確率論と統計学の資料

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1 5 June 015

2 ii ii ii alpha A α beta B β gamma Γ γ delta δ epsilon E ϵ, ε zeta Z ζ eta H η theta Θ θ, ϑ iota I ι kappa K κ lambda Λ λ mu M µ nu N ν omicron O o xi Ξ ξ pi Π π, ϖ rho P ρ, ϱ sigma Σ σ, ς tau T τ upsilon Υ υ phi Φ ϕ, φ chi X χ psi Ψ ψ omega Ω ω

3 a r (a r) a [r] a(a 1)(a ) (a r + 1) a r r 0 a [0] 1 a a r r [r] r(r 1)(r ) 1 r! (r > 0), 0 [0] 0! 1 a r (a r) ( ) a a[r] a(a 1)(a ) (a r + 1) r r! r(r 1)(r ) 1 (r > 0), ( ) a 1 0 n r n r 0 n n [r] n! r!, nc r n! n(n 1)(n ) 3 1 ( ) n r n! r!(n r)! n (n k + 1) (a) n r n r (b) n r n [r] (c) n r ( n r) (d) n r ( ) ( n+r 1 n 1 n+r 1 ) r k1 n k k1 ( ) ( ) n n (e) 1 0 n ( ) ( ) n n (f) r n r ( ) ( n n 1 (g) r r ) + ( ) n 1 r 1 (Pascal s triangle) (Binomial Theorem) n ab (a + b) n k0 ( ) n 0 ( ) n a k b n k k b n + ( ) n a 1 b n ( ) n a n n

4 1 ( ) ( ) n n (h) (1 + t) n + t ( ) n (i) (1 t) n ( 1) k t k k k0 ( ) n (j) n k k0 ( ) n t n n ( ) n (k) 0 ( 1) k k k0 (l) n+1 1 ( ) n n ( ) n r + 1 ( ) a + b ( )( ) a b (m) n k n k k0 k0 ( ) n t k k ( ) n r n + 1 ( ) n n 1.1. (Multinomial Theorem) nk t 1, t,, t k (t 1 + t + + t k ) n n! r 1!r! r k! tr1 1 tr trk k r 1 0, r 0,, r k 0, r 1 + r + + r k n (r 1, r,, r k ) k k1 n(n + 1)(n + 1) 6 k k1 k k1 k1 n(n + 1) n(n + 1)(n + 1) 6 k 3 n (n + 1) 4 k 4 n 30 (n + 1)(n + 1)(3n + 3n 1) k1 k 5 n 1 (n + 1) (n + n 1) k 6 n 4 (n + 1)(n + 1)(3n4 + 6n 3n + 1) k1 k1 (k + 1) 3 k 3 3k + 3k + 1 k 1 n (n + 1) 3 n 3 (n + 1) 3 1

5 1. 3 k k1 n(n + 1) { } 3k + 3k k + 3 k + 1 k1 1 n k1 (n + 1) 3 n k1 k1 k1 k1 k n(n + 1) n (n + 1) k n(n + 1) n k1 3 k (n + 1) 3 n(n + 1) 1 3 n k1 n + 1 { } n + 4n + 3n n(n + 1)(n + 1) 1..1 (1 + 1 n )n, (n 1,, ) lim n + (1 + 1 n )n Napier s constante e π e x e x exp(x) 1..1 (l Hospital s rule) f (x) g(x) x a lim f (x) lim g(x) 0, x a x a f (x) lim x a g (x) f (x) lim x a g(x) f (x) lim x a g(x) lim f (x) x a g (x)

