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6 6.1 6.1 (1 Z ( X = e Z, Y = Im Z ( Z = X + iy, i = 1 (2 Z E[ e Z ] < E[ Im Z ] < Z E[Z] = E[e Z] + ie[im Z] 6.2 Z E[Z] E[ Z ] : E[ Z ] < e Z Z, Im Z Z E[Z] α = E[Z], Z = Z Z 1 {Z } E[Z] = α = α [ α ] [ α α = E α Z = E Z α α Z ] [ ( α ] ( = E Z e α Z α ] + ie[ Z Im α Z ( α ( = e α 1 ( E[ Z ] α Z α Z 6.3 X ϕ X (t X (characteristic fuctio ϕ X (t = E[e itx ] = E[cos(tX] + i E[si(tX], t. 6.4 (i X (ii t ϕ X (t 1 (iii ϕ X ( = 1 ϕ X (t = ϕ X ( t (iv t ϕ X (t : (i, (ii e itx 2 = cos tx + i si tx 2 = cos 2 tx + si 2 tx = 1 6.2 (iii ϕ X ( = E[e ] = E[1] = 1, ϕ X ( t = E[e X ] = E[e X ] = E[e itx ] = ϕ X (t (iv {h } sup s ϕ X (s + h ϕ X (s ( sup s E[ e isx (e ihx 1 ] = E[ e ihx 1 ]. e ihx 1 e ihx + 1 = 2 E[2] = 2 < Lebesgue E[ e ihx 1 ] E[ e 1 ] = 6.5 X a, b ϕ ax+b (t = e itb ϕ X (at. : ϕ ax+b (t = E[e iatx e itb ] = e itb E[e iatx ] = e itb ϕ X (at. 6.6 (1 X B(, p, q = 1 p, ϕ X (t = E[e itx ] = ( e itk p k q k = k k= k= ( (pe it k q k = (e it p + q. k 13

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t = E[e itx ] = e itx 1 2π e 1 2 x2 dx = e 1 2 t2 1 2π e 1 2 (x2 dx (6.1 itx 1 2 x2 = 1 2 (x it2 1 2 t2 > 4 C ( C,1 :, C,2 : it, C,3 : it it, C,4 : it. e 1 2 z2 C Cauchy e 1 2 z2 dz = C 4 e 1 2 z2 dz = e 1 2 z2 dz C C, =1 C,1 e 1 2 z2 dz = C,2 z = + iy dz = i dy e 1 2 z2 dz C,2 = C,4 z = + iy = e 1 2 z2 dz C,4 t t = e 1 2 x2 dx e 1 2 (+iy2 i dy t e 1 2 x2 dx = 2π. e 1 2 (2 y 2 +iy i dy e 1 2 (2 y 2 dy t e 1 2 (2 t 2. C,3 z = x it dz = i dx C,3 e 1 2 z2 dz = 2π ( 17 t e 1 2 ( +iy2 i dy t e 1 2 (2 t 2. e 1 2 (x2 dx = e 1 2 (x2 dx e 1 2 (x2 dx. e 1 2 (x2 dx = (6.1 ϕ X (t = e 1 2 t2 1 2π 2π = e 1 2 t2 6.8 X N(m, σ 2 ϕ X (t = e imt 1 2 σ2 t 2 : Z = X m σ Z X = σz + m 6.5 ϕ X (t = ϕ σz+m (t = e imt ϕ Z (σt = e imt e 1 2 (σt2 = e imt 1 2 σ2 t 2 14

