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2 () Statistik19 Statistik () 19 ( ) (18 ) ()

3 19 () () () 19 Biometrics() Biostatistics ( ) R.A. () ( ) Tukey, Joh Wilder AT&T Bell Laboratories 1965 The future of data aalysis196mathematics of Computatio1965Cooley Software Bit Statistics

4 GDP (i) (ii) : 4

5 : : statistics statisticum (EBM) (iii)

6 globalizatio CD 1 URL(Uiform Resource Locator) Web http(hypertext Trasfer Protocol) URL R (Wi, Mac, Liux)R o Widows: takeshou/r/ Wikipediahttp://ja.wikipedia.org/wiki/ 6

7 yasuda/idex-j.htm

8 .1 (set) > 0 1 a A b A a A a A a A b A b / A U U A B A B d a c b A B (a A, a / B) (1) ; {, 4, 6,, 0} () ; {x x =, = 1,,, 10}, { = 1,,, 10}, { 1 10 } U U A B A B A B A B 8

9 U U A B A B A c A A \ B = A B A,B x A x B A B B A A B A B B A A A A AA,B A B, B A A = B (empty set) {} ϕ ( ) ϕ A A, B A B (itersectio) A B A, B A B (uio) A B U Ω U (Uiverse set) U A U A (complemet) A c A (empty set).1 () : (1) A B = A B () A B = A B. (1) (A B) C = A (B C) () (A B) C = A (B C) (3) A B = B A (4) A B = B A (5) (A B) C = (A C) (B C) 9

10 .1,A 1, A,, A A i = A i = A 1 A A i=1 i A i = A 1 A A i=1.1 {A i, i = 1,,, }. a, b, c {0, 1} A, B, C 1 a A; a = 1 (1 a) (A) ; max{a, b} A B; max{a, b, c} A B C; mi{a, b} A B; mi{a, b, c} A B C..3 x i A i i x A i x i A i i x A i [] (exist) (all) x i x i A i x A i, i A i x A i, i.4 x i {1, 0}, i = 1,,, (i) max{x 1, x,, x } = 1 (1 x 1 ) (1 x ) (ii) mi{x 1, x,, x } = x 1 x x 10

11 . 5 5 =?.3 ( 1) ( )... 1 ( 1) 1 (exclamatio, )!!! = ( 1)( ) 3 1 ( 1), 0! = 1 11

12 10 3! = 6, 4! = 4, 5! = 10, 6! = 70, 7! = 5040, 8! = 4030, 9! = 36880, 10! = r 1. : r = r {}}{ (.1) (permutatio) r {}}{ ( 1)( ) ( (r 1)) =! ( r)! (.). : ( ) ( ) ( + 1)( + ) ( + (r 1)) + r 1 + r 1 = = r! r 1 (.3) (combiatio) ( ) ( ) ( 1)( ) ( (r 1)) = = (.4) r! r r. r {}}{ () r = ( 1)( ) ( r + 1) r {}}{ [r] = ( + 1)( + ) ( + r 1) (a 1, a, a 3,, a r ), {a 1, a, a 3,, a r } r! { } = r! { } (a, b, c) {a, b, c} (i) (ii) (iii) (iv) (v) 1

13 .6 {0,1,,3,5} 5 0 (i) 5 (ii) 5 (iii) 5 (iv) 5 5 (v) ( ) b a + b a O O = (0, 0) (North) (East) m, = 0, 1,, (m, ) N(m, ) = ( m+ ) = (m+ m ) N(0, 3) = 1, N(, 3) = N(3, ) = 10, N(3, 3) = 0 N(m, ) = k k (m, ) k = 15 (m, ) = (, 4), (4, ) k = 5 (m, ) = (5, 5) N(m, ) + N(m 1, + 1) = N(m, + 1) ( m+ ) + (m+ +1 ) = (m ) (x + 1)m+ (x + 1) = (x + 1) m++1 x.8 (m, ) N(m, ) = ( m+ ) = (m+ m ) : a, b ;(a + b) (a + b) = {}}{ (a + b) (a + b) (a + b) (a + b) a b a b a ( ) ( r b ) r a r a r b b r = = () r = () r r r r! ( r)! ( ) ( ) ( ) ( ) ( ) (a + b) = a b 0 + a 1 b a r b r + + a 1 b 1 + a 0 b 0 1 r 1 13