6 F(t) h(t) g(t) f (x ; t) dt f (x ; t)g(t)h(t) df(t) dt h(t) g(t) f (x ; t) t dt + f (h(t) ; t) dh(t) dt f (g(t) ; t) dg(t) dt 1..3 (Taylor expansion) f (x) k f (k) (x) (0 k n 1) [a, b] f (n 1) (x) (a, b) c (a, b) f (b) n 1 k0 f (k) (a) (b a) k + R n, k! R n f (n) (c) (b a) n n! x > 0 Γ(x) 0 t x 1 e t dt Γ(x) Γ function 1.3. x > 0y > 0 B(x, y) 1 0 t x 1 (1 t) y 1 dt B(x, y) B function (a) x > 0 Γ(x + 1) xγ(x) (b) n Γ(n + 1) n! (c) n ( Γ n + 1 ) ( n 1 )( n 3 ) 3 1 (d) x > 0y > 0 B(x, y) Γ(x)Γ(y) Γ(x + y)

7 1.3 5 (e) ( 1 1 Γ ) (Stirling s formula) x Γ(x) πx x 1/ e x x 1 n n! πn n+1/ e n (n ) Γ ( 1 ) π d dx Sin 1 (x) 1 1 x ( 1 B, 1 ) [ 0 x 1/ (1 x) 1/ dx 1 x(1 x) dx 1 dx ( 1 ) (x 1 ) 1 Sin 1 ( x 1 1 ) ( 1 B Γ, 1 π ) π ( 1 ] 1 ) π 0

8 Ω sample spaceb Ω (i) Ω B (ii) A B A c B (iii) A 1, A, B i1 A i B B σ-fieldb (event).1. Pr{ } B (i) A B Pr{A} 0 (ii) Pr{Ω} 1 (iii) B A 1, A, { } Pr A i i1 Pr{A i } i1 Pr{ } probability(ω, B, Pr{ }) probability space.1.3 A B (Ω, B, Pr{ }) B Pr{B} > 0 Pr{A B} B A conditional probaility Pr{A B} Pr{A B} Pr{B}.1.1 (Theorem of total probabilities) (Ω, B, Pr{ }) B 1, B,, B n (i) Ω n k1 B k (ii) Pr{B k } > 0 (k 1,,, n) A B Pr{A} Pr{A B k } Pr{B k } k1

9 (Bayes formula) (Ω, B, Pr{ }) B 1, B,, B n (i) Ω n k1 B k (ii) Pr{B k } > 0 (k 1,,, n) Pr{A} > 0 A B Pr{B k A} Pr{A B k } Pr{B k } nj1 Pr{A B j } Pr{B j } Pr{B k } B k prior probabilitypr { B k A } B k posterior probability.1.4 A B B A B Pr{A B} Pr{A} Pr{B}...1 (Ω, B, Pr{ }) X X( ) Ω X x {ω : X(ω) x} B X (a, b] Pr{a < X b} Pr{X (a, b]} x Pr{X x} x F(x) X cumulative distribution function: cdf (a) F( ) lim F(x) 0, x F(+ ) lim F(x) 1 x + (b) F(x) F(x) F(y) (x y) (c) F(x) lim x a F(a) F(a) X x 1, x, X discrete random variable Pr{X x i } f X (x i ) f X (x i ) 1 X i i F(x) Pr{X x} {x i:x i x} f X (x i )

10 8 f X (x i ) (probability function) f (x) 0 f (x) dx 1 f (x) F(x) Pr{X x} X continuous random variable f (x) (probability density function: pdf ) A Borel X A Pr { X A } Pr { X A } p i, Pr { X A } f (x) dx x f (t) dt x i A A pdf..1 X f (x) g(x) X Y 1 1 X h(y) h( ) Y f { h(y) } h (y) X F(x) x df(x) X expectation E[X] x df(x) Stieltjes E[X] x i f X (x i ) i x f X (x) dx (X ), (X ).3. X E[X] µ E[(X µ) ] σ Var(X) E[(X µ) ] variance σ Var(X) standard deviation