6.9 X Cauchy f(x = 1 π ϕ X (t = e t : ϕ X (t = E[e itx ] = e itx 1 π 1 ( < x < 1 + x2 1 dx (6.2 1 + x2 1st step t > > 1 2 C ( C,1 :, C,2 : z =, Im z. g(z = eitz z i z + i 1 e 2πi C itz 1 + z 2 dz = 1 g(z e t dz = g(i = 2πi C z i 2i. (6.3 C,1 e itz C e itz 1 + z 2 dz = 2 1 + z 2 dz = =1 C, e itz C,2 z = e iθ, θ π, dz = ie iθ dθ C,2 e itz 1 + z 2 dz = = π π e iteiθ 1 + 2 e 2iθ ieiθ dθ t si θ e 2 e 2iθ + 1 dθ dz (6.4 1 + z2 e itx 1 + x 2 dx e itx 1 + x 2 dx π π e it(cos θ+i si θ 2 e 2iθ dθ + 1 t si θ e 2 1 dθ π 2 1. 2 2 e 2iθ + 1 2 e 2iθ 1 = 2 1 θ π si θ t > e t si θ 1 (6.3, (6.4 1 e itx e t dx = 2πi 1 + x2 2i 2i (6.2 ϕ X (t = 1 π 2d step t = ϕ X ( = 1 6.4(iii t < (6.2 y = x ϕ X (t = e it( y 1 π e itx 1 + x 2 dx = e t 1 1 + ( y 2 dy = e i( ty 1 1 π 1 + y 2 dy = e ( t = e t. 3 t > 1st step 6.1 X E[ X k ] < ϕ X (t C k - ϕ (k X (t = ik E[X k e itx ] : 16 15

Cauchy t = Cauchy 6.11 p X = (X 1,..., X p p ϕ X (t X { ϕ X (t = E[e it X ] = E[exp i p j=1 t j X j } ], t = (t 1,..., t p p 6.12 p X p A p b ϕ AX+b (t = e it b ϕ X (A t. : ϕ AX+b (t = E[e it AX e it b ] = e it b E[e i(a t X ] = e it b ϕ X (A t. 6.13 X = (X 1,..., X p p N(m, Σ m = (m 1,..., m p p, Σ = (σ ij ϕ X (t = e it m 1 2 t Σt : P = (p ij D = (λ ij P ΣP = D Y = (Y 1,..., Y p = P (X m 4.12 Y N((,...,, D ( 19 Y 1,..., Y p Y j N(, λ jj 4.6 ϕ Y (t = E[e it 1Y1 e it py p ] = E[e it 1Y 1 ] E[e it py p ] = e 1 2 λ11t2 1 e 1 2 λppt2 p = e 1 2 t Dt. X = P Y + m 6.12 ϕ X (t = e it m ϕ Y (P t = e it m e 1 2 (P t D(P t = e it m e 1 2 t P DP t = e it m 1 2 t Σt. 6.2 Dyki 6.14 X µ X : µ X (A = P (X A, A B(. (6.5 B( Borel µ X X (distributio 6.15 µ X X (, B( µ (Ω, F, P µ ( X F X (x F X (x = P (X x = µ X ((, x] 6.16 X, Y X, Y : µ X = µ Y, µ X (A = µ Y (A ( A B( F X (x = F Y (x ( x σ- Dyki 6.17 (1 S P π (a S P, (b A, B P A B P 2 (2 S D Dyki 16