14 .4 (evet) (probability).3 : N A #(A) A P(A) P(A) = #(A) N a A, {a} P({a}) = 1, a N P({a 1, a,, a }) = N, a i A : ,,3,4,5,6 6 : Ω() #(Ω) < 1 A 0 P(A) 1 () ϕ P(ϕ) = 0, 3 Ω P(Ω) = 1. : A,B A B = ϕ A B (mutually exclusive) (mutually disjoit) A B (exclusive evet) A B A B A B = ϕ P(A B) = P(A) + P(B) A B = ϕ P(A B) =

15 : A A A A A (complemet) A A P(A) = 1 P(A) (coditioal probability); A B A B (coditioal probability) P(B A) P(A) = 0 P(B A) = P(A B P(A) A B A A B Ω A A B B A A B P(Ω) = 1, P(A B) P(B A) = P(A B) P(A) AB A B B A (idepedet), P(B A) = P(B A) P(A B) = P(A) P(B) A P(B A) = P(B) A B P(A B) = P(A) P(B) (.5) {A 1, A,, A } (i), A i A j (ii) P(A 1 A A ) = P(A 1 )P(A ) P(A ) (i) (pairwise idepedet) 15

16 A3 B A = {, 4, 6, 8, 10, 1, 14}, B = {3, 6, 9, 1, 15} 3 p p 7 6 A B = {6, 1}. A B A B P(A B) P(B A) = = P(A) 7 {B 1, B,, B } Ω (partitio), (i) B i B j = ϕ (i = j), (ii) Ω = i B i.3 {B 1, B,, B } Ω P(A) = P(A B 1 )P(B 1 ) + P(A B )P(B ) + + P(A B )P(B ) : 1 A3 B 4 C A = {, 4, 6}, B = {3, 6}, C = {4, 5, 6}., P(B A) = P(B) A B P(C A) = P(C) A C A, B A B A B A B P(B A) = P(B) : E p E r ( r )pr (1 p) r, (r = 0, 1,,, ) Biom(, p) % 4% ab ab 1 16

17 (1) A 1 B 1 AB () A 1 B 1 AB 17

18 3 3.1 ():{H}{T} {1, 0} (radom variable) X H (Head),T (Tail) 1 X 4 X X {0, 1, } (1, ) X (H, H) p (H, T) 1 p(1 p) (T, H) 1 p(1 p) (T, T) 0 (1 p) p X = 0 (1 p) X = 1 p(1 p)x = p X 0 1 (1 p) p(1 p) p 1 X x p X (x) (1 p) if x = 0 ( ) p X (x) = p(1 p) if x = 1 p p X (x) = p x (1 p) x, x = 0, 1, if x = x ( ) p X (x) = p x (1 p) x, x = 0, 1,,, (3.1) x X Biom(, p), p (Biomial distributio) X x 1, x,, x p i = P(X = x i ), i = 1,,, (i) 0 p k, (ii) p i = 1 x i p i i x x 1 x x P(X = x) p 1 p p 1,pdf(probability desity fuctio) (discrete desity) 18

19 X (probability distributio) {X = x k } p X (x k ) = p k, k = 1,,, {x 1, x,, x } {X x}, F X (x) = P(X x), < x < (3.) (distributio fuctio) (i), (ii) 0 1, (iii) f X (x) = d dx F X(x) F X (x) = x f X (t) dt (discrete type) P(a X b) = P(X = x i ) = p X (x i ) a x i b a x i b (cotiuous type) P(a X b) = b a f X (x) dx 3. c X r x 1, x,, x r c X 1 + X + + X c c (X 1 + X + + X ) = 0, c c = 1 (X 1 + X + + X ) P(X = x i ) = f X (x i ), i = 1,,, r N (x) x X 1, X,, X x N (x) = i 1 {Xi =x} X 1 + X + + X = r i=1 x in (x i ), X 1 + X + + X = r N (x x i ) i i=1 *1 N (x)/ f X (x) = P(X = x) (3.3) *1 19