11 m r E[X r ] r (rth momentµ r E[(X µ) r ] r (rth central moment E[(X µ) 3 ] σ 3 skewnessx X 0 4 E[(X µ) 4 ] σ 4 kurtosisx X 3 E[X] m r µ r (a) X g(x) g(x i ) f X (x i ) (b) X g(x) i E[g(X)] g(x i ) f X (x i ) i g(x) f (x) dx < E[g(X)] g(x) f (x) dx (c) ab 1 b b k g 1 (x)g (x) g k (x) k E[a + b 1 g(x) + b g(x) + + b k g(x)] a + b 1 E[g 1 (X)] + b E[g (X)] + + +b k E[g k (X)] E[ a + b 1 g(x) + b g(x) + + b k g(x) ] (d) ab Y a + bx E[Y] a + be[x], Var(Y) b Var[X], µ r (Y) b r µ r (X) pdf.3.5 F(x) q q-quantile F(ξ) q ξ { } inf F(x) q x

12 F(x) median F(ξ) 0.5 ξ X F(x) X moment generating function t E[e tx ] e tx df(x) F(x) Laplace transformation E[e tx ] e txi f X (x i ) X i e tx f (x) dx X t ii 1 characteristic function e itx df(x) t F(x) Fourier transformation (Schwart s inequality) X ε > 0 { } Pr X ε E[X] ε.4. (Qebywv s inequality) X ε > 0 { } Pr X ε E[X] ε.4.3 (Qebywv s inequality) X µ σ ε > 0 { } Pr X µ εσ 1 ε

13 k(u) u 1 u α(0 α 1) k(αu 1 + (1 α)u ) αk(u 1 ) + (1 α)k(u ).4.4 Minkowski E[g 1 (X)] E[g (X)] { E[ g 1 (X) ± g (X)} ] E[g 1 (X)] + E[g (X)].4.5 (Jensen s inequality) k(x) X E[X] E[k(X)] k(e[x]).5 X 1, X, X.5.1 X { } Pr lim X n X 1 n {X n } n1 X almost sure convergence X n a.s. X.5. ε > 0 } lim { X Pr n X > ε n {X n } n1 X convergence in probability X n in P X 0

14 1.5.3 F n (x) X n F(x) X F(x) x lim F n(x) F(x) n {X n } n1 X convergence in distribution X n in d X.5.1 a.s. in P (i) X n X X n X in P in d (ii) X n X X n X k k (X 1, X,, X k ) F X1,X,,X k (x 1, x,, x k ) Pr{X 1 x 1, X x,, X k x k } (X 1, X,, X k ) simultaneous probability distribution function x 1, x,, x k k (X 1, X,, X k ) (X 1, X,, X k ) f X1,X,,X k (x 1, x,, x k ) Pr{X 1 x 1, X x,, X k x k } f X1,X,,X k (x 1i1, x i,, x kik ) 1 i 1,i,,i k (X 1, X,, X k ) i 1,i,,i k F X1,X,,X k (x 1, x,, x k ) Pr{X 1 x 1, X x,, X k x k } f X1,X,,X k (x i1, x i,, x ik ) i 1,i,,i k x i1 x 1, x i x,, x ik x k i 1, i,, i k i 1,i,,i k (X 1, X,, X k ) f (x 1, x,, x k ) 0 f (x 1, x,, x k ) dx 1 dx k 1 f (x 1, x,, x k ) x1 xk F X1,X,,X k (x 1, x,, x k ) f (t 1, t,, t k ) dt 1 dt k

15 .6 13 f (x 1, x,, x k ) (probability density function: pdf ).6. X Y (Ω, B, Pr{ }) F X,Y (x, y) F X (x) F X,Y (x, + ), F Y (x) F X,Y (+, y) X Y marginal probability distribution X Y F X,Y (x, y) F X,Y (x, y) {x 1:x 1 x} {x :x x} F X (x) F X,Y (x, + ) f X,Y (x, y) y f X,Y (x 1, x ) Y X y F Y (y) F X,Y (+, y) f X,Y (x, y) X Y F X,Y (x, y) F X,Y (x, y) f X,Y (s, t) dt ds {s:s x} x {t:t y} F X (x) F X,Y (x, + ) f X,Y (x, t) dt Y y y x.6.3 X Y f X,Y (x, y) Y y X conditional discrete probability function f X,Y (x, y) f (x y) ( f Y (y) 0), f Y (y) X Y ( f Y (y) 0)

