i

i 1 1 1.1..................................... 1 1.2........................................ 3 1.3.................................. 4 1.4..................................... 4 1.5...................................... 5 2 7 2.1...................................... 7 2.2......................................... 9 2.3 (random variable)........................... 11 2.4........................................ 12 2.5..................................... 15 2.6........................... 17 2.7........................... 19 2.8..................................... 22 2.9....................................... 29 2.10...................................... 32 3 37 3.1....................................... 37 3.2....................................... 39 3.3..................................... 39 3.4................................. 41 3.5.................................. 47 3.6........................... 50 3.7 (renewal process)........................... 51 4 54 4.1.................................. 54 4.2............................... 55

ii 4.3............................. 64 4.4......................... 71 4.5..................................... 74 4.6.................................. 77 5 83 5.1................................... 83 5.2..................................... 84 5.3....................................... 91 5.4....................................... 98 5.5................................. 101 5.6 branching process............................. 105 6 112 6.1................................. 112 6.2................................... 114 6.3..................................... 115 6.4............................. 116 6.5....................................... 118 6.6..................................... 119 7 126 7.1................................. 126 7.2 M/M/1............................... 127 7.3 M/M................................. 131 7.4............................ 136 7.5.................................. 140 7.6........................... 144 7.7......................... 149

1 1.1 1.1 225 n n X n {X 0, X 1, X 2,...} 1 225 1.2 1.3

2 1.4 1.5 2 1.6 +1 1 3

3 1.2 t X(t) T {X(t), t T } 1 0 X(t) S T 1.2.1 T = [0, ) {X(t), t 0} T = [a, b] T = (, ) T = {0, 1, 2,...} {X n, n = 0, 1, 2,...} T = {t 0, t 1, t 2,...} X n n 1 0 n X(t) t t 1.2.2 {X(t), t T } S 1 0 X(t) = s S t s X(0) 1.2.3 {t n } {X(t n ), n = 0, 1, 2,...} n {X n, n = 0, 1, 2,...}

4 {X(t), t 0} 1.3 [0, t] C t + s A P (X(t) A C) P (A) = P (A C i ) = P (A C i )P (C i ) (1.1) i i {X(u) = x(u)(0 u s)} t {X(t) A} s Occam s razor 1.3.1 1.4 {X(t), t T } {x(t), t T } t {X(t), t T } 1.7 X(t) t

5 t (0, X(0)), (1, X(1)), (2, X(2)),... 1.8 X(t) t t X(t) = 0 X(t) = 1 X(t) 1 0 1.1 X(t) t X(t) 1.2 {X(t), t T } 1.5 1.9 4 1 1 n X n {X n, n = 0, 1, 2,...} 0, 1, 2, 3, 4 ±1 k(= 0, 1, 2, 3) k + 1 k(= 1, 2, 3, 4) k 1 8 4 X(t) t (= 1) (= 0) 2 0, 1 0 1, 1 0 1.3 5 3

6 1.1 {X(t), t T } 1.2

7 2 2.1 2.1.1 Ω 2.1 2 4 4 {, } 2 1 { } 2 A A A A c Ω Ω φ A, B A, B A B A, B A, B A B A B A c B c (A B) c = A c B c 2.1 A, B A A c = φ A = (A B) (A B c ) A B = A (B A c ) 2.1.2

8 Ω 2 Ω 2 Ω,, c 2 Ω A, B A B, A B, A c 2 Ω 2 Ω 2.2 (H) (T ) Ω = {H, T } 2 Ω = {Ω, {H}, {T }, φ} 2.3 2 Ω = {HH, HT, T H, T T } {Ω, {HH}, {HH} c, φ} 2.1.3 σ F σ σ σ sigma algebra 1. Ω F 2. A Ω A c Ω 3. A 1, A 2,... Ω A 1 A 2 Ω σ 2 3 σ,, c σ G σ(g) G σ 2 {HH} σ 1 Ω 2 {HH} c, Ω c (= φ) F = {Ω, {HH}, {HH} c, φ} σ F {HH} σ Ω = R [a, b] G σ(g) B F = B 2.4 A {Ω, A, A c, φ} A σ

9 2.5 a < b [ a, b n] 1 = [a, b) [a, b] G [a, b) σ(g) σ(g) = B 2.2 A, B, C A, B, C Ω = A B C A, B, C σ 2.2 2.1 σ F 0 1 P (.) F P (Ω, F, P ) 1. A F 0 P (A) 1 2. P (Ω) = 1 3. A, B F P (A B) = P (A) + P (B) (2.1) 3 P 2.6 2 Ω = {HH, HT, T H, T T } F F σ P (HH) = 0.4, P (HT ) = P (T H) = P (T T ) = 0.2 F P (A) A P (A) = P ({HT, T H}) = P (HT ) + P (T H) = 0.4 2.7 2 2 F = {Ω, {HH}, {HH} c, φ} P (HH) = 0.4 {T T } Ω F P (T T ) 2.3 P (A A c ) = 0 P (A) = P (A B) + P (A B c ) P (A B) = P (A) + P (B) P (A B) 2.2.1 1,2,3 Ω = {ω 1, ω 2,...}

10 1 {p 1, p 2,...} P ({ω i }) = p i A P (A) P (A) = p i (2.2) i:ω i A F = 2 Ω Ω 3 {p 1, p 2,...} 2.8 Ω = {1, 2,..., 6}, p 1 = = p 6 = 1 6 2.9 0 < p < 1 p Ω = {1, 2,...}, p i = (1 p)p i 1 (i = 1, 2,...) p 2.2.2 F Ω = R F = B f(x) f(x)dx = 1 (, x] F (x) F (x) = P ((, x]) = x f(x)dx (2.3) F F (x) a < b [a, b] (, b] = (, a] (a, b] P ([a, b]) = P ((a, b]) = F (b) F (a) = b a f(x)dx (2.4) F (.) f(x) 2.10 Ω = R f(x) = 1 2π e x2 /2, < x < (2.5) f(x) 2.11 Ω = [0, ) F (x) = 1 e λx, x 0 (2.6) f(x) = λe λx λ F (x) 1 F (x)

11 2.3 (random variable) ω X (ω) {ω} X(ω) 2.12 X {1, 2, 3, 4, 5, 6} X(1) = X(3) = X(5) = 1 X(2) = X(4) = X(6) = 1 {ω; X(ω) = 1} = {2, 4, 6} Ω, {ω; X(ω) = 1} = {1, 3, 5} Ω X 1 (1) = {2, 4, 6} Ω, X 1 ( 1) = {1, 3, 5} Ω 2.3.1 P (X = a) = i:x(ω i)=a p i (2.7) X = a X = 1 P ({2, 4, 6}) = 3 6 X = 1 1 2 1 5 2 P ({2, 4, 6}) = 3 8 X = 1 3 8 2.2 (probability mass function) X a 1, a 2,... P (X = a k ), k = 1, 2,... X a k P (X = a k ) a k f X (a k ) = P (X = a k ), k = 1, 2, 3,... (2.8) 2.3 (cummulative distribution function) X x x P (X x) X x F X (x) = P (X x) (2.9)

12 X [a, b] P (a X b) = F X (b) F X (a 0) 2.4 (probability density function) f X (x) = d dx F X(x) (2.10) 2.13 i p i {p i } X X P (a < X a + h) = F X (a + h) F X (a) = f X (a)h F X (b) F X (a) = b a f X (x)dx (2.11) 2.14 1 a b(> a) a b 0 < a < b < 1 P ([a, b]) = b a X X [0, 1] u log u Y 10 log u 10 u e 10 X e 10 P (Y 10) = e 10 2.4 2 X X X X 2.4 (expectation) {a 1, a 2,..}