20 X x 1 x x r N (x)/ N (x 1 )/ N (x )/ N (x r )/ 1 f X (x) = P(X = x) f X (x 1 ) f X (x ) f X (x r ) 1 (3.3) µ = r i=1 x i f X (x i ) c µ > c µ = c X µ = i x i P(X = x i ) = x i p X (x i ) (3.4) i f X (x) = d dx F X(x) = d dx P(X x), ( < x < ) µ = x f X (x) dx (3.5) : X :{p X (x i ), i = 1,, }, :{ f X (x), < x < } µ = E(X) (expectatio) (mea) average E(X) = x i p X (x i ) : E(X) = i x f X (x) dx : x m (x m) E[(X m) ] X m X m X (variace) V(X) V(X) = E[(X m) ] = i (x i m) p X (x i ) : V(X) = (x m) f X (x) dx : X (stadard deviatio) σ(x) σ(x) = V(X) V(X) (X m) = X mx + m V(X) = E[(X m) ] = E[X mx + m ] = E[X ] me(x) + m = E[X ] m V(X) = E[X(X 1)] + m m 0

21 3.1 (1) () 1 X X [0, 1] X (Uiform distributio) X Ui f {1,,, 6}, X Ui f [0, 1] P(X 1 + X = a) = P(X 1 = k)p(x = a k) k 3 () X () 3 X 1 P(X 1 + X = 7) = k P(X 1 = k)p(x = 7 k) = P(X 1 = 1)P(X = 7 1) + P(X 1 = )P(X = 7 ) P(X 1 = 6)P(X = 7 6) = (1/6)(5/36) + (1/6)(4/36) + (1/6)(3/36) + (1/6)(/36) + (1/6)(1/36) = 15/16 7 X 1 = 1,,, 6, X =, 3,, 1 X ():X 1 () 1 (1/6) (1/6) 3 (1/6) 4 (1/6) 5 (1/6) 6 (1/6) (1/36) (/36) (3/36) (4/36) (5/36) (6/36) (5/36) (4/36) (3/36) (/36) (1/36) [] X a,b Y = ax + b Y (i) E(Y) = ae(x) + b (iii) σ(y) = a σ(x) (ii) V(Y) = a V(X) (b ) X,Y (X, Y) 1

22 {X = a} {Y = b} (X, Y) = (a, b) f X,Y (a, b) = P(X = a, Y = b) f X (a) = b f X,Y (a, b), f Y (b) = a f X,Y (a, b) P(X 1 + X = a) = f X,Y (x, y) (3.6) {(x,y) x+y=a} x + y = a (x, y) E(X) = i x i p X (x i ) x f X (x)dx E[(X a) ] = i (x i a) p X (x i ) = i xi p X(x i ) a i x i p X (x i ) + a E[(X a) ] = (x a) f X (x)dx = x f X (x)dx a x f X (x)dx + a V(X) a = E(X) i xi p X(x i ) {E(X)} V(X) = x f X (x)dx {E(X)} (3.8) (X, Y) h(x, Y) h(x i, y j )p X,Y (x i, y j ) i j E[h(X, Y)] = h(x, y) f X,Y (x, y) dxdy {p X (x i, y j )} (x i, y j ) {p X (x i, y j )} p X,Y (x i, y j ) = p Y (y j ), p X,Y (x i, y j ) = p X (x i ) i j A i = {X = x i }, B j = {Y = y j } (Margial distributio) (3.7) (3.9) 3.1 F X,Y (x, y) = P({X x} {Y y}) = P(X x, Y y) Y X x y {X x, Y y} (x, y) Y y (0, 0) x X 3. X Y (covariace) cov(x, Y) = E[(X µ X )(Y µ Y )] = E[XY] µ X µ Y