(a S D (b A, B D A B A\B D (c A D, A A +1 ( N =1 A D 3 S C C Dyki L(C D λ (λ Λ Dyki λ Λ D λ Dyki ( {D λ } λ Λ C Dyki L(C = λ Λ D λ λ Λ D λ D λ ( λ Λ C Dyki λ Λ D λ = λ Λ D λ 6.18 (Dyki P π L(P = σ(p *4 : σ- Dyki ( 2(2 L(P σ(p L(P σ(p L(P σ- 1st step A P G A = {B; A B L(P} G A P Dyki π B P A B P L(P. B G A, P G A. (a S P G A (b B 1, B 2 G A, B 1 B 2 A B 1, A B 2 L(P A B 1 A B 2 A (B 1 \B 2 = (A B 1 \(A B 2 L(P. B 1 \B 2 G A. (c B G A, B B +1 ( N A B L(P, A B A B +1 A ( =1 B = =1 A B L(P. =1 B G A. L(P G A A P, B L(P A B L(P 2d step A L(P G A = {B; A B L(P} 1st step P G A (a S P G A (b, (c 1st step G A P Dyki A, B L(P A B L(P L(P π 3rd step L(P σ- (i S L(P (ii A L(P (i S L(P A c = S\A A c L(P. (iii A L(P ( N B = k=1 A k (ii L(P π B = ( k=1 A c c k L(P (c k=1 A k = k=1 B k L(P 6.16 : (= A = (, x] ( = J = {(, x]; x } {(, } A = {A B(; µ X (A = µ Y (A} J π A Dyki ( 2(3 x (, µ X ((, x] = F X (x = F Y (x = µ Y ((, x] x = µ X ((, = µ Y ((, = 1. J A 6.18 σ(j = L(J A. J σ- σ(j Borel B( B( A 6.19 f(x f f(x µ X (dx < E[f(X] = f(xµ X (dx. *4 σ(p P σ- 17

: f(x f(x = a i 1 Ai (x E[f(X] = i a i P (X A i = i a i µ X (A i = f(xµ X (dx 4.2 4.4, 4.5 f(x {f q (x} f q f ( 5.12 f(x = f + (x f (x f(x = e f(x + i Im f(x 6.2 (, B( µ, ν f C b ( f(xµ(dx = f(xν(dx µ = ν C b ( : a ( f (x = 1 (x a, = 1 (x a (a < x < a + 1, = (a x f C b ( f (xµ(dx = f (xν(dx f (x 1 (,a] (x ( f (x 1 ( x Lebesgue 1 (,a] (xµ(dx = 1 (,a] (xν(dx, µ((, a] = ν((, a] 6.16 µ = ν 6.3 6.21 2 ϕ X (t = ϕ Y (t, t, µ X = µ Y ( F X (x = F Y (x, x 6.22 Dirichlet f T (α = T si αt t dt, T >, α (6.6 (1 sup f T (α <, (2 lim f T (α = π { 1(, (α 1 (, (α } T,α T 2 : (1 u = αt f T (α = αt M sup αt π f T (α π 2 2. M π/2 si u u si u u du si u u ( u π 2 si u u du ( 21 (2 α = α > 6.23 α < s = t α > 6.23 si x x dx = π 2. : < ε < Im z 4 C ε, ( C ε,,1 : ε, C ε,,3 : ε, C ε,,2 : z =, Im z, C ε,,4 : z = ε, Im z ε ε 18

Cauchy = Cε, e iz z dz = 4 =1 Cε,, e iz z dz. (6.7 Cε,,1 e iz z dz + Cε,,3 e iz z dz = ε e ix ε x dx + e ix x dx = e ix e ix si x dx = 2i ε x ε x dx si x 6.22(1 ε +, 2i x dx C ε,,2 z = e iθ, θ π, Cε,,2 e iz z dz = π e ieiθ e iθ ieiθ dθ π π e i(cos θ+i si θ dθ = e si θ dθ e si θ, e si θ 1 ( < θ < π Lebesgue C ε,,4 Cε,,4 e iz z dz = π e iεeiθ εe iθ εieiθ dθ = i π e iεeiθ dθ ε + e iεeiθ 1, e iεeiθ = e ε si θ 1 ( < θ < π Lebesgue i π dθ = iπ (6.7 si x = 2i dx iπ x ( 22 6.24 (Lévy X F ϕ(t a, b (a < b F F (b F (a = 1 T 2π lim T T x F X lim y x F X (y = F X (x x : X µ X 6.19 T T e b e a ϕ(t dt = g(x = e x eb e a T T e b e a ϕ(t dt. (6.8 e b e a e itx µ X (dx dt e b e a = g (ξ(b a = e ξ (b a (a < ξ < b b a Fubii ( 5.14 = ( T T e b e a T T eb e a e itx µx (dxdt b a 2T < ( T e itx si t(x a si t(x b dt µ X (dx = dt µ X (dx T t 19