13 X E(X) = k a k f(a k ) (2.12) f(x) X E(X) = E(X) = xf(x)dx (2.13) xdf (x) (2.14) Laplace-Stieltjes X 2 2.4.1 X g(x) g(x) g(x) X {b; g(a) = b, a X E(g(X)) = k g(a k )f(a k ) (2.15) E(g(X)) = g(x)f(x)dx E(aX + by ) = ae(x) + be(y ) (2.16) 2.4.2 2.5 E( X E(X) ) = x E(X) f(x)dx (2.17) 2.6 (varinace) 2 V (X) = E((X E(X)) 2 ) = E(X 2 ) (E(X)) 2 (2.18) 2.7 (standard deviation) S(X) = V (X) (2.19) X

14 2.8 (coefficient of variation) CV (X) = S(X) E(X) (2.20) V (ax + b) = a 2 V (X) (2.21) S(aX + b) = as(x) CV (ax) = CV (X) X E(X) 0 V (X) X E(X) X 0 1 ( ) ( ) X E(X) X E(X) E = 0, V = 1 (2.22) S(X) S(X) 2.15 A 1 0 1 A A P (1 A = 1) = P (A), P (1 A = 0) = 1 P (A) (2.23) E(1 A ) = P (1 A = 1) = P (A) 2.5 P (A) = p A 1 A 1 A 2.6 X f(x) = { ae ax, x 0 0, x < 0 2.4.3 2.9 X n n (moment) E(X n ) = k a n kf(a k ) (2.24) E(X n ) = x n f(x)dx (X E(X)) n n

15 E ((X E(X)) 3) = E ( X 3 3E(X)X 2 + 3(E(X)) 2 X (E(X)) 3) 2.7 X n 2 3 { 1, 0 x 1 f(x) = 0, 2.5 2.5.1 (generating function) {a k, k = 0, 1, 2,...} G(z) = a k z k (2.25) k=0 {a k, k = 0, 1, 2,...} generate a k = 1 k! 2.16 {1, 1, 1,...} z < 1 G(z) = a k = 1 d k dz k G(z) = 1 z=0 k! G(k) (0) (2.26) k=0 G (k) (z)) = 2.17 { 1, 1, 1 2, 1 3!, 1 4!, 1 5!,...} G(z) = a k z k = 1 1 z k=0 k! 1 z k+1 1 k! zk = e z G (k) (z)) = e z a k = 1 k! 2.5.2 (probability generating function) X {f(k), k = 0, 1, 2,...} {f(k), k = 0, 1, 2,...} X X f(k)z k z X G X (z) = E(z X ) = f(k)z k (2.27) k=0

16 f(k) = 1 k! d k dz k G X(z) (2.28) z=0 z = 1 d n dz n G X(z) = k(k 1) (k n + 1)f(k)z k n k=n d n dz n G X(z) = E (X(X 1) (X n + 1)) (2.29) z=1 n G X(1) = E(X(E 1)(X 2)) = E(X 3 ) 3E(X 2 ) + 2E(X) n n 2.8 n 1 p 0 < p < 1 n, p 2 f(k) = { ( n ) k p k (1 p) n k, k = 0, 1,..., n 0, 2.9 a > 0 1 f(k) = ak k! e a, k = 0, 1, 2,... 2.5.3 (moment generating function) e θx X M X (θ) = E(e θx ) (2.30) X X k m k {m k /k!, k = 0, 1, 2,...} k=0 m k k! θk = k=0 ( E(X k ) ) θ k X k = E k! k! θk = E ( e θx) k=0 n θ = 0 n E(X n ) = dn dθ n M(θ) (2.31) z=0

17 2.1 X, Y 2.10 X M X (θ) ax + b e bθ M X (aθ) 2.6 X X n, p np 2.6.1 Bernoulli trial p p n n 2.6.2 (Bernoulli distribution) p 1 p X X 0 1 P (X = 1) = p, P (X = 0) = 1 p (2.32) p p(1 p) 1 p + pz 2.6.3 (binomial distribution) p n n, p 2 X X 0 n ( ) n P (X = k) = p k (1 p) n k, k k = 0, 1, 2,..., n (2.33) n n, p = 0.5 2 np np(1 p) (1 p + pz) n 2.6.4 (geometric distribution) p X X X X

18 p P (X = k) = (1 p) k 1 p, k = 1, 2, 3,... (2.34) p X X p p 1 p 2 (1 p) pz/(1 (1 p)z) 2.11 p pz/(1 (1 p)z) 2.6.5 (Poisson distribution) n, p np = λ n X X λ P (X = k) = λk k! e λ, k = 0, 1, 2,... (2.35) 0 n, p np = λ, n 1 ( n! P (X = k) = k!(n k)! pk (1 p) n k λk 1 λ ) n λk k! n k! e λ λ 2.12 λ X λ 2.6.6 (uniform distribution) n X {1, 2,..., n} P (X = k) = 1, k = 1, 2,..., n (2.36) n 2.13 X {1, 2,..., 6} 2.6.7 (multinomial distribution) m n k X k (X 1, X 2,..., X m ) (X 1, X 2,..., X m ) (X 1, X 2,..., X m ) n, p 1,..., p m

19 n! P (X 1 = n 1, X 2 = n 2,..., X m = n m ) = n 1!n 2! n m! pn 1 1 pn 2 2 pnm m (2.37) n 1, n 2,..., n m n 1 + n 2 + + n m = n m = 2 X k np k np k (1 p k ) X j, X k np j p k 2.7 2.7.1 (normal distribution) X X µ, σ f(x) = 1 e (x µ)2 /(2σ 2) = 1 ) (x µ)2 exp ( 2πσ 2πσ 2σ 2, < x < (2.38) µ σ 2 X µ, σ (X µ)/σ Φ(z) Φ(z) = z 1 2π e x2 /2 dx (2.39) Excel =NORM.S.DIST(z,true) 2.18 µ, σ 2 µ, σ 2 X µ, σ 2 Z = X µ σ Z M Z (θ) = E(e θz ) = e θx 1 2π e x2 /2 dx = e θ2 /2 1 2π e (x θ)2 /2 dx = e θ2 /2 θ, 1 1 X = σz + µ M X (θ) = E(e θ(σz+µ) ) = E(e (θσ)z )e θµ = e θµ M Z (θσ) = exp ) (µθ + σ2 2 θ2 2.14 (1)Excel R µ = 0, σ = 1 Excel NORM.S.DIST R dnorm, pnorm (2) µ = 4, σ = 4 Excel R curve(dnorm(x),-3,3)

20 2.7.2 (exponential distribution) λ > 0 λ f(x) = λe λx, x 0 (2.40) λ 1 λ 2 p t T ( t) p/ t t 0 P (T ( t) > t) = (1 p) t/ t = (1 λ t) t/ t e λt X P (X > t + s X > s) = P (X > t), t, s > 0 (2.41) 2.15 λ λ 1, λ 2 2.16 (1)Excel R λ = 1 [0, 3] Excel EXPON.DIST R dexp (2) [5, 8] 2.7.3 (gamma distribution) a, b > 0 a, b f(x) = ab Γ(b) xb 1 e ax, x > 0 (2.42) Γ(t) Γ(t) = 0 x t 1 e x dx (2.43) t Γ(t) = (t 1)! b = 1 b > 1 0 < b < 1 x 0 y J b/a b/a 2 b b b = m f(x) = a m (m 1)! xm 1 e ax, x > 0 2.19 a, b b/a, b/a 2

