..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A
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- かんじ ももき
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1 .. Laplace ). A... i),. ω i i ). {ω,..., ω } Ω,. ii) Ω. Ω. A ) r, A P A) P A) r... ).. Ω {,, 3, 4, 5, 6}. i i 6). A {, 4, 6} P A) P A) 3 6. ).. i, j i, j) ) Ω {i, j) i 6, j 6}., 36. A. A {i, j) i j }. A 36 5 A A ) A P A) /36.,. i) A, B A B A B. A B A B. ii) A, B, P A B) P A) + P B) P A B). A, B, A B P A B) P A) + P B). ),. i) A, A. ii) A, B A B A + B A B. A B A B P A) + P B).,., Kolmogorov ).
2 ..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A j ) P A i ) P A i ) A i, A i {ω Ω i ω A i }.,F Ω ), A F P A) A..4. ) Ω {,, 3, 4, 5, 6}, F Ω P A) A 6 ) Ω {i, j) i 6, j 6} F Ω, P A) A 36 Ω, F, P ). Ω {ω,..., ω }, F Ω, ω i p i P A) {i ω i A} p i Ω, F, P ). p i /, p i.. 3), ), ) ).. 6.
3 . Ω Ω {a, a,..., a,...) a i,..., 6}. F P. Kolmogorov. 3) Ω [a, b]. F [a, b] P A) A b a.,[a, b]..),..,.,... ) Ω R, [a, b], {x, x,..., x } R. P,, )., R A P A). ) R [a, b]. 3) {a, a,...} ). 4) P F z) P, z]), z] ) F z) R. P..,... ). ) P {i}) 6 i 6) R. A R P A) A,..., 6.. ) λ > 0. λ λk P {k}) e k! k 0,,...) λ. λk k0 e λ k! ). Poisso)., 3
4 i) ii) iii) iv) R P ) fx) i) x fx) 0 ii) fx)dx R iii) A R. P A) P A). fx)dx f A. A A fx)dx P fx), F z) fx) F z) P, z]) z fx)dx. F x) fx).,.4. P fx),.5 ). ) F x) fx). m, > 0. ) fx) exp x m) π, Nm, ). N0, ) 0, ). R x m) e π 4
5 ) e x dx π R., m, Nm, ),.. i) ii) m. F z) z e x π dx,.., 39. Iz) z 0 e x π dx z > 0 F z). ) a, b a < b). A R P A) F z) + Iz) A [a, b]) b a [a, b]., a < c < d < b P [c, d]) d c b a. [a, b] fx) [a,b]x) b a. [a,b] x) { a x b 0 x < a x > b,[a, b] 0. b a P A) A. 3) λ > 0. { λe λx x 0 fx) 0 x < 0 λ. R λe λx dx. 0.., 5
6 i) ii),..,.,, ) ) Ω, F, P ). Ω. Ω ω Ω Xω) ) X X a b P {ω Ω a Xω) b}). R A X A P {ω Ω Xω) A}). P {ω Ω a Xω) b}), P {ω Ω Xω) A}) P a X b), P X A). 3.. ) Ω {i, j) i 6, j 6}. ω i, j) i, j. X ω) i, X ω) j ω i, j) X, X. P X i) i i 6 i,, 3, 4, 5, 6). Y ω) X ω) + X ω), Zω) X ω)x ω), W ω) maxx ω), X ω)). X {,..., 6}, X + X {,..., }. Ω, Xω),. ) X [a, b].. [a, b] {ω a Xω) b} F ),. {ω a Xω) b}. 6
7 3. Ω ) X Ω, F, P ). A R P X A) P X A) P X A) P X. P X X,, ). ) X F X F X x) P X x). X P, P X, X P ) X {a i } N P X a i) p i.,{a i } N. X ) N E[X] a i p i 3.). ) X f E[X]. xfx)dx 3.) ).,, P X,.,. X P X P X X F X. 7