e iξ = cos ξ + i si ξ T T cos t(x a cos t(x b dt = t 6.22 f T (α = 2 (f T (x a f T (x b µ X (dx 6.22(1, (2 Lebesgue T π [ 2 1(, (x a 1 (, (x a 1 (, (x b + 1 (, (x b ] µ X (dx 2 [ = π 1(a, (x 1 (,a (x 1 (b, (x + 1 (,b (x ] µ X (dx = π [µ X ((a, µ X ((, a µ X ((b, + µ X ((, b] = π [1 F (a F (a (1 F (b + F (b ] a, b F (6.8 = 1 π [1 F (a F (a (1 F (b + F (b ] = F (b F (a (6.9 2π 6.21 : F X F Y c c c 6.24 a, b c F X, F Y F X (b F X (a = F Y (b F Y (a. {a } c, a F X (b = F Y (b x {b } c b x + F X (x = F Y (x 6.16 µ X = µ Y 6.25 X 1,..., X X j Poisso P (λ j Y = X 1 + + X Poisso P (λ 1 + + λ ϕ Y (t = E[e itx1 e itx ] = E[e itx 1 ] E[e itx ] = e λ 1(e it 1 e λ (e it 1 = e (λ 1+ +λ (e it 1 2 X 1,, X 6.6(2 6.6(2 Poisso P (λ 1 + + λ 6.21 6.26 X 1, X 2,... Cauchy f(x = 1 1 π 1 + x 2 ( < x < 6.9 ϕ Xi (t = e t Y = 1 (X 1 + X 2 + + X ϕ Y (t = E[e i t X1 e i t X ] = E[e i t X1 ] E[e i t X ] = e t e t = e t Y Cauchy Cauchy 6.27 X ϕ(t f X (x ϕ(t dt < X f X (x = 1 e x ϕ(t dt. (6.1 2π 2

: 6.24 eb e a b a (6.8 (, (6.9 F (x a, b (a < b 1 1 {F (b + F (b F (a F (a } = 2 2π e b e a b F (x b a + 1 {F (a F (a } 2 F (x F (x + h F (x = 1 2π h > h x+h x h e (x+h e x e y ϕ(t dy dt 2h F (x + h F (x = 1 2π ϕ(t dt = 1 2π x+h x ϕ(t dt < Lebesgue F (x F (x = f X (x (6.1 ϕ(t dt b a 2π x+h x e y ϕ(t dy dt ϕ(t dt ϕ(t dt < Fubii e y ϕ(t dt dy e y ϕ(t dt y 6.9 : 15(2 f(x = 1 2 e x ϕ(t = 1 ϕ(t dt = π < 6.27 1 2 e x = 1 e x 1 2π 1 + t 2 dt x t, t x Cauchy ϕ X (t 1+t 2 ϕ X (t = e itx 1 π 1 1 + x 2 dx = 2 1 e i( tx 1 2π 1 + x 2 dx = 2 1 2 e t = e t 6.4 6.28 ( f C b ( lim E[f(X ] = E[f(X] X X (covergece i law (covergece i distributio X = X C b ( (cf. [D] p.15 pp.26 27. 6.29 {X } X 21