21 X a, b M X (θ) = = e θx ab 0 ) b ( a a θ Γ(b) xb 1 e ax dx = ( a ) b a θ 0 (a θ) b x b 1 e (a θ)x dx Γ(b) a θ, b 1 d dθ M X(θ) = b ( ) b+1 ( ) b+2 a, d2 a a θ dθ 2 M b(b + 1) a X(θ) = a 2 a θ E(X) = d dθ M X(0) = b a V (X) = d2 dθ 2 M X(0) (E(X)) 2 = b a 2 2.17 Excel R (a, b) = (1, 0.5), (1, 1), (1, 2), (1, 5), (1, 10), (2, 5) Excel GAMMA.DIST R dgamma 2.7.4 [a, b] f(x) = { 1 b a, a x b 0, otherwise (2.44) (a + b)/2 (b a) 2 /12 2.18 [0, 1] X [a, b] Y X Y Y 2.7.5 p, q f(x) = { Γ(p+q) Γ(p)Γ(q) xp 1 (1 x) q 1, 0 < x < 1 0, otherwise (2.45) p, q > 0 p = q = 1 (0, 1) p, q < 1 U p, q > 2 2.19 Excel R (p, q) = (0.3, 0.4), (1, 1), (1, 3), (2, 4), (5, 10) Excel BETA.DIST R dbeta

22 2.7.6 (log-normal distribution) µ σ 2 X Y = e X P (Y y) = P (X log y) = f Y (y) = 1 exp ( 2πσy log y (log y µ)2 2σ 2 1 2πσ e (x µ)2 /2σ 2 dx ), y > 0 (2.46) ( ) e µ+σ2 /2 e 2µ+σ2 e σ2 1 2.20 log Y µ, σ 2 Y 2.21 µ, σ 2 ( ) e µ+σ2 /2, e 2µ+σ2 e σ2 1 2.22 Excel R (µ, σ) = (0, 1) (µ, σ) = (1, 1), (0, 2) Excel LOGNORM.DIST R dlnorm 2.8 A H H A H A P (A H) P H (A) P (A H) = P H (A) = P (A H) P (H) (2.47) H P (A H) = P (A H)P (H) (2.48) P (.) P P H P (. H) P H (A A c ) = 0 P (A A c H) = 0 P H (A) = P H (A B) + P H (A B c ) P H (A B) = P H (A) + P H (B) P H (A B)

23 A, B P (A B) = P (A) + P (B) P (A B) A A H B B H P ((A H) (B H)) = P ((A B) H) = P (A H) + P (B H) P (A B H) P (H) 2.23 2.8.1 B 1, B 2,..., B n n k=1 B k A n n P (A) = P (A B k ) = P (A B k )P (B k ) (2.49) k=1 k=1 2.8.2 A B B A P (A B) = P (A) P (A B) = P (A)P (B) (2.50) A, B A, B, C P (A B) = P (A)P (B) P (A C) = P (A)P (C) P (B C) = P (B)P (C) P (A B C) = P (A)P (B)P (C) n A 1, A 2,..., A n n 1 P (A 1 A 2 A n ) = P (A 1 )P (A 2 ) P (A n ) m; i 1, i 2,..., i m P (A i1 A i2 A im ) = P (A i1 )P (A i2 ) P (A im )

24 X\Y 0 1 2 0 (1 p) 3 p(1 p) 2 0 (1 p) 2 1 p(1 p) 2 p(1 p) p 2 (1 p) 2p(1 p) 2 0 p 2 (1 p) p 3 p 2 2.8.3 (joint distribution) f X,Y (a j, b k ) = P (X = a j, Y = b k ) (2.51) X = a j Y = b k, F X,Y (x, y) = P (X x, Y y) (2.52) f X,Y (x, y) = 2 x y F X,Y (x, y) (2.53) f X,Y (x, y)dxdy = P (x < X x + dx, y < Y y + dy) f X,Y (a j, b k ) = P (X = a j ) (2.54) k f X,Y (a j, b k ) = P (Y = b k ) j F X,Y (x, ) = x ( ) f X,Y (x, y)dy dx = P (X x) (2.55) f X,Y (x, y)dy = f X (x) (2.56) 2.20 p 3 2 X 2 Y (1) X, Y (2) X (3) Y (1)(2) (3) Y 0, 1, 2 P (Y < 0) = 0, P (Y < 1) = (1 p) 2, P (Y < 2) = 2p(1 p), P (Y 2) = 1

25 2.8.4 X, Y Y = b k X = a i P (X = a i Y = b k ) = P (X = a i, Y = b k ) P (Y = b k ) f X Y (a i b k ) = P (X = a i Y = b k ) (2.57) f X Y (a b) = f X,Y (a, b) f Y (b) (2.58) 2.8.5 X, Y F X,Y (x, y) X, Y F X (x), F Y (y) x, y X, Y F X,Y (x, y) = F X (x) F Y (y) (2.59) {X x}, {Y y} n X 1, X 2,..., X n X 1, X 2,..., X n F X1,X 2,...,X n (x 1, x 2,..., x n ) = F X1 (x 1 ) F X2 (x 2 ) F Xn (x n ) (2.60) n n 1 n X 1, X 2,..., X n i.i.d. (independent, identically distributed) 2.8.6 X, Y X + Y X, Y f X+Y (n) = f X (k)f Y (n k), (2.61) k

26 f X+Y (x) = f X (u)f Y (x u)du = (f X f Y ) (x), (2.62) (convolution) X 1, X 2,..., X n i.i.d. X 1 + X 2 + + X n f ( n) (x) ( ) ( ) ( ) E(z X+Y ) = f X+Y (i)z i = E(z X )E(z Y ) = f X (i)z i f Y (k)z k i i k z n f X+Y (n) = n f X (i)f Y (k) = f X (i)f Y (n i) i+k=n i=0 2.8.7 E(g(X, Y )) = j,k g(a j, b k )f X,Y (a j, b k ) (2.63) E(g(X, Y )) = X, Y g(x, y)f X,Y (x, y)dxdy (2.64) E(X + Y ) = E(X) + E(Y ) (2.65) E(XY ) = E(X)E(Y ) (2.66) X 1, X 2,..., X n i.i.d. S n = X 1 + X 2 + + X n X n M Sn (θ) = E(e θ(x1+x2+...+xn) ) = n M Xk (θ) = (M X (θ)) n (2.67) 2.21 X 1, X 2,..., X n p S n = X 1 + X 2 + + X n M Sn (θ) = E ( e θs n ) = n k=1 k=1 E ( e θx ) ( k = pe θ + 1 p ) n

27 Z n, p M Z (θ) = n ( ) n e θk p k (1 p) n k = ( pe θ + 1 p ) n k k=0 S n n, p 2.24 2.25 X 1, X 2,... f(x) = p x (1 p) 1 x, x = 0, 1 X 1 + X 2 + + X n 2.8.8 2.10 (covariance) C(X, Y ) = E(X E(X))(Y E(Y )) = E(XY ) E(X)E(Y ) (2.68) 2.11 (correlation coefficient) ( X E(X) R(X, Y ) = E S(X) 1 1 ) Y E(Y ) = C(X, Y ) S(Y ) S(X)S(Y ) (2.69) 0 0 0 C(X, X) = V (X) C(X, Y ) = C(Y, X) C(X, ay + b) = ac(x, Y ) ( ) V X i = V (X i ) + 2 i, X j ) i i i>jc(x 2.26 1 1 z V (zx + Y ) 0

28 2.8.9 Y = y X E(X Y = y) = xf X Y (x y)dx y Y = y g(y) = E(X Y = y) g(y ) E(X Y ) g(y )(= E(X Y )) E(E(X Y )) = E(X Y = y)f Y (y)dy ( ) = xf X Y (x y)dx f Y (y)dy = = = x ( f X Y (x y)f Y (y) ) dxdy xf X,Y (x, y)dxdy = xf X (x)dx = E(X) ( ) xdx f X,Y (x, y)dy E(X) = E(E(X Y )) (2.70) A 1 A X E(1 A Y = y) = P (A Y = y) P (A) = E(P (A Y )) (2.71) V (X) = E(V (X Y )) + V (E(X Y )) (2.72) 2.27 V (X) = E(X 2 ) E(X) 2 (2.70) 2.8.10 X, Y X, Y µ X, σ 2 X, µ Y, σ 2 Y, ρ ( 1 f X,Y (x, y) = 2πσ X σ exp 1 Y 1 ρ 2 2(1 ρ 2 ) ( ) 2 ( ) ( x µx x µx y µy η(x, y) = 2ρ σ X σ X σ Y ) η (x, y), < x, y < (2.73) ) ( ) 2 y µy + σ Y