23 µ X, µ Y 3.3 : X a Y b {X = a} {Y = b} P({X = a} {Y = b}) = P(X = a, Y = b) P(X = a) P(Y = b) X Y 3.1 X Y F X,Y (x, y) = F X (x)f Y (y), x, y 3. X, Y, E(X + Y) = E(X) + E(Y) (3.9) h(x, y) = x + y 3.3 X, Y, E(X Y) = E(X) E(Y) X Y = X + 3 [] P(X = i) = 1 6, i = 1,,, 6. E(X) = 6 i=1 i 1 6 = ( ) 1 6 = 7 ( ) 7 V(X) = E(X ) = ( ) = = 35 1 Y E(Y) = E(X + 3) = E(X) + 3 = = 10, Y V(Y) = V(X) = = X,Y 3.4 (i) V(X + Y) = V(X) + V(Y) (ii) V(X Y) = V(X) + V(Y),cov(X, Y) = 0 V(X + Y) = V(X) + V(Y) + cov(x, Y) V(X Y) = V(X) + V(Y) cov(x, Y) 3. A,B 1 A

24 1 B A,B X,Y E(X + Y) E(XY) X Y X + Y XY {X = 1} 4/6 = /3 {X = 0} /6 = 1/3 {Y = 1} 1/6 {Y = } 3/6 = 1/ {Y = 3} /6 = 1/3 E(X) = 1 / /3 = /3, V(X) = 1 / /3 (/3) = /9 E(Y) = 1 1/6 + 1/ + 3 1/3 = 13/6, V(Y) = 1 1/6 + 1/ + 3 1/3 (13/6) = 17/36 {XY = 0} /6 = 1/3 {X + Y = } /6 = 1/3 {XY = 1} 1/6 {X + Y = 3} 3/6 = 1/ {XY = } /6 = 1/3 {X + Y = 4} 1/6 {XY = 3} 1/6 E(X + Y) = 1/ / + 4 1/6 = 17/6, V(X + Y) = 1/ / + 4 1/6 (17/6) == 17/36. E(XY) = 0 1/ /6 + 1/ /6 = 4/3, V(XY) = 0 1/ /6 + 1/ /6 (4/3) = 3 16/9 = 11/9. cov(x, Y) = E(XY) E(X)E(Y) = 4/3 /3 13/6 = 1/9 V(X) + V(Y) + cov(x, Y) = /9 + 17/36 + ( 1/9) = 17/36 V(X + Y) 3. X, Y X + Y XY X + Y 3.3, X X {0,1,,3} {X = 1} /6 5/6 3 3(1/6)(5/6) X X (5/6) 3 3(1/6)(5/6) 3(5/6) (1/6) (1/6) A p A X X q = 1 p 4

25 X 0 1 k ( ) ( ) ( ) ( ) ( ) q p 1 q 1 p q p k q k p k (biomial distributio) Biom(, p) ( ) P(X = k) = p k q k, k = 0, 1,,, (3.10) k X X X q = 1 p ( ) ( ) 1 X k =, k = 1,,, 1 k k 1 ( ) E(X) = k p k q k 1 ( ) 1 = p k p k 1 q k = p k k=0 k=1 ( ) ( ) X X = X(X 1) + X k(k 1) = ( 1), k =, 3,, k k 1 X V(X) = k=0 k(k 1)( k )pk q k + p = ( 1)p = ( 1)p + p = pq k= ( k )pk q k + p 3.5 X q = 1 p X Biom(, p) = E(X) = p, V(X) = pq ( ) ( ) ( ) Biom(, p) = ,, 8,, 8, : ( ) p(k) = p k (1 p) k, k = 0, 1,,, k p 3 (p = 3 10 ), (p = 10 5 = 1 ), (p = 8 10 ) :,p = 1 = 8, 16, 3 3 B(8, 1 ),B(16, 1 ),B(3, 1 ) 5

26 X 6

27 4 4.1 : kg kg ±1.5kg 3.0kg : : 1 : 0 7

28 ()Average 1 {x 1, x,, x } x r i (i = 1,,, r) a i f i = f 1 + f + + f r a a 1 a a r x 1 + x + + x = a 1 f 1 + a f + + a r f r = i a i f i x f f 1 f f r (I) (II) X = 1 k=1 x k X = 1 i a i f i c x i c x i y i = x i c c 4.1 X Y 4.. (, ) Media {x 1, x,, x }, {x (1) x () x () } Me / ( + 1)/ Me = x ( ) + x ( +1 ) Me = x ( +1 ) 50% 50% 4..3 () Mode Mo 8

29 L Mo Me X L Box Whisker Chart) 1 (1) (mi), () 5% (:Q 1 ), (3) (average), (4) 50% (;Me = Q ), (5) 75% ( :Q 3 ),(6) (max) Rage R = max i x i mi i x i 5% 50% 5% (box chart) 4.3. Mea Deviatio x x i, i = 1,,, x i X (MD:mea deviatio) X = MD = = i (x i X) = 0 9

30 4.3.3 Variace Stadard Deviatio []: (x i X), i = 1,,, (variace) s s = 1 i = 1 i x i X = i x i ( i x i ), x 1 + x + + x = a 1 f 1 + a f + + a r f r = i a i f i s = 1 i (a i X) f i = 1 i a i f i X = i (x i X) = 0 1 (ubiased variace) u = 1 1 i (x i X) = 1 ( ) (x i x i ) i<j 1 i<j {(i, j) ; 1 i < j } ( ) (x i x i ), i, j = 1,,, cm cm u u = u u x (stadard deviatio) []: u = 1 1 i=1 x i 1 X u = u xi ( x i ) = ( 1) X AVERAGE() :{x 1, x,, x } ()Me ()Mo MEDIAN() MODE() Q QUARTILE() 0: 1: 1 (5%), : (50%)=3: 3 (75%)4: u STDEV() ( -) = (-1) s STDEVP() u VAR() (-1) s VARP() 30

31 4.1 ; X 61.0 s 6.04 (SD) u 5.38 u 8.93 (Me) (Mo) No available 4 (Q 0 ) (Q 1 ) (Q ) (Q 3 ) (Q 4 ) Score {x 1, x,, x } x i z i x a, b y = ax + b b = 0 y i = x i X, i = 1,,,, : a = 1 u u, b = X u (4.1) z i = 10y i % (quatile)10% 10 (decitile)100 (percetile) (rage) (quatile rage) 3/4(75%) 1/4(5%),skewess) (,kurtosis) kg cm xkg ycm 31

32 x, y {(x i, y i ); i = 1,,, } x X, y Y x s x = 1 i=1 (x i X) = 1 i=1 x i X y s y = 1 i=1 (y i Y) = 1 i=1 y i Y s xy = 1 i=1 (x i X)(y i Y) = 1 i=1 x iy i XY x y (x i X) (y i Y) x y x y (x, y) x, y ( ) s x,s y x i X, y i Y, i = 1,,, s X s Y x, y r 4.1 x y r r = s XY s X s Y (4.) r r 1 r 1 r 1 r 0 ( kg cm r = CORREL(, ) = CORREL(, ) = CORREL(, ) =

33 (regressio aalysis) 1 y ax b Y X (regressio lie of Y o X) y = ax + bx:y:x X Y (1) () coefficiet of reversio coefficiet of regressio 33

34 1888 co-relatio r (x) (y) (x) (y) a b (x 1 ) (y 1 ) y = ax + b a b () Fittig the Regressio Lie (x i, y i ) : i = 1,,, (x i, y) : y = ax i + b d i = y i y = y i (ax i + b) : i = 1,,, i d i Galto, F. (1886) x, y y = ax + b y (x, y ) y = ax + b a d i (x i, y i ) (x i, ax i + b) (x 3, y 3 ) (x, y ) i (x 1, y 1 ) x (x 1, y 1 ), (x, y ), (x i, y i ),, (x, y ) () y = ax + b ( ) a b x y y i (ax i + b) = δ i, i = 1,, δi = (y i ax i b) i δ i δ i i δ i a b a b 34