: 1st step {X } X f C b ( f(x f(x f M f(x M ( x f(x (ω M (ω Ω Lebesgue ( 5.13 X X lim E[f(X ] = E[f(X] 2d step {X } X f C b ( a = E[f(X ] f(x M (x a M {a } a = E[f(X] {a } a {a } {X } 5.6 {X } X 1st step {a } a {a } a lim E[f(X ] = E[f(X] f C b ( {X } X 6.3 6.29 {X } N P (X = 1 = P (X = 1 = 1/2 f C b ( E[f(X ] = 1 2 f(1 + 1 2 f( 1 lim E[f(X ] = E[f(X 1 ] {X } X 1 ( X k < ε < 1 2 P ( X X 1 ε = P (X 1 = 1, X = 1 + P (X 1 = 1, X = 1 = 1 2 1 2 + 1 2 1 2 = 1 2 {X } X 1 6.31 µ, = 1, 2,..., µ ( S ( µ µ f C b (S lim S f(xµ (dx = S f(xµ(dx {X } X {µ X } µ X µ ν f(x µ(dx = f(x ν(dx, f C b ( µ = ν 6.2 6.32 {X } X Y F Y (x = P (Y x (1 {X } X (2 F X x lim F X (x = F X (x 22

: (1 (2 x F X 1 (,x] (y f + δ, f δ δ > 1 y x f + δ (y = 1 1 δ (y x x < y < x + δ y x + δ ( 1 y x δ, f δ (y = 1 1 δ (y (x δ x δ < y < x y x 1 (,x δ] (y f δ (y 1 (,x](y f + δ (y 1 (,x+δ](y, y F X (x = P (X x = E[1 (,x] (X ] E[f + δ (X ] C b(, f + δ C b( {X } X lim sup F X (x lim E[f + δ (X ] = E[f + δ (X] E[1 (,x+δ](x] = P (X x + δ = F X (x + δ δ + lim sup F X (x lim F X(x + δ = F X (x (6.11 δ + F X (x = E[1 (,x] (X ] E[f δ (X ] f δ C b( lim if F X (x lim E[f δ (X ] = E[f δ (X] E[1 (,x δ](x] = P (X x δ = F X (x δ δ + lim if F X (x lim δ + F X(x δ = F X (x x F X (6.11 lim F X (x = F X (x (2 (1 F X ε > F X a, b (a < b F X (a ε, 1 ε F X (b (2 N N = F X (a 2ε, 1 2ε F X (b δ > f C b ( a = a < a 1 < < a K = b a j (1 j K 1 F X max a j 1 x a j f(x f(a j δ (1 j K 2 f [a, b] h f (x = K f(a j 1 (aj 1,a j](x j=1 23

f = sup x f(x y / (a, b] f(y h f (y = f(y f N E[f(X ] E[h f (X ] K E[ f(x h f (X 1 (aj 1,a j ](X ] + E[ f(x h f (X 1 (a,b] c(x ] j=1 K δp (X (a j 1, a j ] + f P (X / (a, b] j=1 = δp (X (a, a K ] + f (F X (a + 1 F X (b δ + 4ε f. E[f(X] E[h f (X] δp (X (a, a K ] + f (F X (a + 1 F X (b δ + 2ε f. a j F X (2 F X (a j F X (a j ( E[h f (X ] = K K E[h f (X 1 (aj 1,a j](x ] = f(a j (F X (a j F X (a j 1 j=1 j=1 K f(a j (F X (a j F X (a j 1 = E[h f (X] ( j=1 E[f(X ] E[f(X] E[f(X ] E[h f (X ] + E[h f (X ] E[h f (X] + E[h f (X] E[f(X] lim sup E[f(X ] E[f(X] 2δ + 6ε f ε, δ ε, δ > δ +, ε + lim E[f(X ] = E[f(X] 6.33 lim sup E[f(X ] E[f(X]. α > X 1, X 2,... i.i.d. f(x = α(x + 1 α+1 1 (, (x (Parate Y = 1/α max{x 1, X 2,, X } {Y } F Z (z Z F Z (z = (z, F Z (z = e z α (z > (Fréchet : z lim P (Y z = z > Y P (Y z = P (X 1 1/α z,, X 1/α z = P (X 1 1/α z P (X 1/α z ( 1/α z = α(t + 1 dt α 1 = (1 ( 1/α z + 1 α ( = 1 1 (z + 1/α α e z α ( {Y } Z S µ 24