29 E(X) = µ X, V (X) = σ 2 X E(Y ) = µ Y, V (Y ) = σ 2 Y C(X, Y ) = ρσ X σ Y, R(X, Y ) = ρ X, Y ( ) 1 ρ Σ = Σ 1 = 1 ρ 1 1 ρ 2 ( 1 ρ ρ 1 ) ( ) t x = x µx σ X, y µ Y σ Y 1 2(1 ρ 2 ) η(x, y) = 1 2 xt Σ 1 x X 1, X 2,..., X n a 1, a 2,..., a n n k=1 a kx k X k X 1, X 2,..., X n X, Y X, X + Y 2.28 X, Y (1)X µ X σ 2 X (2)X, Y ρ (3)Y = y X µ X + ρσ X (y µ Y )/σ Y σ 2 X (1 ρ2 ) 2.29 Excel R Excel µ Y, σ Y =NORM.INV(RAND(),µ Y, σ Y ) y µ X + ρσ X (y µ Y )/σ Y σ X 1 ρ 2 x (x, y) ρ R y=rnorm(n) x=rnorm(n,y*r,sqrt(1-r^2)) (x,y) r ρ [ 1, 1] 2.9 1 X 1, X 2,... µ ε > 0 ( ) lim P X 1 + X 2 +... + X n µ n n < ε = 1 (2.74)

30 5 2 2 X 1, X 2,... µ X 1 + X 2 +... + X n lim n n = µ w.p.1 (2.75) w.p.1 (with probability one) ( ) X 1 + X 2 +... + X n P lim = µ = 1 (2.76) n n 3 X 1, X 2,... µ σ 2 lim P n ( X1 + X 2 +... + X n nµ nσ Φ(z) Z n = X 1 + X 2 +... + X n nµ nσ = ) < z = Φ(z) (2.77) n i=1 (X i µ) nσ Z n X i µ M X (θ) M Zn (θ) = E(e θz n ) = ( E e θ(x i µ)/ ) ( ) n n/σ θ = M X nσ ( ) θ n exp 2 2 log M Zn (θ) θ2 2 ( ) θ log M Zn (θ) = n log M X nσ n x = n 0.5 lim log M log M X Z n (θ) = lim n x 0 x 2 ( θ σ x) = lim x 0 g(x) x 2

31 1 g (x) = θ σ d dx M ( θ X σ ( x) M θ X σ x) g (0) = 0 d dx M X (0) = E(X i µ) = 0 g (x) = ( θ 2 d 2 σ 2 dx M θ 2 X σ x) ( θ ( M θ X σ x) σ g (0) = θ 2 d dx M X ( θ σ x) M X ( θ σ x) d2 dx 2 M X (0) = V (X i µ) = σ 2 2 2 lim log M G n n (θ) = θ2 2 4 S n n, p lim P n ( ) S n np < z np(1 p) ) 2 = Φ(z) (2.78) X 1, X 2,... P (X i = 1 p) = p, P (X i = p) = 1 p Z n = S n n np i=1 = X i np(1 p) np(1 p) Z n X i M X (θ) M Zn (θ) = E(e θzn ) = ( E e θxi/ ) ( ) n np(1 p) θ = M X np(1 p) ( ) θ n exp 2 2 log M Zn (θ) θ2 2 A = ( ( ) ( )) θ(1 p) θp log M Zn (θ) = n log p exp + (1 p) exp np(1 p) np(1 p) 1 p p ( ( ) ( θa = n log p exp n + (1 p) exp θ )) A n n x = n 0.5 ( ( lim log M log p exp (θax) + (1 p) exp θ A Z n (θ) = lim x)) g(x) n x 0 x 2 = lim x 0 x 2

32 1 g (x) = pθa exp (θax) (1 p) θ A exp ( θ A x) p exp (θax) + (1 p) exp ( θ A x) g (0) = 0 g (x) = pθ2 A 2 exp (θax) + (1 p) θ 2 A exp ( θ 2 A x) p exp (θax) + (1 p) exp ( θ A x) ( pθa exp (θax) (1 p) θ A exp ( θ A x)) 2 ( ( p exp (θax) + (1 p) exp θ A x)) 2 g (0) = θ 2 2 2 lim log M Z n (θ) = θ2 n 2 2.30 Excel R 100 1000 10000 R mean(sample(2,10000,replace=t))-1 10000 2.31 Excel R n 0.5 1/ 12n n = 100, 1000, 10000 n 0.5 1/ 12n 1 0 Excel 2.10 2.10.1 (a + b) n = n k=0 ( ) n a k b n k (2.79) k ( ) n 1 + k 1 ( ) n = m n 1 k=m 1 ( ) n 1 = k ( ) k m 1 ( ) n k (2.80) (2.81)

33 0 < m min{n, M} ( m N )( M ) k m k ) = 1 (2.82) k=0 ( N+M m 2.10.2 e x = 1 + x + x2 2! + x3 3! + = e x + e x x 2k = 2 (2k)! k=0 e ix = cos x + i sin x = k=0 k=0 ( 1) k x 2k (2k)! x k k! + i k=0 ( 1) k x 2k+1 (2k + 1)! (2.83) (2.84) (2.85) r < 1 r n = 1 1 r n=0 (2.86) (n + 1)r n 1 = (1 r) 2 (2.87) n=0 (n + 2)(n + 1)r n = n=0 2 (1 r) 3 (2.88) 2.10.3 f(x), g(x) x = a f(a) = g(a) = 0 f(x) lim x a g(x) = lim f (x) x a g (x) (2.89) F (x) lim x(1 F (x)) = 0 (2.90) x ( lim 1 + a ) n = e a n n (2.91) n n! nn 2πn e n n 1 2.10.4 F (x)g(x)dx = G(x)F (x) F (x)g (x)dx (2.92)

34 ae ax dx = e ax a 2 xe ax dx = axe ax + ae ax dx = axe ax e ax a k x k 1 e ax dx = a k 1 x k 1 e ax + (k 1) a k 1 x k 2 e ax dx 2.10.5 Γ(t) = I 2n = 0 x t 1 e x dx = (t 1)Γ(t 1) (t > 1) (2.93) x 2n e x2 /2 dx I 2n = (2n 1)I 2n 2 = (2n 1)!! 2π (2.94) t Γ(n) = (n 1)! Γ(0.5) = π I 2n = I 2n 1 = 0 n!! n(n 2)(n 4) x 2n e x2 /2 dx I 2n = (2n 1)I 2n 2 = (2n 1)!! 2π 2.10.6 d d (f(x)g(x)) = g(x) dx dx d dx f(g(x)) = d dy f(y) d dx g(x) d dx f 1 1 (x) = f (f 1 (x)) f(x) + f(x) d dx g(x) (y = g(x)) d dx d dx d dx x a a x g(x) a f(u)du = f(x) f(u)du = f(x) f(u)du = f(g(x))g (x)

35 1. 2. 3. 2.1 n, p 2 X m, p 2 Y X + Y n + m, p 2 2 2.2 15 2 2 2.3 2 3 12 7 11 4 4 7 4 7 5, 6, 8, 9, 10 4 7 7 2.4 1 ux + Y u 2.5 X, Y λ X + Y = z X 2.6 λ k X k X 1, X 2,..., X n Y J Y J 2.7 X 1, X 2,... λ N p X 1 + X 2 +... + X N 2.8

36 t B A X f(x) Z, W { A if Z < t Z = min{t, X}, W = B if Z = t (1)Z (2)W (3)Z (4)W (5)X λ E(W )/E(Z) t 2.9 (1) X n, p 2 Y X = m m, r 2 Y = k (2) X a Y X = m m, r 2 Y = k (3) np = a n (1) (2) 2.10 X i, Y i p, r Z i = X i Y i n i=1 Z i

