[ ] =. =3.5 3 =.3 =. =0.30 : (f i ) u i u i f i u i f i
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1 [ ] ( x s, x + s) 5% x s = 3, x + s = 7 s = = 70 5 = 0 a = 3, b = = = % 5 A = 56.5, = 60 6 x = 70.7, s = 5. 7 :
2 [ ] =. =3.5 3 =.3 =. =0.30 : (f i ) u i u i f i u i f i ū = = 0.0 x = = { } 6.0 s u = 589 ( ) =. 0 0 s x = 7. = < < ( x s, x + s) = (7.39, 8.7) % ( x s, x + s) = (.85, 6.0) % ( x 3s, x + 3s) = ( 3.69, 79.55) 0 00%
3 [ ] 3 ( x s, x + s).. 3 ( x s, x + s) 3 3: () ( ) ( ) ( ) 0 = =3 3 == =0.3 x = 3.95 s = 5.56 =0.7 ( x s, x + s) = (7.39, 8.5) 78 ( x s, x + s) = (.83, 6.07) 8 ( x 3s, x + 3s) = ( 3.73, 79.63)
4 [ ] 8 : =3.8 =.5 3 =55.9 =. =0.8 5 ū = 36 = 0.36 x = = ( ) 3.0 s u = = s x =.83 = < < 5 ū = = 0. x = =.3 98 ( ) 3.0 s u = =.53 s x =.53 =
5 [ ] 5 5: (f i ) u i u i f i u i f i ( x s, x + s) = (7.8, 6.) ( x s, x + s) = (0.65, 77.97) ( x 3s, x + 3s) = ( 6.8, 9.80) x s % 6.5%
6 [ ] ( x s, x + s) = (5.65, 65.93) : 79 (7.6, 60.8) : 66 ( x s, x + s) = (5.5, 86.07) : 96 (0.77, 77.53) : 9 ( x 3s, x + 3s) = (.63, 06.) : 98 ( 5.9, 9.) : B A B A a: 9% b: 8% c: 78% d: % e: 83% f: 7% ψ = 3. Q = 0.5
7 分 ) [ ] 7 7 i 0 5(.8) 0(0.) (0.) 8(7.8) iii ψ =.05, Q = (35.6) 8(7.) (.) 6(6.6) ψ = 0.897, Q = 0.05 ii 0 35(35.8) 5(.) (7.) 8(8.8) iv ψ = 0.875, Q = (76.) 33(39.6) 6 0 9(35.6) 5(8.) ψ =.7, Q = i ii iii iv x = y = 7 (, ), (, ), (3, 5) 9 5 r = 0. 待ち時間 ( 予約時刻 - 到着時刻 ( 分 ) I ū = ( x a)c, v = (ȳ b)d s uv = n n (u i ū)(v i v) i=
8 [ ] 8 = = n cd n n {(x i a)c ( x a)c}{(y i b)d (ȳ b)d} i= n (x i x)(y i ȳ) = cds xy i= u v r uv x y r xy.5. I s u = cs x, s v = ds y r uv = s uv s u s v = cds xy cds x s y = r xy Ω = { AAAA, AAAD, AADA, AADD, ADAA, ADAD, ADDA, ADDD, DAAA, DAAD, DADA, DADD, DDAA, DDAD, DDDA, DDDD } { AAAA, AAAD, AADA, AADD }. { DDAA, DADA, DAAD, ADDA, ADAD, AADD } Ω = { ABCD, ABDC, ACBD, ACDB, ADBC, ADCB, BACD, BADC, BCAD, BCDA, BDAC, BDCA, CABD, CADB, CBAD, CBDA, CDAB, CDBA, DABC, DACB, DBAC, DBCA, DCAB, DCBA }. A B C D i ii iii iv = = 0.5=( ) + ( ) 0.5 = = = P (A G) = P (G) P (A G) !!! 5! = 3,560,!!!! =,30 6 = 3
9 [ ] C C = 55 9 C 3 3 C C = x p(x) x 3 p(x) 3 = 6 3 x 0 p(x) E(x) = 0 3, E(x) = 5, V (x) = 3 5 x 3 3 p(x) 3 = 3 3 = 6 µ =, σ = V (x) = 5 9 ( ) 3 x p(x) =, 7 90 x =,, 3,..., , ,000 8 = 3,50 00,500 0, B(, 0.9) m 5m 8 8 x ( B 8, ) x = 35 8
10 [ ] P o() p(x) = x x! e p(0) = e, 0.35 p() = e, 0.7 p() = e, (e + e + e ) = 5e P o() p(x) = x x! e p(0) + p() = 3e, e p(x) = 3x x! e 3 p(0) = e 3, p() p(0) = 3e e 3 ( ) x ( ) 00 x ( 00 9 B(00, 00 ) p(x) = 99 00C x 00 00, p(0) = 99 00) = , p() = ( ) 99 = , p() = ( 00 ) ( 99 00) 98 = 0.89 P o() : p(x) = x! e, p(0) = e = , p() = e = , p() = e = p(x) = x x! e 0 p(0) + 7,000 p() +,000 p() +,000 [ {p(0) + p() + p()}] 8,000 =,7,000,003 =.5, Q(.5) = % 998,003,00,003 =.5, Q(.5) = ( ) = % = 0.5, Q(0.5) = =.5, =.0, µ u(0.) = µ % u(0.) =.8, = 0.8, Q(0.8) = 0.9, a 65 5 =.5, Q(.5) = % =.0, {Q(.0) + Q(.0)} = % = µ = = 0.3, Q(0.3) = 0.38, = =.8 a = cm