8 ) Ω {ω,..., ω } X E[X] Xω i )P {ω i }) 3.3). P {ω i }) ω i. 3.) 3.3),. 3.3) X 3.). 3).. X fx). N. k A k { } k k+ ω Ω N Xω) < N. k A k Ω, A k A l k l) A k., A k k N X N ω) X N ω) Xω) N N X N X. X N k E[X N ] N P X N k ) N k k k k N P N X < k + ) N k k k+)/n fx)dx N k/n k k R k N f xfx)dx. k N E[X] xfx)dx. R X. g R gx). gx).. ), ),, X, g R. ) X 3.4 ) EgX) ) X 3.4 ) EgX) ) N N ga i )p i. gx)fx)dx.. ) Ω {ω,..., ω } 3.5 ) E[gX)] gxω i ))P {ω i }). 3.4) 8
9 A i {ω Xω) a i } N A i Ω, A i A j i j) p i P A i ). gxω i ))P {ω i }) N gxω))p {ω i }) ω A i N ga i ) P {ω}) ω A i N N ga i )P A i ) ga i )p i. 3.5) 3.7 ). X, Y, a, b E[aX + by ] ae[x] + be[y ]. g : R R E[gX) + hx)] E[gX)] + E[hX)].. Ω {ω,..., ω }. 3.5 ) E[aX + by ] axω i ) + by ω i ))P {ω i }) axω i )P {ω i }) + by ω i )P {ω i }) ae[x] + be[y ]. 3.6) 3.4,.,,., X V [X] E[X m) ] X [X] V X) X. m X E[X ] X. 3.6 N ) P X a i ) p i {a,..., a N } ) V [X] a i m) p i, ) X f V [X] x m) fx)dx,. R 9
10 3.9., i) ii). fx) π+x ),. E[gX) + hx)] E[gX)] + E[hX)] ) V [X] E[X ] E[X]. ) V [ax + b] a V [X]. X m.. 3. Chebyshev ) ). X, m P X m r) r , A 5 B. 3, C D A, B P A B) P A)P B). 0
11 A, B Ω, F, P ) A, B Ω F ) A B Ω P A B). 4...,. R, R, R 3, W, W R, W ). Ω {R, R ), R, W ), W, R ), W, W ) R,, W,, 3 } ,.. C {R, ), R, ) 5 } 4.) D {, W ),, W ),, W 3 ) 5 } 4.) C D {R i, W j ) i,, j,, 3} 4.3) P C) C P D) D C D) P C D) ) 4.5) P C)P D) 4.6) 5. A, 5 B.,. 3. 3,,, C D.. 4. A, B A c, B c.. C Ω, F, P ). B. P A B) P A B), A F P B) P B) B. P B) B.
12 4.4. P B) > 0, P A) > 0. A, B P A B) P A), P B A) P B) A B B A, A B A,..., A {i,..., i k } {,..., } P A i A ik ) P A i )P A i ) P A ik ). A, B, C P A B C) P A)P B)P C), P A B) P A)P B), P A C) P A)P C), P B C) P B)P C) A B, A C, B C A, B, C. 4.. X, X, A {X i} { i } 4.7) B {X j} { j } 4.8) A, B, P A B) P A)P B). 4.9) 4.9) P X i, X j) P X i) P X j) Ω, F, P ) X, Y a < b, c < d P X [a, b], Y [c, d]) P X [a, b])p Y [c, d]) 4.7. X, Y a < b, c < d {ω Xω) [a, b]} {ω Y ω) [c, d]}
13 4.8. Ω {i, j) i 6, j 6} A Ω P A) A /36 X ω) i ω i, j) ) X ω) j ω i, j) ) X X X +X X X P X + X 5, X X 0) 0 P X +X 5) > 0, P X X 0) > X, X Y X + X, Z X X, W maxx, X ). Y, Z, W., X,..., X. 4.9 ). X,..., X I,..., I [a i, b i ] ) P X I,..., X I ) P X I ) P X I )., I i {X I },..., {X I }. 5,,,.,.. 5. ) ) p, q p). X, B, p).. X i i ) P X i ) p, P X i 0) p, 0 ). S X + + X 5.) 3
