∂p 2 = 2xp − x − mp 2 [p(1 − p)] 2 ∂ 2 ln L(x 1 , , x m , p) ∂p 2 p=x/m = 2 x 2 m − x − x 2 m [p(1 − p)] 2 = x( x m − 1) [p(1 − p)] 2 < 0 x m = f < 1 X ∼ N(µ, σ 2 = 2 2 ) µ σ 2 X − σ √ n u α/2 ; X + σ √ n u α/2 x = 8, 5; σ = 2; n = 600; α = 0, 05 ⇒ u α/2 = u 0,025 = 1, 96 8, 5 − 2 √ 600 · 1, 96; 8, 5 + 2 √ 600 · 1, 96 = (8, 34; 8, 66) X ∼ N(µ, σ 2 ) µ σ 2 n 30 X − S √ n · t (n−1) α/2 ; X + S √ n · t (n−1) α/2 n = 30; α = 0, 05 ⇒ t (n−1) α/2 = t (29) 0,025 = 2, 045 x = 304 30 = 10, 1333 s 2 = 1 29 (3082, 14 − (10, 1333) 2 .30) = 0, 0554 ⇒ s = 0, 0554 = 0, 235 10.1333 − 0, 235 √ 30 · 2, 045; 10.1333 + 0, 235 √ 30 · 2, 045 = (10, 045; 10, 221) µ σ 2 X − S √ n · t (n−1) α/2 ; X + S √ n · t (n−1) α/2 n = 25; α = 0, 05 ⇒ t (n−1) α/2 = t (24) 0,025 = 2, 064 x = 538 25 = 21, 52 s 2 = 1 24 (11716 −21, 52 2 .25) = 5, 76 s = 5, 76 = 2, 4 21, 52 − 2, 4 √ 25 · 2, 064; 21, 52 + 2, 4 √ 25 · 2, 064 = (20, 53; 22, 51) µ σ 2 X − S √ n · t (n−1) α/2 ; X + S √ n · t (n−1) α/2 n = 25; α = 0, 05 ⇒ t (n−1) α/2 = t (24) 0,025 = 2, 064 x = 962, 5 25 = 38, 5 s 2 = 1 24 (37706, 25 −38, 5 2 .25) = 27, 0833 s = 27, 0833 = 5, 204 38, 5 − 5, 204 √ 25 · 2, 064; 38, 5 + 5, 204 √ 25 · 2, 064 = (36, 352; 40, 648) µ σ 2 X − S √ n · t (n−1) α/2 ; X + S √ n · t (n−1) α/2 n = 25; α = 0, 05 ⇒ t (24) 0,025 = 2, 064; x = 40; s = 5 40 − 5 √ 25 · 2, 064; 40 + 5 √ 25 · 2, 064 = (37, 936; 42, 064) µ σ 2 X − S √ n · t (n−1) α/2 ; X + S √ n · t (n−1) α/2 n = 25; α = 0, 05 ⇒ t (n−1) α/2 = t (4) 0,025 = 2, 776 x = 1 5 5 i=1 x i = 2, 018 s 2 = 1 4 5 i=1 x 2 i − n. x 2 = 2.10 −5 ⇒ s = 0, 0045 2, 018 − 0, 0045 √ 5 · 2, 776; 2, 018 − 0, 0045 √ 5 · 2, 776 = (2, 0124; 2, 0236) [...]... 916, 5; 1 x1 = 463; s1 = 30, 274 (4) (n 1) 1 1 = 0, 95 t/2 = t0,025 = 2, 77 6 30, 274 30, 274 KQ 463 2, 77 6; 463 + 2, 77 6 = (425, 416; 500, 584) 5 5 425, 416 < à1 < 500, 584 ẻ ẻ ừ ỉề í ỉ ẹ n2 = 5; x2 = 4 67; s2 = 839, 5; 2 s2 = 28, 974 (4) (n1) 1 = 0, 95 t/2 = t0,025 = 2, 77 6 28, 974 28, 974 KQ 4 67 2, 77 6; 4 67 + 2, 77 6 = (431, 03; 512, 97) 5 5 431, 03 < à1 < 512, 97 ẻ à ặ ỉ ểũề ỉề í ỉ é... i=1 1 ã 1 175 , 5 = 11, 75 5 100 1 (13820, 05 100.11, 75 52) = 0, 02 07 99 s2 = s= 0, 02 07 = 0, 144 ỉ ếũ 0, 144 0, 144 ã 1, 96; 11, 75 5 + ã 1, 96 = (11, 72 7; 11, 78 3) 11, 75 5 100 100 ủ ắằẵ é à ề ỉ 2 ểũề ỉ ề í éủ S S (n1) (n1) X ã t/2 ; X + ã t/2 n n ỉ ủ ỉ n = 100; = 0, 05 ẹ ỉ x= s2 = (n1) t/2 (99) = t0,025 u0,025 = 1, 96 1 ã 9 072 = 90, 72 100 1 (82 473 2 100.90, 72 2) = 17, 375 4 99 s... n1 n2 c= k= s2 1 n1 s2 1 n1 + s2 2 n2 28,6381 15 28,6381 + 25,26 67 15 16 = 0, 5313 (n1 1)(n2 1) 14.15 = = 28, 73 (n2 1)c2 + (n1 1)(1 c)2 15.0, 53132 + 14.0, 46 872 k = 29; Kq = 28, 6381 25, 26 67 + = 7, 342 15 16 /2 = 0, 025 (29) (k) t/2 = t0,025 = 2, 045 (91, 75 84, 73 33) 7, 342.2, 045; (91, 75 84, 73 33) + 7, 342.2, 045 = (7, 9 977 ; 22, 0311) ủ à ằẵ ễ éủ ỉ é ẹủề ề f f (1 f ) ã u/2; f + n é... 3363 n1 n2 c= k= s2 1 n1 s2 1 n1 + s2 2 n2 = 0,45 47 6 0,45 47 + 0,2984 6 8 = 0, 67 (n1 1)(n2 1) 5 .7 = = 79 , 5 (n2 1)c2 + (n1 1)(1 c)2 7. 0. 672 + 5.0.332 k = 10; ểũề ỉ ề í (x y) 1 = 0, 95 ĩề à1 à2 (k) (10) t/2 = t0,025 = 2, 228 ỉ ề í éủ s2 s2 (k) 1 + 2 t/2 ; (x y) + n1 n2 s2 s2 (k) 1 + 2 t/2 n1 n2 = (4, 76 67 5, 2125) 0, 3363.2, 228; (4, 76 67 5, 2125) 0, 3363.2, 228 = (1, 195; 0, 303) ủ... 1, 092 = 1, 1 071 ; 2 ỉ 2 X1 N (à1, 1 ) é ề éủ 2 2 S1 S2 (k) + t/2 n1 n2 x1 = 7, 66 n2 = 64; ề ề x2 = 7, 32 s2 s2 1 + 2 = n1 n2 c= k= s2 1 n1 s2 1 n1 + s2 2 n2 = 0, 525 67. 63 (n1 1)(n2 1) = = 129, 95 (n2 1)c2 + (n1 1)(1 c)2 63.0, 5252 + 67. 0, 475 2 k = 130; KQ 1, 2996 1, 1 071 + = 0, 191 68 64 1 = 0, 95 (k) (130) t/2 = t0,025 u0,025 = 1, 96 (7, 66 7, 32) 0, 191.1, 96; (7, 66 7, 32) + 0, 191.1,... 4.24, 7 4.24, 7 ; 9, 488 0, 71 07 = (10, 413; 139, 018) ằẵ éủ é ủ 2(n1) 1/2 = 0,95 = 0, 71 07 3, 23 < < 11, 79 ủ s2 = 24, 7 /2 = 0, 05; ề é ì ề ỉ ề í ẹ 2 2 ỉ ểề ỉệểề ẹ ỉ ỳỉ ì à í 1 éủ ỉ ỉ ề (n 1)S 2 (n 1)S 2 ; 2(n1) 2(n1) /2 1/2 n = 15; 2(n1) /2 1 = 0, 95 2(14) = 0,025 = 26, 12; (n 1)S 2 (n 1)S 2 ; 2(n1) 2(n1) /2 1/2 = 0, 05; 2(n1) s2 = 46 078 53, 4 2(14) 1/2 = 0, 975 = 5, 629 = 14.46 078 53,... 