35 i δ i 0 a b 0 a (y i ax i b) = ( ) (y i ax i b)x i = 0 b (y i ax i b) = ( ) (y i ax i b) = 0 a b { ( x i )a + ( x i )b = x i y i ( x i )a + b = y i (4.3) a b a b = x iy i x i y i x i ( x i ) = x i y i x i x i y i x i ( x i ) = σ xy σ x = (σ x + x ) y x (σ xy + xy) σ x = y x σ xy σ x (4.4) y x y y = σ xy (x x) σ x a b x i y i δ i δ i = (y i ax i b) = y i b y i + b + a x i + ab x i a x i y i a b σ a σ b σa = /( xi ( x i ) ) δi /( ) (4.5) σb = x i /( x i ( x i ) ) δi /( ) r a r b σ a σ b y x y = y i x = x i y = ax + b y = ax + b y σ y = 1 (y i y) ax + b ŷ i = ax i + b, i = 1,,, ax + b σ ŷ = 1 (ŷ i y) = 1 (ax i + b ax b) = a (x i x) = a σ x 35

36 a (x i x) (y i y) a = x iy i x i y i x i ( x i ) a (x i x) (y i y) = a ( ) x i ( x i ) y i ( y i) = x i y i x i y i x i ( x i ) xi ( x i ) y i ( y i) = ( x i y i x i y i ) { x i ( x i ) } { y i ( y i) } = r r = x i y i x i y i x i ( x i ) y i ( y i) = σ xy σxx σyy = σ xy σ x σ y r r r a r a = σ y σ x r ( ) 36

37 5 Microsoft Excel, OpeOffice calc 5.1 1,,3, A,B,C, C 3 C3 A B C D E A B3 35, [] ; SUM [] ; {+,-,*,/,%,^} = X1 Y1 =X1+Y1 X1 Y1 =X1-Y1 X1 Y1 =X1*Y1 X1 Y1 =X1/Y1 X1 Y1 =X1%Y1 X1 Y1 =X1^Y1 5. x y 1 X1 X1 37

38 X1 =SQRT(X1) X1 =ABS(X1) X1 =INT(X1) X1 =ROUND(X1) X1 =ROUNDUP(X1) X1 =ROUNDDOWN(X1) X1XX3X X1 XX3XY1YY3Y (X1:Y) (X1;Y) : Excel Opeoffice =SUM(X1:X) =AVERAGE(X1:X) =MEDIANE(X1:X) =MODE(X1:X) 5.3 =if(, 1, ) =ad( 1,,...) =or( 1,,...) =coutif(, ) (true) 1 ; =IF(X1>Y1,"","OK") X1 Y1 () (OK) 1 =if(ad( 1,,...), 1, ) 1 =if(or( 1,,...), 1, ) =IF(X1>=100,SUM(Y1:Y),""), X1 100 Y1:Y ( ) =COUNTIF(X1:X,"") COUNTIFS 1 3 (Histgram) (cout) (frequecy) coutif Frequecy Ctrl Shift Eter 38

39 (cross tabulatio) 3 1 ( ) 1 coutifs MS pivot Microsoft Office Excel [] [ ] / Microsoft Office Excel 007 [ ] [] [ ] [] [ ] Tally mark ; ***** ::: *** ; ; ; *************** ; ; :::: ************** ; ::: ******** ; ; :: ************ ::: *** (%) :/ : [] A1:A10 9 ;{ 150, 00, 50, 30, 330, 360, 380, 40, 480} (1) (4) 39

40 A : (1) 00 () (3) (4) / < [] 1. (A1:A10). []-[] 3. [Excel ] [] [] 4. [] $A$1:$A$10 [] 5. [] 6. [] [] 7. [] 8. [] 9. [OK] 10. [] D1 11. [] 1. D -[] 13. [] [] [] 100 [OK] frequecy [] (1) =max{}=mi{} () =cout{ } (3) (4) D13 D17 [](5) ( E13 E17 (6) =frequecy (7) FREQUENCY(, ) help (8) A A10 (9) D13 D17 (10) OK CTRL+SHITFT OK 5.4 [][] 5.0kg 55.0kg

(1) () () D, (3) 1,,3, (4) 1,,3,,OK (5) (6) (0%) OK JPEG A B C 1 5.0-55.0 53.5 1 3 55.0-58.0 56.5 3 4 58.