14 S. k {S k}, i,...,i k, A i,...,i k P A i,...,i k ) P X i i i,, i k ), X j 0 j i,, i k )) p k p) k. i,..., i k A i,...,i k {S k} P S k) P A i,...,i K ) i,,i k C k p k p) k. 5.) {0,,..., } X P X k) C k p k q k 0 k B, p). 5.) S. E[X i ] p + 0 p) p. 5.3) [ ] E[S ] E X i E[X i ] p. 5.4) 5.. X B, p). ) E[X] p, ) V [X] pq, 6. 3). ) λ > 0 ) 0 {0,,,...} {k} λ λk P {k}) e k! k 0,,... λ ) ) 0 X λ λk P X k) e k! X λ 4
15 X B, p) p λ ) P X k) C k p k p) k! λ k! k)! ) k λ ) k + ) k lim x + x) x e ) k λ ) k λ k λ ) λk k! k! e λ, 5.3. X λ E[X] V [X] λ. E[X ] <. 3) p 0 < p < ). k + p) k p. p. X p P X k) p) k p, k 0,,,.... X X p. E[X] p p, V [X] p p. E[X ] <... { E[X] k p) k p p p) k p) k p p) d )} p) k dp k0 k0 k0 { p p) d )} p dp p p. 5. ) [a, b] fx) { b a a x b 0 x > a, x < b.. x a, b 0 /b a)., a+b b a),. ) fx) { λe λx x 0 0 x < 0 5
16 λ.. [0, ), t k k 0,,... A p λ λ. λ > 0. p λ ). T A. P T k/) ) λ k λ λ/ ). t [k t /, k t + )/). P t / A ) λ ) kt λ λ )t λ ) λe λt ) P T dt) λe λt dt,t fx) λe λx x 0). t [k t /, k t + )/) k A p k p k kt+c k λ ) k λ ) kt + k. Bk t +, p) p λ/). k t + ) λ λt k t ) lim p k λt)k k! e λt. λt. I). λ. II) I) λ X λ E[X] λ, V [X] λ, E[X ]! λ. 3) m R, > 0. ) fx) exp x m) π R Nm, ) X Nm, ). ) X Nm, ). E[X] m, V [X]. ) p 0), q. px + q Npm + q, p ). ), ). Proof. ) E[X] R R R x exp π y + m) exp π y + m) exp π ) x m) dx y y ) dyy x m ) ) dy m R π exp ) y dy ). 6
17 V [X] E[X E[X]) ] E[X m) ] R R R x m) π exp y) π exp ) x m) dx ) y dy, 5.5) ) y exp y dy 5.6) π. 5.6)!). ) p > 0. p < 0. x u q p P px + q z) P P px + q z) z z X z q p z q p ) exp π exp π π p exp ) x m) dx. 5.7) u q ) p m) p du u q mp) p ) du. 5.8) 5.8) px + q. px + q mp + q, p. X m. T F x) P T x) X. a m P a X b) P X m b m ) a m P T b m ) ) ) b m a m F F. 6 ).,,3,.. 7
18 ) X, Y Ω, F, P ) R ) X X, Y ) A R P X A) P X A) R X X, Y ) X, Y ) X, Y R X, Y 6. ) X, Y P X a i ) p i i M), P Y b j ) q j j N) X, Y ) {a i, b j )} i M, j N P X, Y ) a i, b j )) p ij i M, j N p ij a i, b j ) p ij R 6. ). ) N j p ij p i i M), M p ij q j j N),,.. ) X, Y X, Y ) fx, y) < a < b < +, < c < d < + P a X b, c Y d) 6. b d a c fx, y)dxdy 6.3. X, Y ) fx, y) X Y f x), f y) f x) fx, y)dy, f y) fx, y)dx 6.4. X, Y P X a i ) p i i M), P Y b j ) q j j N), P X, Y ) a i, b j )) p ij X, Y p ij p i q j i M, j N) X, Y f x), f y) fx, y). X, Y x, y) fx, y) f x)f y). 8