16 X Y f X,Y (x, y) Y y X conditional discrete cumulative distribution F X Y (x y) Pr{X x Y y} f X Y (x, y) {x i:x i x} Y f Y (y) f Y (y) 0 y.6.5 X Y f X,Y (x, y) Y y X conditional continuous probability function f X,Y (x, y) f (x y) ( f Y (y) 0), f Y (y) X Y ( f Y (y) 0).6.6 X Y f X,Y (x, y) Y y X conditional continuous cumulative distribution F X Y (x y) Pr{X x Y y} f X Y (t, y) dt Y f Y (y) f Y (y) 0 y {t:t x}.7 (X 1, X,, X k ) (X 1, X,, X k ) Pr { X 1 x 1i1 } p1i1, Pr { X x i } pi,, Pr { X k x kik } pkik.7.1 i 1,, i k p i1 i k Pr { } X 1 x 1i1, X x i,, X k x kik Pr { } { } { } X 1 x 1i1 Pr X x i Pr Xk x kik p1i1 p i p kik X 1, X,, X k mutually independent (X 1, X,, X k ) (X 1, X,, X k ) f 1 (x 1 ), f (x ),, f k (x k ).7. (x 1, x,, x k ) f (x 1, x,, x k ) f 1 (x 1 ) f (x ) f k (x k ) X 1, X,, X k mutually independent

17 (X 1, X,, X k ) g(x 1, x,, x k ) i 1,i,,i k g(x 1i1, x i,, x kik ) p i1 i k g(x 1, x,, x k ) (X 1, X,, X k ) g(x 1, x,, x k ) E[g(X 1, X,, X k )] g(x 1, x,, x k ) E[g(X 1, X,, X k )] i 1,i,,i k g(x 1i1, x i,, x kik )p i1 i k g(x 1, x,, x k ) f (x 1, x,, x k ) dx 1 dx k g(x 1, x,, x k ) f (x 1, x,, x k ) dx 1 dx k.8.1 a, b 1,, b r g 1 (x 1, x,, x k ), g (x 1, x,, x k ),, g r (x 1, x,, x k ) r E[a + b 1 g 1 (X 1, X,, X k ) + b g (X 1, X,, X k ) + + b r g r (X 1, X,, X k )] a + b 1 E[g 1 (X 1, X,, X k )] + b E[g (X 1, X,, X k )] + + b r E[g r (X 1, X,, X k )].8.1 (X 1, X,, X k ) α r1 r k E[X r1 1 Xr Xrk k ] r i 1r j 0 ( j i) α r1 r k X i E[X i ] m i µ r1 r k E[(X 1 m 1 ) r1 (X m ) r (X k m k ) rk ] r i r j 0 ( j i) µ r1 r k X i Var(X i ) r i r j 1r k 0 (k i, j) σ i µ (X i ) E[(X i m i ) ] σ i j µ 11 (X i, X j ) E[(X i m i )(X j m j )] X i X j i j µ 11 (X i, X j ) σ (X i ) 0 < σ i <, 0 < σ j < ρ i j ρ(x i, X j ) σ i j σii σ j j X i X j

18 16.8. k k Σ (σ i j ) (X 1, X,, X k ) variance-covarinace matrixk k (rho i j ) correlation coefficient matrix.8. k (X 1, X,, X k ) a 1, a,, a k (i) k Var( a i X i ) i1 k k a i a j Cov(X i, X j ) i1 j1 k a i Var(X i) + a i a j Cov(X i, X j ) i1 1 i< j n (ii) (X 1, X,, X k ) k Var( a i X i ) i1 k a i Var(X i) i k (X 1, X,, X k ) t 1, t,, t k φ(t 1, t,, t k ) E[exp(i(t 1 X 1 + t X + + t k X k ))] (X 1, X,, X k ) characteristic function.9.1 (i) k (X 1, X,, X k ) φ(t 1, t,, t k ) φ(t 1 )φ(t ) φ(t k ) (ii) k (X 1, X,, X k ).9. k (X 1, X,, X k ) f (x 1, x,, x k ) k y i g i (x 1, x,, x k ) (x 1, x,, x k ) (y 1, y,, y k ) x 1, x,, x k x i g i (y 1, y,, y k )