41 (1) () () D, (3) 1,,3, (4) 1,,3,,OK (5) (6) (0%) OK JPEG A B C

42 5. 1 D D B C = B * C B11 SUM 3 E E B B11 = B - B11 4 F G9 F = (B - $B$11)^ * C F3F8 $ $ 5 B1 B13 A B C D E F G

43 (x i ) (z i ) (w i ) x i :z i = x i X s (), :w i = x i X u ( ) ( 1 ) (x i ) 1(z i ) (w i ) X s mediame modemo #N/A() #N/A() #N/A() ( ) stdev SD = u varp() s = (x i X) 4 u = (x i X) 3 u rage =max -mi mi max sum cout CORREL( 1, ) COVAR( 1, ) COVAR(x, y) VARP( 1, ) CORREL = VARP(x) VARP(y)

44 5.3 () 0.76 (, ) : (60.3, 161.) (57.9, 154.3) (65.4, 16.8) (56.1, 160.4) (53.6, 155.7) (6.7, 163.5) (70.0, 17.5) (55.8, 166.4) (67.1, 173.) (63.1, 164.0) 5.5 pseudoradom umbers (Mersee twister) [0, 1] U i, i = 1,,, 1 X = i=1 U i 6 X N(0, 1) = rad()

45 6 6.1 [] ; [] ; Garbage i, garbage out 6. 45

46 6.1 µ σ σ X i Z = c 1 X 1 + c X + + c X? {c i } 1. E(Z) = (c 1 + c + + c )µ. V(Z) = (c 1 + c + + c )σ [] ; 6. []: X ; 1. E(X) = µ. V(X) = σ µ σ X Z = X µ σ/ Z N(0, 1) 6.1 µ = 10 σ = 16 = 5 X (1) P(X < ), () P( X 10 > 3 5 ) 6. (i) X, Y N(0, σ ) X + Y X Y X (ii) X = i +X j i<j, u = ( ) 1 (X i X j ) i<j (iii) X i, i = 1,,, 6.3 m m σ( σ ) () {X 1, X,, X } X = 1 i X i X N(m, σ ) Z = X m σ/ N(0, 1) P( Z c) = P( c Z c) = c 46

47 1.96 { Z c } = { X m c σ } { = X c m σ m X + c [ X c σ } σ m X + c σ ] 95% c = 1.96 c.58 95% 99% []: σ X m 95% X 1.96 σ m X σ 99% p.58 σ p p +.58 σ σ σ s = 1 i X i X s = s > cm 6cm 1 m 95% : A p A X X X m = p σ = p(1 p) p = X Z = { Z c } = X p p(1 p) = p p p(1 p)/ N(0, 1) { } { } X p c p(1 p) = p c p(1 p)/ p p + c p(1 p)/ p 1/ [ ] p(1 p) p(1 p) p c p p + c 6.4 () p p 95% 99% p 1.96 p.58 p(1 p) p(1 p) p p p p p(1 p) p(1 p)

48 A A 95% 6.4 H 0 H 1 1 ( 1 1 (1) () (3) (4) (5) 1. N(µ, σ ) σ = 15 = x = 38.0 H 0 : µ = 40 H 0 : µ = 40 α =

49 σ = 15 = 5 z = x µ 0 σ / = x 40 = 1.33, α = /100 α/ = 0.05 =.5%, z α/ = z = z > z = x = 38.0 N(40, 5) 9 o C o C 5% H 0 : µ = 1455 H 1 : µ = 1455 x 1, x,, x 9 = 9, i x i = 1986, i = ( 1400 α = 0.05, ν = 9 1 t 0.05 (8) =.306 t >.306 (.306 < t <.306) t = x µ = = 1.78 u / 416.1/9 H 0 ( Kg, 3.Kg 6.8Kg,.5Kg ( 5% 1% µ 1 µ H 0 : µ 1 µ = 0 H 1 : µ 1 µ > 0 1 = = 36 z = x 1 x s 1 + s 1 = =.51