19 ) X, Y E[XY ] E[X]E[Y ]. ) X, Y CovX, Y ) E[X E[X])Y E[Y ])]. X, Y CovX, Y ) 0. 3) X, Y V [αx + βy ] α V [X] + β V [Y ].. ) X, Y. {x,..., x }, {y,..., y m } X, Y. E i {ω Xω) x i }, F j {ω Xω) y j } Xω) m x i Ei ω), Y ω) y j Fj ω), A A ω) ω A ), A ω) 0 ω A c ) A. ) m E[XY ] E x i Ei y j Fj ω) i, j m i, j m i, j m i j x i y j E[ Ei Fj ] j x i y j P E i F j ) 6.) x i y j P E i )P F j ) 6.) x i P E i ) j m y j P F j ) E[X]E[Y ]. 6.),6.) E[ Ei Fj ] E[ Ei F j ] P E i F j ), P E i F j ) P {X x i, Y y j }) P X x i )P Y y j ) P E i )P F j ). ) ). 3). [ { } ] V [αx + βy ] E αx + βy ) E[αX + βy ] [ { } ] E αx E[X]) + βy E[Y ]) α E[X E[X]) ] + β E[Y E[Y ]) ] + αβe[x E[X])Y E[Y ])] α V [X] + β V [Y ] 6.3) 6.7. ) X, Y E[XY ] E[X]E[Y ]. X, Y E[XY ] E[X]E[Y ] X, Y. ) X, Y φ, ψ E[φX)ψY )] E[φX)]E[ψY )]. 9
20 3) X,..., X [ ] V X i V [X i ]. 4) 6.6. X i P X i ) p, P X i 0) p S X + + X B, p). E[X i ] p + 0 p) p 6.4) E[X i p) ] p) p + p) p) p p) { p + p} p p). 6.5) [ ] E[S ] E X i E[X i ] p 6.6) ) [ ] E[S E[S ]) ] E X i E[X i ]) E X i E[X i ]) p p). 6.7) 6. R 6.8. ) X,..., X. X X,..., X ) R P X A) P X A) A R ). P X X, ). ) X X,..., X ) fx,..., x ), A R P X A) A fx,..., x )dx dx. X. X ), X X,..., X ) m E[X ],..., E[X ]) 6.8) X ). i, j) E[X i E[X i ])X j E[X j ])] X... 0
21 6.0. A - m. fx,..., x ) fx,..., x ) exp π det A A ) ij x i m i )x j m j ) i j Nm, A). A ) ij A A i, j).. fx,..., x ) fx,..., x )dx dx R. m, A. 6.. X Nm, A) X m, A ) X X,..., X ) X ) {a,..., a N }, P X a i ) p i i N) R ϕ N E[ϕX)] ϕa i )p i. ) X X,..., X ) fx,..., x ) R ϕ E[ϕX)] ϕx,..., x )fx,..., x )dx dx. R X,..., X X i, X,..., X, X,..., X ) X i {X,..., X }. i X i f i x). {X,..., X } X X,..., X ) fx,..., x ) f x ) f x )., 6.4. t 8.6). t- William Gosset ) Gosset Studet Studet t-..,,., ) X X,..., X ) Nm, A). P k, ) k, v k. Y t P t X + v )
22 k t ). Y k ). ) X X,..., X ). {X,..., X } i, j i j) X i, X j. 3) X, Y. X Y CovX, Y ) ). ) X,..., X ) 6.4 ) X i, X j {X,..., X },. 3, 3. ). ) X, X. X + X X X. CovX + X, X X ) E[X + X )X X )] ). i X i X X X + X X X X i i ) Nm i, i ). ) X,..., X ) Nm, A) m m,..., m ), A i, i) i. P X,..., X ) A) A π i exp x i m i ) i ) dx dx. ) i a i > 0, i b i > 0. a,..., a ), b,..., b ) Y a i X i, Z b i X i. Y, Z) Y, Z). 3) ) Y Z a ib i i 0.. ) 6.3. ) 6.4 ). 3) 6.4 3).. ) CovY, Z) E a i X i E[X i ]) b j X j E[X j ]) j a i b j E [X i E[X i ])X j E[X j ])] i,j a i b i E[X i E[X i ]) ] i a i b i i 6.9)