19 .9 17 x i, (i 1,,, k) y i x 1 x 1 x 1 y 1 y y k x x x (x 1, x,, x k ) (y 1, y,, y k ) y 1 y y k x k x k x k y 1 y y k Y 1, Y,, Y k Y i g i (X 1, X,, X k ), (i 1,,, k) (Y 1, Y,, Y k ) { } (x 1, x,, x k ) f h 1 (y 1, y,, y k ), h (y 1, y,, y k ),, h k (y 1, y,, y k ) (y 1, y,, y k ).9.3 Convolution X Y f (x) g(y) Z X + Y h(z) h(z) f (z y)g(y) dy g(z x) f (y) dy

20 pdf N equally likely) N Pr { X x } 1, (x 1,,, N) N N + 1 N 1 1 µ 3 N(N + 1) 4 µ 4 (N + 1)(N + 1)(3N + 3N 1) 30 N j1 1 N e jt µ r E[X r ] r µ r E[X µ r ] r 1 p q 1 p p Pr { X 0 } p, Pr { X 1 } 1 p q p pq µ r p q + pe t n B(n, p) n n p X (n, p) X Bi(n, p) Pr { X k } ( ) n p k (1 p) n k (k 0, 1,, n) k np npq µ 3 npq(q p) (q + pe t ) µ 4 3n p q + npq(1 6pq) X Y X Bi(n 1, p), Y Bi(n, p) X + Y Bi(n 1 + n, p)

21 X Bi(n, p) n X np npq in d N(0, 1 ) in d 1 N(0, 1 ) 1 X Bi(n, p) np λ λ n X in d Po(λ) Po(λ) λ K M K M n x Pr { X k } ( K )( M K x n x ) ( M n ) (x 0, 1,,, n) n K M n K M M K N M n M 1 E[X(X 1) (X r + 1)] )( n r) r! ( K r ( M r ) Poisson K M K M n x Pr { X k } ( K )( M K x n x ) ( M n ) (x 0, 1,,, n) n K n K M K M n E[X(X 1) (X r + 1)] M M N M 1 )( n r) r! ( K r ( M r )

22 , r , , , (Qebywv) , , r , , 19 r , r Poisson r , 15 r , r r

23 1 A almost sure convergence B Bayes formula Bernoulli distribution Beta function binomial distribution Binomial Theorem C characteristic function , 16 Qebywv s inequality conditional continuous cumulative distribution conditional continuous probability function conditional discrete cumulative distribution conditional discrete probability function conditional probability continuous random variable convergence in distribution , 19 convergence in probability convex function convolution correlation coefficient correlation coefficient matrix covariance cumulative distribution function D discrete random variable discrete uniform distribution E equally likely event expectation F Fourier transformation G Gamma function H hypergeometric distribution I independence J Jensen s inequality K kurtosis L Laplace transformation l Hospital s rule M marginal probability distribution median Minkowski s inequality moment rth central , 15 rth , 15 moment generating function Multinomial Theorem mutually independence N Napier s constant normal distribution , 19 P Pascal s triangle Poisson distribution posterior probability prior probability probability density function probability function probability space probability Q quantile R random variable rth central moment , 15 rth moment , 15 S sample space Schwart s inequality σ-field simultaneous probability distribution function skewness standard deviation Stirling s formula T Taylor expansion Theorem of total probabilities V variance , 16 variance-covarinace matrix

基礎数学I

基礎数学I I & II ii ii........... 22................. 25 12............... 28.................. 28.................... 31............. 32.................. 34 3 1 9.................... 1....................... 1............