77 , 3) 0, 543.1, 96; (68, 1 77 , 3) 0, 543.1, 96 = (10, 263; 8, 1 37) ẻ í ủ ỉề í ỉ ằẵ 8, 1 37 < à2 à1 < 10, 263 ẵ á éủ éú ì ỉ ề ề ủề ề ễ ụỉ ỉệ ề ủ ề ề ễ ụỉ ỉệ ề 2 X N (à1 , 1 ); n1 = 7; 2 Y N (à2 , 2 ); s2 s2 1 + 2 = n1 n2 c= k= s1 1 n1 s2 1 n1 + s2 2 n2 = x = 17, 5; n2 = 11; y = 15, 3; s1 = 3, 2 s2 = 2, 9 3, 22 2, 92 + = 1, 493 7 11 3,22 7 3,22 7 + 2,92 11 = 0, 6 57 6.10 (n1 1)(n2 1) = = 11,... 1)c2 + (n1 1)(1 c)2 10.0, 6 572 + 6.0, 3432 k = 12; 1 = 0, 95 (k) (12) t/2 = t0,025 = 2, 179 ẵ ểũề ỉ ề í ĩề à1 à2 ỉ ề í éủ s2 (k) s2 1 + 2 t/2; (x y) + n1 n2 (x y) s2 (k) s2 1 + 2 t/2 n1 n2 = ( 17, 5 15, 3) 1, 493.2, 179 ; ( 17, 5 15, 3) + 1, 493.2, 179 = (1, 053; 5, 453) ủ ằẵ ẵ á éủ ẹ éừẹ ễ ụỉ ể ụ ề ủ ề ỉ ắ 2 X N (à1 , 1 ); 14, 3 4, 76 67; s2 = 0, 45 47 1 3 41, 7 y= = 5, 2125; s2 = 0, 2984... /2 (X 1 X 2 ) è 2 2 S1 S2 (k) + t n1 n2 /2 x1 = 78 1 n1 1 x2 = 99 1 s2 = 2 n2 1 (x1i x1)2 = 1 ã 1805 = 275 , 875 7 (x2i x2)2 = 1 ã 11520 = 256 45 1 = 0, 95 s2 = 1 s2 s2 1 + 2 = n1 n2 c= k s2 1 n1 s2 1 n1 s2 2 n1 = /2 = 0, 025 2 57, 875 8 2 57, 875 + 256 8 46 = 0, 8528 (n1 1)(n2 1) = 9, 58 10 (n2 1)c2 + (n1 1)(1 c)2 (k) ểũề ỉ ề í + 2 57, 875 256 + 6, 148 8 46 (10) t/2 = t0,025 = 2, 228 ĩề... 191.1, 96 = (0, 034; 0, 71 4) ủ ắằẵ ễ éủ ỉ à é ủ ỉểụề ễ éủ f ỉ éừề é ề ề ụ ỉệ ỉ ỉ ễ ẻ f (1f ) u/2 n ắ ỉ ề í (1 ) ỉ ụ ỉệ ỉ ỉ è ủ n = 170 ; p>f 12 0, 070 6; u = u0,05 = 1, 64 170 f (1 f ) 0, 070 6.0, 9294 ã 1, 64 = 0, 0384 u/2 = 0, 070 6 n 170 f= à ặ ẵẳẳẳ ẻ ỉ ề í ỉ ỉ é ỉ éừề p . = 1 n k i=1 n i x i = 1 100 · 1 175 , 5 = 11, 75 5 s 2 = 1 99 (13820, 05 −100.11, 75 5 2 ) = 0, 02 07 ⇒ s = 0, 02 07 = 0, 144 11, 75 5 − 0, 144 √ 100 · 1, 96; 11, 75 5 + 0, 144 √ 100 · 1, 96 = (11, 72 7; 11, 78 3) µ σ 2 X. u 0,025 = 1, 96 x = 1 100 · 9 072 = 90, 72 s 2 = 1 99 (82 473 2 −100.90, 72 2 ) = 17, 375 4 ⇒ s = 4, 168 90, 72 − 4, 168 10 · 1, 96; 90, 72 + 4, 168 10 · 1, 96 = (89, 903; 91, 5 37) µ σ 2 x = 3 2 X − σ √ n u α/2 ; X. t (n−1) α/2 = t (4) 0,025 = 2, 77 6 µ D = µ 2 − µ 1 d − s √ n t (n−1) α/2 ; d + s √ n t (n−1) α/2 = 1, 8 − 4, 025 √ 5 · 2, 77 6; 1, 8 + 4, 025 √ 5 · 2, 77 6 = (−3, 1 97; 6, 79 7) X 1 , X 2 µ 1 − µ 2 σ 1 ,