6.34 S {µ } µ 4 (1 (4 2 (1 {µ } µ (2 S G lim if µ (G µ(g (3 S F lim sup µ (F µ(f (4 S A B(S µ( A = lim µ (A = µ(a A A compact 6.35 (Prohorov {X α } (1, (2 (1 {X α } compact, {X α } {X α } {X αk } X {X α } X (2 ε > M > if α P (X α [ M, M] 1 ε {µ α } (2 if α µ α([ M, M] 1 ε {µ α } tight (2 {X α } tight 6.35 S [ M, M] S compact ( Prohorov 6.36 (Helly {F (x} {F k (x} F (x (F (x F x : 1st step lim F k (x = F (x (6.12 k Q = {x 1, x 2, } F (x = F, (x {F, (x 1 } [, 1] Bolzao-Weierstrass {F 1, (x 1 } F (x 1 {F 1, (x 2 } [, 1] Bolzao-Weierstrass {F 2, (x 2 } F (x 2 j = 1, 2,... {F j+1, (x} {F j, (x} {F j, (x j } F (x j F k (x = F k,k (x j = 1, 2,... {F k (x j } k j {F j, (x j } 1 {F k (x j } k F (x j ( x i < x j F k (x i F k (x j F (x i F (x j 2d step 1st step F (x (x Q F (x (x F (x = if{ F (y; y Q, y > x} (6.13 F F (x (??? x F (6.12 ε > z 1, z 2, z 3 Q z 1 < z 2 < x < z 3 25

F (x ε < F (z 1 F (z 2 F (x F (z 3 < F (x + ε x (6.13 k F k (z 2 F (z 2 F (z 1, F k (z 3 F (z 3 F (z 3 k F (x ε < F k (z 2 F k (x F k (z 3 < F (x + ε, F k (x F (x < ε (6.12 6.35 : (1 (2 (2 ε > M > if α P (X α [ M, M] < 1 ε {X α } {X α } N P (X α [, ] < 1 ε (6.14 (1 {X α } {X αk } X {X αk } X x F X lim F X α (x = F X (x {x m }, {y m } k k lim x m =, lim y m = x m, y m F X m k m m k < x m, y m < k (6.14 F X (y m F X (x m = lim k (F X α k (y m F Xα k (x m = lim k P (x m < X αk y m lim if k P ( k X αk k 1 ε lim {F X(y m F X (x m } 1 ε. lim F X(y =, lim F X(x = 1 m y x (2 (1 F Xα F Xα Helly ( 6.36 F Xα F Xαk F F x ε > M > lim F X α (x = F (x k k if k P (X α k [ M, M] 1 ε F (x x, y x < M, M < y F (y F (x = lim (F X α (y F Xα (x = lim P (X k k α k (x, y] k k if P (X α k [ M, M] > 1 ε k F (x F X (x = F (x X 6.5 6.37 {X } X ϕ (t (1 {X } X t ϕ (t ϕ X (t (2 t ϕ (t ϕ(t ϕ(t t = ϕ(t X {X } X 26

: (1 f(x = e itx cos x, si x (2 1st step 6.35 {X } compact X µ a 1 = si a a si 1 = si a a c = 1/(1 si 1 > M > [ P ( X M E c (1 = c si X M X M ( 1 si x M x M ( = c µ (dx 1 2 2 2 ] [ ( 1 { X ce 1 M 1} µ (dx = c 1 e ix t M 1 si 1 = 1 si a a ( 1 1 1 2 1 dt µ (dx si X M X M ] 1 si 1 e it x M dt µ (dx ( = c 1 1 2 1 1 [ 1 e itx dt = 2 1 2ix eitx] 1 = eix e ix = si x t= 1 2ix x, 1 1 ϕ ( t M dt 3 e ix t M 1 Fubii I (M s = t/m ( I (M = c 1 1 1/M 2 1/M ( = c 1 M 2 1/M 1/M I (1 (M + I (2 (M ϕ (sm ds ϕ(s ds + c M 2 1/M 1/M (ϕ(s ϕ (s ds ε > ϕ(s s = ϕ( = 1 M > I (1 (M < ε 2 M lim ϕ (s = ϕ(s ( s ϕ(s ϕ (s ϕ(s + ϕ (s 2 Lebesgue lim I(2 (M = = I (2 (M < ε 2 if P ( X M 1 ε 6.35 {X } compact 2d step {X } {X } 1st step {X } X {X } X (1 ϕ (t ϕ X (t ϕ X (t = ϕ(t Y ϕ X (t = ϕ Y (t 6.21 µ X = µ Y µ X µ X {X } X 27