37 3 T i i N(t) T i 1 N(t) T i t T i [0, t] {N(t), t 0} 3.1 2 3.1.1 (Bernoulli trial) 1 p 3.1 X 1, X 2,... P (X n = 1) = p, P (X n = 0) = 1 p, n = 1, 2, 3,... (3.1) {X n, n = 1, 2, 3,...} p {X n, n = 1, 2, 3,..., m} m X n = 1 n X n = 0 {X n, n = 1, 2, 3,...} {0, 1} X n, n = 1, 2, 3,...} ( ) n P (X 1 = x 1,..., X n = x n ) = p m (1 p) n m, m = x k (3.2) 3.1 k=1

38 3.1.2 (binomial process) 3.2 {X n, n = 1, 2, 3,...} p {X i } S 0 = 0 S n = X 1 + X 2 + + X n = S n 1 + X n, n = 1, 2,... (3.3) {S n, n = 0, 1, 2,...} p S n n {S n, n = 0, 1, 2,...} n 0 n 3.1 S n n, p ( ) n P (S n = k) = p k (1 p) n k, k k = 0, 1,..., n (3.4) 3.2 S n n {S n } 1000 S 1000 1000 S n = 1000 n 3.1 S n n, p 3.2 S n = i, S n+m = k m > 0, k i 3.1.3 k T k T k k T 0 = 0 Y k = T k T k 1, k = 1, 2,... (3.5) Y k k 1 k Y 1 > n n P (Y 1 > n) = (1 p) n P (Y 1 = n) = P (Y 1 > n 1) P (Y 1 > n) = (1 p) n 1 p Y 1 p Y 2, Y 3,...

39 3.2 p p 3.1.4 (memoryless property) Y p P (Y > n + m Y > m) = P (Y > n + m) P (Y > m) = (1 p) n = P (Y > n) (3.6) Y m n t 2 3.2 counting process 2 2 1, 2, 3,... ATM A N(t) [0, t] A {N(t), t 0} N(t) A 1 {N n, n = 0, 1, 2,...} 3.3 3.3.1 {N(t), t 0} t ((n 1) t, n t] n

40 {Ñ(n t), n = 0, 1, 2,...} t n t t Ñ(n t) N(t) 2 t t = n t 2 E(Ñ(n t)) = np = t t p V (Ñ(n t)) = np(1 p) = t p(1 p) t p t E(Ñ(n t)) np n Ñ(n t) N(t) n n np {N(t), t 0} λ E(N(t)) = λt E(Ñ(n t)) = p t t t 0 n p 0 t t {N(t), t 0} {Ñ(t), t 0} t = n t P (N(t) = k) P (Ñ(n t) = k) = ( n k ) p k (1 p) n k = t 0 n k ( ) n (λ t) k (1 λ t) n k k ( ) k P (N(t) = k) nk λ (1 k! n t λn ) n t (1 (λt)k λn ) n k! t n ( lim 1 + a ) n = e a n n P (N(t) = k) = (λt)k e λt, k = 0, 1, 2,... k! t = n t n t λ {N(t), t 0} N(t) λ Poisson P (N(t) = k) = (λt)k e λt, k = 0, 1, 2,... (3.7) k!

41 t 0 0 t( 0) 3.3.2 3.3 {N(t), t 0} λ (1) N(0) = 0 (2) 0 t 1 < t 2 t 3 < t 4 N(t 2 ) N(t 1 ) N(t 4 ) N(t 3 ) (3) t > 0, s 0 N(s + t) N(s) λt P (N(s + t) N(s) = k) = (λt)k e λt, k = 0, 1, 2,... (3.8) k! N(s + t) N(s) (s, t + s] x s + t increment 3.3 3.4 3.5 3.4 3.4.1 t t

42 P (N(s + t) N(s) = 0) = e λt = 1 λ t + = 1 λ t + o( t) P (N(s + t) N(s) = 1) = λ te λt = λ t(1 λ t + ) = λ t + o( t) P (N(s + t) N(s) 2) = 1 P (N(s + t) N(s) < 2) = o( t) 0 t t λ t [0, t] N(t) λt (law of rare events) 3.6 3.7 3.3 A4 30 6 20 1 10 3 1 3.4.2 N(t) = 0 T 1 Y 1 t P (T 1 > t) = P (Y 1 > t) = P (N(t) = 0) = e λt (3.9) λ λ 1 T 1 λ P (Y > t + s Y > s) = P (T > t) (3.10) Y Y

43 ipod ipod 3.4 Y λ Y 3.5 30 1 (1)9 12 (2) 3.6 2 30 1 3.4.3 5 X, Y λ, µ min{x, Y } λ+µ X 1, X 2,..., X n λ 1, λ 2,..., λ n min{x 1, X 2,..., X n } λ 1 + λ 2 +... + λ n {min(x, Y ) > t} X > t Y > t X, Y P (min{x, Y } > t) = P (X > t, Y > t) = P (X > t)p (Y > t) = e (λ+µ)t (3.11) min{x, Y } λ + µ 6 X, Y λ, µ P (X < Y ) = λ λ + µ (3.12) λ : µ X < Y X > Y X P (X < Y ) = 0 e µt λe λt dt = λ λ + µ 7 N p X 1, X 2,... λ X 1 + X 2 +... + X N λp

44 E(e θx ) = λ/(λ θ) X 1 + X 2 +... + X N λp/(λp θ) E(e θ(x 1+X 2 +...+X N ) ) = E(E(e θ(x 1+X 2 +...+X N ) N)) = ( E(e θx ) ) n (1 p) n 1 p = = λp λp θ pe(e θx ) 1 (1 p)e(e θx ) λp X 1 + X 2 +... + X N λp 3.7 MD 5 MD 2 MD 3.8 2 A, B, C 3 A, B C C (1) 3 (2) 1 3 5 1/3 (3) 3 3.9 2 3 A, B C 3.4.4 n 1 N(t) = 1 [0, t] 1 [0, t/2] 5 P (T 1 < x N(t) = 1) = x t, 0 x t (3.13) [0, t] 2 U 1, U 2,..., U n [0, t] U (1) U (2)... U (n) N(t) = n T 1, T 2,..., T n U (1), U (2),..., U (n) 3.8 100 [0, t]

45 β N(t) N(t) E 100e βt k = E E 100e βt k N(t) k=1 k=1 ( n ) E 100e βt k = ne ( 100e βu ) 1 = 100n k=1 k=1 = 100n βt N(t) E 100e βt k = n=0 = 100λ β ( 1 e βt ) 100n βt t 0 ( 1 e βt ) (λt) n ( 1 e βt ) n! e βu du t e λt 3.10 T 1 P (T 1 < x N(t) = 1) = x t (N(t) = 1) (T 1 t < T 1 + T 2 ) 3.11 2 λ {N(t), t 0} 1 2 T 1, T 2 U 1, U 2 [0, 1] U (1) = min{u 1, U 2 }, U (2) = max{u 1, U 2 } N(1) = 2 (U (1), U (2) ) (T 1, T 2 ) 3.12 3 20 1 3.4.5

46 3.9 3.4.6... N(t), M(t) λ, µ E(z N(t)+M(t) ) = E(z N(t) )E(z M(t) ) = e λt(1 z) e µt(1 z) = e (λ+µ)t(1 z) N(t) + M(t) λ + µ 3.13 N(t), M(t) λ, µ N(t) + M(t) λ p 1 p N 1 (t) N 1 (t) λp p N 2 (t) N 2 (t) N 1 (t), N 2 (t) N(t) = N 1 (t) + N 2 (t)