23 CovX i, X j ) 0 i j) Cov a i X i, b i X i ) a i b i CovX i, X i ) X, X,..., X,. X X + + X 7.. ) X,..., X m E[ X ] m. m. ) X,..., X m, X. X, X,..., X, ) {X, X,..., X,...} {X,..., X }. ) {X, X,..., X,...} idepedet ad idetically distributed, i.i.d. ) X i {X i }. 7.3 ). X, X,... m,. S X + + X, X S. ε lim P X m > ε ) 0. X m ) {X i } i.i.d.. ) X i i X i. ) m X+ +X 3) X i i, 0. X S. X. 7.3 Chebyshev ). P X m ε ) ε 3
24 7.,.,. X, X,..., X,... m,. S X + X + + X. X m X m) T S m 0 E[T ] 0, V [T ] T ). X, X,..., X,... m,. T 7.). a < b b lim P a T b) e x dx. a π X m) 7.) X i i,,...) P X i ) p, P X i 0) p E[X i ] p, V[X i ] p p) lim P a ) S p b b e x dx p p) a π. de Moivre-Laplace - ). - Stirlig ) lim! π e. S p+ p p)t B, p) Np, p p)) , S P 45 S 55). ) 00, p / S N50, 5) S. ) P 45 S 55) P 44.5 S )
25 S T S P 44.5 S 55.5) ) P T 5 5 P. T.) P T ) ) P 45 S 55) 55 k45 00C k ) ), ).... ),. 3).. X, X,..., X,.... ).. 3) X, X,.... a) X X+ X. b) s Xi X ). c) u Xi X ). 5
26 {X,..., X,...}. m, lim P lim P X m ε ) 0 ) ) Xi X ) ε 0 ), m,. ) X m 7.3). 8.. u X, s u s u E[u ] ). 8. χ, t X, X,..., X,... s X i X ), u X i X ). s, u., Z,..., Z. χ Z i χ 0 x 0 f x) Γ ) x e x x > 0. Γs) 0 x s e x dx s > 0) ) Γ) )!,,...) 8.3 ϕ E[ϕ Zi )] R ϕx)f x)dx 6
27 ). E[ϕZ + Z )] R ϕ x + y ) e x +y π r0 0 0 θ0 ϕx),z] x) π dxdy ϕ r ) e r π rdrdθ ϕr )e r rdr r t ) ϕt) e t dt P Z + Z z ) z 0 x r cos θ, y r si θ ) e t/ dt. t > 0 f t) e t/, t 0 f t) 0. R χ χ. χ X,..., X,... Nm, ) ) Xi m χ X,..., X Nm, ). s )u Xi X ) χ Y i Xi X. 6.4 Y i 0. E[ Y i ] E[Y i ] χ. Y i 0 Y,..., Y χ.. Z i X i m. a ij i ) j a ij a kj 0 k j) j {Z i } 7
28 {a ij } i, j U i a ij Z j i ) j Xi X ). 6.4 U,..., U Xi X ) χ. U i Nm, ) X,..., X X m) T.., s X i X ) T T X m) S m s X i X ). s. T., William Gosset ). T t. s u t X, Y χ X Y. f x) t) X Y/ / B/, /). t) t. Bp, q). 0 ) + + x x p x) q dx p > 0, q > 0 8
29 .. x + t t ), t si θ π/ θ π/) B/, /) / 4 / dt ) π/ 3/. 4t π/ dθ π f x) + x 3/ π ϕ [ )] X E ϕ ϕt)f t)dt Y. [ 0 E ϕ e y+t ). )] X Y { <x<,y>0} { <t<,y>0} ϕ x/y) e x e y π dxdy y+ t ϕt) e ) 4 π ydtdy ) ϕt) 4 e y+t ) ydy dt, π 0 ydy exp r + t ) ) r dr + t )3/ T. 0 0 x t y ). e r r dr π + t )3/, 8.9. X,..., X Nm, ). X, u, s t-... i) X m ˆT X m) X m) u s ˆT 6.4) X m u ii) )u χ 8.6). t X m )u. Cov X, X i X ) ) X X i X. 9