6.6 (cetral limit theorem 6.38 ( X 1, X 2,... i.i.d. E[X ] = m, V (X = σ 2 > U = 1 σ k=1 (X k m {U } N(, 1 Y : Z = X m σ lim P (a U b = 1 b 2π a e y2 2 dy, < a < b <. (6.15 E[Z ] =, V (Z = 1, U = 1 k=1 Z k U ϕ U (t = E[e i t k=1 Z k ] = k=1 ( ( E[e i t Z k t ] = ϕ ϕ(t Z 1 (Z k ϕ(t = ϕ Zk (t E[Z 2 1] < 6.1 ϕ(t C 2 - Taylor ( t ϕ = ϕ( + t ϕ ( + 1 ( t 2ϕ ( θ t, < θ < 1 2 ϕ( = 1, ϕ ( = E[Z 1 ] = ϕ (θ t = E[Z1e 2 iθ t ] Z 2 1 e iθ t Z 2 1 Lebesgue lim ϕ ( θ t = E[Z 2 1] = 1. ( ( t ( lim ϕ = lim 1 + 1 t 2 2 ϕ ( θ t = e t2 2 Y N(, 1 ϕ Y (t = e t2 2 6.37(2 {U } Y (6.15 Y F Y (x 6.32 6.39 X 1, X 2,... (cf. 5.9, 5.21, 5.25 (cf. [D] p.127, Example3.4.5 6.4 (de Moivre-Laplace p Beroulli X 1, X 2,... : X 1, X 2,... P (X = 1 = p, P (X = = 1 p. S = k=1 X k ( lim P a S p b = 1 p(1 p b e y2 2 dy, < a < b <. (6.16 2π a : E[X k ] = p, V (X k = p(1 p 6.38 28

6.41 (1 6.4 S B(, p (6.16 B(, p S (cf. 6.42 (2 de Moivre-Laplace S B(, p Stirlig (6.16 p.17 (cf. 34(1 6.42 72 6 13 15 S 6 S B(72, 1 6 P (13 S 15 E[S] = 12, V (S = 72 1 6 (1 1 S 12 6 = 1 Z := 1.1573 ( 13 12 P (13 S 15 = P S 12 15 12 = P (1 Z 3 1 1 1 = P (Z > 1 P (Z > 3 =.1586.1349 =.157251 6.43 2 P (a S b P (a.5 S b + 5 6.42 ( 129.5 12 P (13.5 S 15 +.5 = P S 12 15.5 12 = P (.95 Z 3.5 1 1 1.1699 Maple * 5 = P (Z >.95 P (Z > 3.5 =.171.1144 =.169856 P (13 S 15 = P (S 15 P (S 129 =.9984997678.8292544225 =.1692453453 6.44 X 1, X 2,... Poisso P o(1 6.25 S := X 1 + + X Poisso P o( E[X 1 ] = 1, V (X 1 = 1 S P (S = k= ( lim P S = 1 e y2 1 2 dy = 2π 2 k e k! lim (1 e + + 2 2! + + = 1! 2 *5 S B(, p P (S k stats[statevalf,dcdf,biomiald[,p]](k; 29

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