50 α = 0.05 (z 0.05 = {z : z > 1.645} z =.51 > ( 50

51 7, 51

52 1. (i) H 0 : µ = µ 0 H 1 : µ = µ 0 (ii) H 0 : µ = µ 0 H 1 : µ > µ 0 (iii) H 0 : µ = µ 0 N(µ, σ ), σ z = x µ 0 σ/ ( 30), σ u ( α = 0.05, 0.01 ) N(0, 1) (i) z > z α/ (ii) z > z α/ (iii) z < z α/ H 1 : µ < µ 0 z = x µ 0 u/. () (i) H 0 : µ = µ 0 H 1 : µ = µ 0 (ii) H 0 : µ = µ 0 H 1 : µ > µ 0 (iii) H 0 : µ = µ 0 N(µ, σ ), σ t = x µ 0 u/ t ( ν = 1 ) (i) t > t α (ν) (ii) t > t α (ν) (iii) t < t α (ν) H 1 : µ < µ 0 3. (i) H 0 : σ = σ0 H 1 : σ = σ0 (ii) H 0 : σ = σ0 H 1 : σ > σ0 N(µ, σ ) ν = 1 χ = u σ /ν = s σ / ( ν = 1 ) (i) χ < χ 1 α (ν), χ > χ α (ν) (ii) χ > χ α(ν) 4. ( (i) H 0 : p = p 0 H 1 : p = p 0 (ii) H 0 : p = p 0 H 1 : p > p 0 B(p), ( > 30) (iii) H 0 : p = p 0 H 1 : p < p 0 p, (1 p) > 5 z = p p 0 p 0 (1 p 0 ) p N(0, 1) (i) z > z α/ (ii) z > z α/ (iii) z < z α/ 5. () (i) H 0 : µ 1 = µ H 1 : µ 1 = µ (ii) H 0 : µ 1 = µ N(µ 1, σ1 ), H 1 : µ 1 > µ N(µ, σ ), (iii) H 0 : µ 1 = µ 1, H 1 : µ 1 < µ σ1, σ z = x 1 x u u = σ 1 / 1 + σ / N(0, 1) (i) z > z α (ii) z > z α (iii) z < z α 5

53 7.1 F Excel ; Ui f [0, 1] ; Biom(, p) ; N(µ, σ ) ; N(0, 1) ; χ ( f ) ; F( f 1, f ) RAND() 0 1, [a, b], RAND() (b a) + a BINOMDIST(k,, p, T/F) k:, :, p:, T/F:TRUE, FALSE NORMDIST(x, µ, σ, T/F) x:, µ:, σ:, T/F: TRUE FALSE NORMINV(p,µ,σ) NORMDIST, p:,µ:,σ:, NORMDIST(x,µ, σ, TRUE) = p x NORMSDIST(z) z:, P(Z z) Z N(0, 1), NORMSINV(p) p:, NORMSDIST CHIDIST(x, f ) x;, f :, ( ) :CHIDIST= P(X > x) CHIINV(p, f ) p:, f :, CHIDIST, p =CHIDIST(x, f ), CHIINV(p, f ) = x FDIST(x, f 1, f ), x:, ( f 1, f ): FINV(p, f 1, f ) FDIST, FDIST(x, f 1, f ) = p, FINV(p, f 1, f ) = x, p:, ( f 1, f ): t ; t( f ) TDIST(x, f, B) x:, f :, B:, 1 TDIST(x, f, 1) = P(X > x) TDIST(x, f, ) = P( X > x) = P(X > x or X < x)x < 0, TDIST( x, f, 1) = 1 TDIST(x, f, 1) = P(X > x) TDIST( x, f, ) =TDIST(x, f, ) = P( X > x) t TINV(p, f ) TDIST, p:, f :, TINV P( X > t) = p t, P( X > t) = P(X < torx > t) t p p = 0.05 f = 10 TINV(0.05, 10) =.8139 TINV( 0.05, 10) =

(I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : 2 Typeset by Akio Namba usig Powerdot. 2 / 47

4 Typeset by Akio Namba usig Powerdot. / 47 (I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : 2 Typeset by Akio Namba usig Powerdot. 2 / 47 (I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : 2 (radom variable):