30 8.0. ) 6.4 3) X, X i X. E[ X ] m, E[X i X ] 0 Cov X, X i X ) E[ X m)x i X )] { E X j m) X i m) j E X j m)x i m) E j 0. m)} X k k j, k.. X j m)x k m) Y,..., Y. Z fy,..., Y p ), Z gy p+,..., Y ). Z Z. ) Y Z, Y Z,...Y Z. Y i Z... Y,..., Y, Z) Cov Y i, Z) CovY i, Z) 0 ). X X i X ). ) s. u s. ˆT f x) e x / π. lim x / x) dx π ) 0 lim ) + + x e x. ).. ) e x. ) x t /.. 9 ) ) ) ) 3) 30
31 ) ) km , 8.0, 8.3, 7.7, 8.5, 8.0, 8.6, 7., 8.7, 8. km) x 0 x m. X m, 0.50 X X+ +X. X 0 m T P T.96) 0.95, P T.58) 0.99 X 0 m ) X0 m P X m X X0 x 0 [ x , x ] [ , ].96 [7.8, 8.4] ) 99 [ , ]
32 ) T. 0 < α P T zα)) α zα) 00α.. 5 z0.05).96, z0.0).58 Nm, ) X,..., X T X m N0, ) P zα) X ) m zα) α. X zα) m X + zα) α. x, m [ x zα), x + ] zα) ) α. ) 00 α). ) ) cm, ,. Nm, ). Gosset 8.9) X,..., X Nm, ) ˆT X m s t. X X + +X, s X i X ) t t 4 ) P t.09) 0.95, P t.86) ) X0 m P 9 s
33 X 0 s m X 0 + s X 0, s 0 x 0.5, s 0 0. [ , ].09 [.45,.57] ) X m s t. 0 t ) t t. 0 < α < P t t α)) α t α) t 00α. 9 ) 5 t ).09, t 9 0.0).86. Nm, ) X,..., X t P t α) X m s ˆT X m s ) t α) α. X s t α) m X + s t α) α. α, x, s, m [ x s t α), x + s ] t α). ) 00 α). ) 3) A A )p. p. 33
34 i) ii) ) , 99. p. P X i ) p, P X i 0) p X,..., X S X + + X B, p)). V [X i ] p p) T X p p p) N0, ). X S ). 00, 5.96 X 00 p P ) 00 p p) X p p) 00 p X p p) X p. i) p,. p p) p p p ) p p) 4 x , x ii) p x [ ), ] )
35 X,..., X. 400, X i X 400 m ) 30 65, 0 m 95. ) , 0 m 95. X,..., X. s ) 30 X 30 m s t. ) 300 N0, ). X 300 m s : x p p) 4 95 [ x.96, x + ] ) ). 35
36 HHypothesis H, ) i) ii) 90 9, H ).. H 300 X B300, 0.3). 300 X N90, ) X. P X 90 9) P X ) ) X P P T.07) T ) i), ii).,. ) H. ) H X,..., X T X,..., X ). 3) )α P T X,..., X ) W α ) α α W α. 4) x,..., x T x,..., x ) W α W α. 36
37 300 5 W 0.05 { }. 90. H : H H H : 30, H : 30. X i P X i ) 0.3, P X i 0) 0.7 S 000 X i S B000, 0.3) S 300 S P S a) 0.05 a. S N300, ) N300, 0) S ) P S a) P S a 0.5 ) S 300 P a P T a 0.5 ) 0 ) ) T N0, ). ). P T.65) 0.05 a a 4.4. [34.4, ). a 3.9 [33.9, ). 330 H. 37
38 0.3. ) P T.33) 0.0 a a [334.3, ). ) 5. 4 ). 500, H : 500. H : N500, 30) i.i.d.x i i,,...). X X + +X T X P T.33) 0.0. X X 0 X H P T.58) 0.0 [ , ] A A 5 H: A N595, 50) H : A 595 H A X,..., X X X + X X
39 N0, ) T x 0 P T.65) 0.05 x t ) ? 5 Nm, ) H: m 00 0 X,..., X 0, X 0 X + X 0, s 0 0 X i 0 0 X 0 ) X 0 00 s t t 9 t t T X,..., X 0 ) X 0 00 s 0 P t.09) X0 00 P ) 9 s
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