11 [ ] B(50, 0.8) N(0, ( 8) ) = , Q(0.53) = 0.98, 0.98 = % = , Q(0.88) = 0.89, 9% 5 00, B(00, 0.) N(0, ).5 0 =.875, Q(.88) = % 6 B (,600, u(0.0) =.36, 7 ) N(86, 8.8 ) 5 =.33, Q(.3) = % a =.36 a = 93 P {x = 0, y = 0} = 0., P {x = 0} P {y = 0} = = 0.5 E(x) = 0.5, V (x) = 0.5, E(y) =, V (y) = 0.6. x + y E(x + y) =.5, V (x + y) = 0.65 E(x + y) = E(x) + E(y) V (x + y) = V (x) + V (y) P {x = y} = 0.5, P {y > x} = 0.30 E(x) = 0.88, V (x) = , E(y) = 0.95, V (y) = E(xy) = 0.90 Cov(x, y) = = 0.06, ρ xy = = N(360, (0 ) ) = , Q(0.35) = %
12 [ ] 30 = 0 5 = = 0.65, Q(0.65) = N(60, ) 00 N(60, ( 5) ) =.789, Q(.79) = = % σ u(0.0) =.36 + σ ( ) σ = + σ.0 g.36 = a 5 a 3 {( 5, 5), ( 5, 5), ( 5, 5), ( 5, 5), ( 5, 5), ( 5, 5), (5, 5), (5, 5), (5, 5)}. x \x x x E(x x ) = 0 V (x x ) = 7 V (x ) = V (x ) = 36 3 N(., (0.0 ) ) =., Q(.) = % = 3, = 0 N(3,000, 0 ) 3,00 3,000 0 =.0, Q(.0) = = = 97 N(6,97) =.0, Q(.0) =
13 [ ] ,500, ,000 5, ,000, , µ µ (0.,.8) 95% 95% µ,050 ± t(5, 0.05) 00 = (997,,03) n = = ( ( ) 3 6 x N µ, ) P { 0.3 < x µ < 0.3} = {0.5 Q(.)} = n 0. n 70.6 n 7. ± = (9.50,.90) s = 3.0, t(, 0.05) =.0. ± = (9.9, 3.) σ 3.0 s iii,000,000 B(, 000, p) p 6 n =.96 pq =,0 ˆpˆq 95% = 0.039
14 [ ] 7 n \ p , , , e \ p ,537,07,305, ,58 6,7 8,068 9,0 9,60 % (,050,050,500 ±.96 8 p : p : 990,500 ±.96,500 50,500 ( 990,500 50,500 ) = (0.677, 0.73),500 ) = (0.636, 0.68),500, ,500 ±.96,050,500 50,500, ,500 50,500 = (0.007, 0.073),500 95% 95% 0 0 N(0, ).65 9 =.355 Q(.36) = % u(0.0) = n.36 n n 6 3 µ 0 t(n, 0.05) s x µ 0 + t(n, 0.05) s µ 0 n n x t(n, 0.05) s n µ 0 x + t(n, 0.05) s n [ x t(n, 0.05) s, x + t(n, 0.05) s ] n n x ± t(n, 0.05) s n µ 95% µ 0
15 [ ] 5 N(µ, 3.5 ) 0 x = 5.9 H 0 : µ = 60. H : µ x = 5.9 < 56.0 = A 5 00 x x H 0 : µ = 0 6 x N(50, (/5) ) /5 =.5, /5 =.5, Q(.5) = µ = 9. N(9., (/5) ) [9.5, 50.5] /5 =.5, /5 = 6.5, Q(6.5) 0 Q(.5) Q(6.5) =. Q(.5) = , 0.07 µ = µ = Z = (5 ) + (39 0) 0 + (6 6) 6 = 9.65 χ (, 0.05) = 5.99, χ (, 0.0) = 9., χ (, 0.005) = n = n = 600, ˆp = = 8 5, ˆp = = , ˆp = = ( ˆp ˆp = > = ) 600 p 95% 8 ( 8 5 ± ) = (0.93, 0.573) p 95% 7 ( 7 5 ± ) = (0.7, 0.507) 5 600
16 [ ] 6 H 0 : p = p α c α x = 3 x = 7 α x 7 x 6.5 ˆp ˆp /(0 ) =.333, Q(.33) = , = x x y y nˆα + x i ˆβ = yi () xi ˆα + x ˆβ i = x i y i () () () xi n { x i ( x i ) } n ˆβ = x i y i xi yi n (xi x) ˆβ = (xi x)(y i ȳ) ˆβ = (xi x)(y i ȳ) (xi x) = s xy s x () ˆα = ȳ ˆβ x.
17 [ ] 製作時間 製作個数 6 x i y i x i x (x i x) y i ȳ (y i ȳ) (x i x)(y i ȳ) x i x i y i x = 3, ȳ = 36 5ˆα + 5 ˆβ = 80, 5ˆα + 55 ˆβ = 65 ˆα = 3.5, ˆβ = 7.5 ˆβ = s xy s x = 75 0 = 7.5, ˆα = ȳ ˆβ x = = 3.5. ŷ = x x = 6 ŷ = = y i ŷ i y i ŷ i (y i ŷ i ) S e = 7.5 S T = 570 r = S e = 7.5 S T 570 = 75 = σ s = 7.5 = ± t(3, 0.05) 0 = (5.9, 9.09)
18 [ ] 8 ˆβ = 7.5 >.59 = t(3, 0.05) 読書 音楽鑑賞の時間 テレビを視る時間 7 ŷ = x S e = 0.9, r = ( 0.9, 0.5) ˆβ = 0.35 > 0.77 = t(7, 0.05) 民間最終消費支出 ( 千億円 ) 国内総生産 ( 千億円 ) 8 ŷ = x GDP r = s = ± t(, 0.05) = (0.8, 0.536) ˆα =. > 73.8 = t(, 0.05) 5.6
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