47 P (N 1 (t) = j, N 2 (t) = k) = P (N 1 (t) = j, N(t) = j + k) = P (N 1 (t) = j N(t) = j + k)p (N(t) = j + k) ( j + k = )p j (1 p) k (λt)j+k j (j + k)! e λt = (λpt)j j! (λ(1 p)t) k e λt = P (N 1 (t) = j)p (N 2 (t) = k) k! 3.14 2 10 2 2 3.15 {N(t), t 0} λ {L(t), t 0} µ {N(t) + L(t), t 0} λ + µ 3.16 {N(t), t 0} λ {M(t), t 0} {M(t), t 0} 3.17 {N(t), t 0} λ 3.5 {N(t), t 0} Y 1, Y 2,... Z(t) = Y 1 + Y 2 +... + Y N(t), t 0 (3.14) Z(t) {Z(t), t 0} compound Poisson process Y 3.10

48 1 3.11 1 1 3.3 Z(t) t Y [0, t] E(Z(t)) = E(E(Y 1 + Y 2 +... + Y N(t) N(t)) (3.15) = ne(y ) (λt)n e λt = E(Y )E(N(t)) n! n=0 3.18 1 100 1 24 0.1 1 3.5.1 N(t) Z(t) M Z(t) (θ) = E(e θz(t) ) = E(E(e θ(y1+y2+...+yn(t)) N(t)) (3.16) = E(e θy ) n P (N(t) = n) (3.17) n=0 Y M Y (θ) = E(e θy ) N(t) G N(t) (z) M Y (θ) N(t) at G N(t) (z) = z n (λt)n e λt = exp( λt(1 z)) (3.18) n! Z(t) n=0 E(e θz(t) ) = exp( λt(1 M Y (θ))) (3.19) Y 1, Y 2,... Z(t) E(z Z(t) ) = E(E(z (Y 1+Y 2 +...+Y N(t) ) N(t)) = E(z Y ) n (λt)n e λt = exp( λt(1 G Y (z))) n! n=0 G Y (z) = E(z Y )

49 3.19 Y 1, Y 2,... Z(t) λpt n θ = 0 n ( ) d d dθ M Z(t)(θ) = λt dθ M Y (θ) exp( λt(1 M Y (θ))) θ = 0 E(Z(t)) = λte(y ) E(Z(t)) = E(N(t))E(Y ) ( d 2 ( ) ( d 2 dθ 2 M Z(t)(θ) = λt dθ 2 M Y (θ) + λt d ) ) 2 dθ M Y (θ) M Z(t) (θ) θ = 0 λte(y 2 ) + (λte(y )) 2 E(Z(t) 2 ) (E(Z(t))) 2 = λte(y 2 ) Y H(x) Y 1 + Y 2 + + Y n H(x) n H ( n) (x) P (Z(t) < z) = E(P (Z(t) < z N(t)) = n=0 H ( n) (z) (λt)n e λt n! 3.12 t k Y k a T T > t Z(t) < a G(y) n G ( n) (y) P (T > t) = P (Z(t) < a) = E(P (Z(t) < a N(t)) = G ( n) (a) (λt)n e λt n! n=0 E(T ) = 1 λ G ( n) (a) n=0 µ G ( n) (y) n 1 G ( n) (y) = 1 E(T ) = 1 λ k=0 n=0 k=n (µy) k e µy = k! (µa) k k! (µy) k k=n k! e µy e µa = 1 (1 + µa) λ 3.13 X k X 1, X 2,...

50 N(t) t X 1 + X 2 +... + X N(t) α min t { X1 + X 2 +... + X N(t) > α } 3.14 Y p 3.20 {N(t); t 0} λ Y 1, Y 2,... {N(t); t 0} µ σ 2 {Z(t); t 0} Z(t) = Y 1 + Y 2 +... + Y N(t) Z(t) 3.21 12 1 1000 400 1 3.6 P (N(t + h) N(t) = 1) = λ(t)h + o(h) (3.20) λ(t)h (t, t + h] λ(t) t (non-stationary Poisson process, non-homogeneous Poisson process) λ(t) t intensity function 3.4 λ(t) {N(t); t 0} λ(t) (1) N(0) = 0 (2) (3) (t, t + h]

51 λ(t) Λ(t) P (N(t + h) N(t) = 0) = 1 λ(t)h + o(h) P (N(t + h) N(t) = 1) = λ(t)h + o(h) P (N(t + h) N(t) 2) = o(h) Λ(t) = t P (N(s + t) N(s) = k) = 0 λ(t)dt (3.21) (Λ(s + t) Λ(s))k e (Λ(s+t) Λ(s)), k = 0, 1, 2,... (3.22) k! λ(t) = λ Λ(t) = λt 3.6.1 λ(t) t thinning algorithm λ = max 0 t T λ(t) λ t 1, t 2,... λ t k u k u k < λ(t k )/λ t k t k λ(t) 3.7 (renewal process) Y 1, Y 2,... {N(t), t 0} N(t) [0, t] n T n = Y 1 + Y 2 + + Y n T 1 {M(t), t 0} M(t) = N(t + T 1 ) 1, t 0 T 1 T 1 {N(t)} N(t) n T n N(t) n T n t

52 Y n T n N(t) {S n+t, n = 0, 1, 2,...} {S n, n = 0, 1, 2,...} {S n, n = 0, 1, 2,...} T T N(t + T ) N(T ) T t λt 3.15 T T {N(t + T ) 1, t 0} {N(t), t 0} {N(t), t 0} 3.22 {S n } p n > 0, m > 0, 0 j k n, m, j, k P (S n+m = k S m = j) 3.23 {N(t), t 0} λ {N(t + s) N(s), t 0} λ 3.1 {N(t); t 0} t, s > 0 N(t) N(t + s) 3.2 N(t) = n N(s)(s < t) 3.3 k T 3.4 99% 3.5 X 1, X 2,... λ M p X 1 + X 2 +... + X M

53 3.6 λ 1, λ 2, λ 3 X, Y, Z U = min {X, Z}, V = min {Y, Z} P (U > u, V > v) = exp( λ 1 u λ 2 v λ 3 max{u, v}) 3.7 U, V λ 1, λ 2, λ 3 Mathematica Excel U, V 3.8 10 20 3.9 {Z(t); t 0} + {Y n } Y 1, Y 2,... {Z(t); t 0} P (Y = 1) = p, P (Y = 0) = 1 p 3.10 {Z(t); t 0} + {Y n } Y 1, Y 2,... M Y (θ) 3.11 Z(t) 3.12 3.9 3.13 2% 3.14

54 4 4.1 X 1, X 2,... S n = X 1 + X 2 +... + X n = S n 1 + X n, n = 1, 2,... (4.1) S 0 = 0 X 1, X 2,... 2 P (X = 1) = 1 P (X = 1) = p (4.2) {(k, S k ), k = 0, 1, 2,...} (k, S k ) 4.1.1 S n n n n S n S 2n = 2k 2n u X = +1 d S 2n = u d u, d 2k = u d, 2n = u + d u = n + k, d = n k ( ) ( ) 2n 2n P (S 2n = 2k) = p u (1 p) d = p n+k (1 p) n k, k = 0, ±1, ±2,..., ±n u n + k ( ) 2n + 1 P (S 2n+1 = 2k + 1) = p n+k+1 (1 p) n k, k = 0, ±1, ±2,..., ±n, n 1 n + k + 1 ( ) n P (S n = k) = p (n+k)/2 (1 p) (n k)/2, (n + k)/2 k = n, n 2,..., n (4.3) n + k

55 4.1 P (S 2n+1 = 2k + 1), P (S n = k) X 1 E(X 1 ) = 2p 1, V (X 1 ) = 4p(1 p) S n E(S n ) = n(2p 1), V (S n ) = 4np(1 p) (4.4) p 0.5 2p 1 n 2p 1 0 p = 0.5 p = 0.5 {S n, n = 0, 1, 2,...} 4.2 p = 0.5 {S n, n = 0, 1, 2,...} S n 4.2 4.2.1 S 2n = 0 2n ( ) 2n P (S 2n = 0) = v 2n = p n (1 p) n, n = 0, 1, 2,... (4.5) n p > 0.5 E(S n ) p = 0.5 P (S 2n = 0) = v 2n = ( ) 2n 2 2n = (2n)! n n 4 n (n!) 2 = 2k 1 2k k=1 n 0 n n 4.1 n n! n log k k=1 n+0.5 1 n! n n e n 2πn (4.6) log xdx (n + 0.5) log n n = log ( n n ne n)

56 2π v 2n 1 nπ (4.7) 4.3 n v 2n {v 2n, n = 0, 1, 2,...} 4.2 ( 2n G v (z) = v 2n z 2n = n n=0 n=0 ) (z 2 ) 2n = 1 1 z 2 (4.8) a (1 + x) a (1 + x) a a(a 1) = 1 + ax + x 2 a(a 1)(a 2) + x 3 + 2 3! n 1 1 ( ) a = (a k) x n x n (4.9) n! n n=0 k=0 a ( a n) n > a ( a n) a n 2 G v (z) a 1 2 x z2 ( 2n n ) Gv (z) (2n)! (2n)! = n k=1 n 1 2k n=0 n 1 (2k + 1) = 2 n n! 2 n k=0 ( ) 2n = (2n)! n (n!) 2 = 1 22n n! n 1 k=0 k=0 ( k + 1 ) 2n ( 1)n = 2 2 n! ( k + 1 ) 2 n 1 k=0 ( 12 k ) G v (z) = n=0 1 n! n 1 k=0 ( 12 k ) ( z 2 ) n = ( ) 1/2 ( z 2 ) n n n=0 G v (z) S 2n = 0 1 0 1 {S2n =0} n=0 1 {S2n =0}

57 E ( 1 {S2n=0}) = v2n n v 2n n v 2n G v (z) z = 1 G v (0) = 4.4 (1 z 2 ) 0.5 ( z 2 ) n v 2n 4.2.2 4.3 km > 0 k, m (0, k) (n, m) x (0, k) (n, m) ( ) n (n + k + m)/2 j (0, k) (j, 0) x (0, k) (n, m) (0, k) (n, m) (0, k) (n, m) x x (0, k) (n, m) (0, k) (n, m) x (0, k) (n, m) x (0, k) (n, m) 6

58 x (0, k) x 4.5 (0, k) (n, m) ( ) n (n + k + m)/2 4.2.3 0 n s(> 0) φ s (n) n + s s n s n 1 s 1 (0, 0) (n 1, s 1) s (0, 0) (n 1, s 1) s (0, 0) (n 1, s 1) ( ) n 1 (n + s)/2 1 s (0, 2s) (n 1, s 1) ( ) n 1 (n + s)/2 n s (( ) ( )) n 1 n 1 φ s (n) = 2 n = s ( ) n 2 n (4.10) (n + s)/2 1 (n + s)/2 n (n + s)/2 7

59 s 1 1 s 1 1 φ s (n) = n s+1 m=1 φ 1 (m)φ s 1 (n m) 4.2.4 s (0, 0) (n, k) s(> k) r s,k (n) n + k k (0, 0) (n, k) ( n ) (n + k)/2 s (0, 2s) (n, k) ( ) ( ) n n = (n + k 2s)/2 (n + 2s k)/2 (0, 0) (n, k) (k < s) s (( ) n r s,k (n) = (n + k)/2 ( n (n + 2s k)/2 )) 2 n 4.2.5 T T T P (T = 2n) f 2n f 2n (1, 1) (1, 1) (1, 1) 2n (1, 1) 2n 1 1 φ 1 (2n 1) (1, 1) 1 2n 1 ( 2n 1 ) 2 2n+1 f 2n = 1 2 (φ 1(2n 1) + φ 1 (2n 1)) = φ 1 (2n 1) = n = 2 ( ) 2n 2 2 2n (4.11) n n 1 n f 2n = 1 (2n)! 1 2 2n n!n! (2n 1) 1 1 πn 2n 1

60 n n 3/2 G v (z) T < 2n 2n 1 2m 2m v 2n 2n v 2n = n f 2m v 2n 2m (4.12) m=1 {f 2k }, {v 2k } H(z), G(z) G(z) H(z) = f 2k z 2k, G(z) = v 2k z 2k = k=1 k=0 1 1 z 2 (4.12) z 2n n = 1, 2,... ( n ) v 2n z 2n = v 2n 2k f 2k n=1 = G(z) 1 = f 2k z 2k k=1 n=1 n=k k=1 z 2n v 2n 2k z 2n 2k = H(z)G(z) H(z) = 1 1 G(z) = 1 1 z 2 (4.13) f 2k z ( ) 1/2 ( ) 1/2 H(z) = 1 ( z 2 ) n = ( 1) n 1 n n n=0 n=1 = 1 (2n 3)!! 2 z2 + 2 n z 2n n! n=2 n!! n(n 2)(n 4) z 2n ( ) ( 1/2 1 1 ( 1) n 1 = ( 1) n 1 2 2 1) ( 1 2 2)... ( 1 2 n + 1) n n! = 1 (2n 3) (2n 5)...1 2 n n! = 1 (2n 3)!! 2 n, n 2 n!

61 P (T = 2n) = f 2n = (2n 3)!! 2 n n! = 2 n ( ) 2n 2 2 2n (4.14) n 1 T H (z) = z 1 z 2 z = 1 H(1) = P (T < ) = 1 2n r 2n r s,k (n) (1, 1) (1, 1) (1, 1) (2n, 2k) x r s,k (n) ( ) ( ) ( ) 2n 1 2n 1 2n 1 = n + k 1 n 1 k n + k 1 ( ) 2n 1 n + k k = 1, 2,... 2n ( ) ( ) 2n 1 2n r 2n = 2 2 2n = 2 2n = v 2n n n 4.6 2n 2n v 2n 2n 4.2.6 Excel x 8 2n 2m 2n 2m p m (n) p m (n) = v 2m v 2n 2m (4.15) 2n 2n S 2n v 2m

62 n m n m 2 p 0 (1) = P 1 (1) = 1 2 v 2 = 1 2 n = 1, m n p 0 (n) = p n (n) = v 2n 2n 2n 2k = (0. 2, 4,...) 1 (0, 0) (2n, 2k)(k = 0, 2, 4,...) 1 r 1,2k (2n) 2 2n ( ) ( ) ( ) ( ) 2n 2n 2n 2n =, k = 0, 1, 2,... n + k n + 1 k n + k n + k 1 k = 0, 1, 2,... ( 2n n ) 2 2n v 2n p 0 (n) = v 2n p n (n) = v 2n m > 0 2i 2i 2 1 2n 2i 2m 2i 2i 2n 2i 2m 2n 2i 2m 2i v 2(m i) v 2(n m) p m (n) = 1 2 = 1 2 m f 2i p m i (n i) + 1 2 i=1 i=1 n m i=1 f 2i p m (n i) m f 2i v 2(m i) v 2(n m) + 1 n m f 2i v 2m v 2(n m i) 2 i=1 = 1 2 v 2mv 2(n m) + 1 2 v 2(n m)v 2m = v 2m v 2(n m) 2 3 m f 2i v 2(m i) = v 2m, i=1 n m i=1 f 2i v 2(n m i) = v 2(n m) n 1 ( ) 2n 4πn(2n) 2n e 2n n 2π n 2n+1 e 2n = 1 2 2n (4.16) πn ) (, 2n 2m ) ( 2m m n m p m (n) = v 2m v 2n 2m = ( 2m m )( 2n 2m n m ) 2 2n 1 π m(n m)

63 [0, 1] m/n m n < a < m+1 n m m p m (n) = 1 π m m a 1 m n (1 m n ) n 1 π m m 1 a 0 dx x(1 x) = 2 π arcsin a (4.17) 4.4 x F (x) = 2 π arcsin x f(x) = d dx F (x) = 1 1 π x(1 x) x = 0.5 x = 0.5 x = 0, x = 1 8 1 0 x 4.2.7 2n 2m 2n 2m

64 2m 2n 2m v 2m r 2n 2m r 2n 2m = v 2n 2m v 2m v 2n 2m p m (n) n 2n 2n 4.7 1 1 n(= 100) 0 0 0 1000 Excel R Mathematica R sample(c(-1,1),n,replace=t) cumsum 0 which 4.8 2n w 2n w 2n = 1 ( ) 2n 2 1 n n 1 2 2n = 1 4n v 2n 2 (1, 1) (2n 1, 1) 2 2n 2 0 0 4.9 2n 4w 2n+2 4.10 2n m/n m = 0, 1,..., n 1 n+1 p m (n) m/n q m (n) 0 q m (n) 4.3 4.3.1 f X (x) X g(x) g(x) g(x)f X (x) x

65 X Y f Y (x) f X (x) 0 f Y (x) 0 g(x)f X (x) = x x g(x) f X(x) f Y (x) f Y (x) x g(x)l(x)f Y (x) (4.18) g(x) g(y ) f X(Y ) f Y (Y ) Y X E X (.) Y E Y (.) ( E X (g(x)) = E Y g(y ) f ) X(Y ) E Y (g(y )L(Y )) (4.19) f Y (Y ) L(x) g(x) A 1 0 1 A ( ) f X (Y ) E X (1 A ) = E Y (1 A L(Y )) = E Y 1 A f Y (Y ) E Y (1 A L(Y )) A ϖ(a) ϖ(a) E Y (1 A L(Y )) (4.20) ϖ(a) 3 0 E Y (1 A L(Y )) E Y (L(Y )) = x f X (x) f Y (x) f Y (x) = x f X (x) = 1 0 1 ϖ(ω) = E(1 Ω L(Y )) = E(L(Y )) = 1 (4.21) A, B 1 A B = 1 A + 1 B ϖ(a B) = E(1 A B L(Y )) = E((1 A + 1 B )L(Y )) (4.22) = E(1 A L(Y )) + E(1 B L(Y )) = ϖ(a) + ϖ(b) ϖ(a) ϖ(a) = P (A) E Y (1 A L(Y )) (4.23) 4.1

66 f X (1) = p = 1 f X ( 1) f X (x) = p x + 1 2 + (1 p) 1 x, x = 1, 1 2 X p(0 < p < 1) f Y (x) = p x + 1 2 Y L(x) = + (1 p) 1 x, x = 1, 1 2 p(x + 1) + (1 p)(1 x) p(x + 1) + (1 p)(1 x) A = {X = 1} P (X = 1) = E(1 A L(Y )) = p = p = P (X = 1) (4.24) p p 4.3.2 M Y (θ) = E(e θy ) exponential tilting f Y (x) = 1 M X (θ) eθx f X (x) (4.25) {X 1, X 2,...} f X (x) {U 1, U 2,...} P (X = 1) = p = 1 P (X = 1) (4.26) P (U = 1) = P (U = 1) = 0.5 (4.27) M X (θ) = pe θ + (1 p)e θ (4.28) f U (1) = 1 M X (θ) eθ f X (1) = pe θ pe θ + (1 p)e θ = 1 1 p 2 eθ = p (4.29) M X (θ) = 2pe θ = 2 p(1 p) S n = S n 1 + X n, n = 1, 2,..., P (X i = 1) = p = 1 P (X i = 1) Z n = Z n 1 + U n, n = 1, 2,..., P (U i = 1) = P (U i = 1) = 0.5

67 L(x) = f X(x) f U (x) L({u 1:n}) = (M X(θ)) n e θz = n ( 2 p(1 p) e θz n ) n (4.30) u 1:n {u 1,..., u n } g(x) A 1 A A n P (A) E U (1 A L(Z n )) A 4.2 2n Z 2n = 0, e θz 2n = 1 ( P X (S 2n = 0 S 0 = 0) = 2 2n p(1 p)) PU (Z 2n = 0 Z 0 = 0) ( ) 2n = 2 2n (p(1 p)) n 2 2n n = ( 2n n ) p n (1 p) n ( ) 2n v 2n = p n (1 p) n n 4.11 n s X X < a A A 1 A = g(x) θ = P (A) X x 1, x 2,... ˆθ = 1 n n g(x i ) i=1 g(x i ) = 0 θ a f Y (x) f Y (x) y 1, y 2,... ˆθ 1 = 1 n n g(y i )L(y i ) = 1 n i=1 n i=1 g(y i ) f X(y i ) f Y (y i ) (4.31) θ importance sampling method

68 4.3 µ σ 2 a = µ 3σ µ σ 2 n a a n 0 1/n, 2/n X µ σ 2 ) M X (θ) = exp (µθ + σ2 2 θ2 (4.32) f Y (x) = 1 M X (θ) eθx f X (x) = 1 e (x µ σ2 θ) 2 /(2σ 2 ) 2πσ (4.33) µ + σ 2 θ σ 2 σ 2 θ θ = 3/σ L(x) = e 3x/σ M X ( 3/σ) = e 3(x µ)/σ+4.5 Y µ 3σ σ 2 f Y (x) y µ 3σ g(y) L(y) n i=1 g(y i)l(y i ) µ 3σ σ 2 y 1, y 2,..., y n 1 n n g(y i )L(y i ) = 1 n i=1 y i<µ 3σ L(y i ) 4.3.3 i(> 0) n k(> 0) P (S n = k, min S 1:n > 0 S 0 = i) = P (S n = k i, min S 1:n > i S 0 = 0) min S 1:n min {S 1,..., S n } p = 0.5 i n k i (??) r i,k i (n) P (Z n = k, min Z 1:n > 0 Z 0 = i) (( ) ( )) n n = (n + k i)/2 (n k i)/2 ( ) = m (n) k i m(n) k+i 2 n ( ) n (n) (n+j)/2 m j min Z 1:n min S 1:n p 0.5 (4.30) L(z n ) 2 n

69 S n = Z n = k, S 0 = Z 0 = i P (S n = k, min S 1:n > 0 S 0 = i) = P (S n = k i, min S 1:n > i S 0 = 0) = E(1 A L(Z n ) Z 0 = 0) = P (Z n = k i, min Z 1:n > i Z 0 = 0)e θ(k i) (M U (θ)) n ( ) ( = m (n) k i m(n) k+i 2 n 2 ) ( ) n (k i)/2 p p(1 p) 1 p = ( m (n) k i m(n) k+i ) p (n+k i)/2 (1 p) (n k+i)/2 S n P (S n = k) q (n) k = m (n) k p(n+k)/2 (1 p) (n k)/2 ( ) i P (S n = k, (M U (θ)) n > 0 S 0 = i) = q (n) 1 p k i q (n) k+i (4.34) p ( ) i (p = 0.5) 1 p p p {S n} n p (n) ik = P (S n = k S 0 = i), k = 1, 2,... 4.12 n k(> 0) s(> k) q (n) k ( 1 p p ) s k q (n) 2s k 4.3.4 S 0 = i(> 0) n T 0 T 0 1 1 0 {T 0 = n S 0 = i} {S n = 0, min S 1:(n 1) > 0 S 0 = i} {X n = 1, S n 1 = 1, min S 1:(n 1) > 0 S 0 = i} ( ( ) ) i P (T 0 = n S 0 = i) = (1 p) q (n 1) 1 p 1 i q (n 1) 1+i p (4.35a)

70 P (T 0 = n S 0 = i) = P (U n = 1, Z n 1 = 1, min Z 1:(n 1) > 0 Z 0 = i)e iθ (M U (θ)) n ( ) = m (n 1) 1 i m (n 1) 1+i 2 n 2 n p (n i)/2 (1 p) (n+i)/2 m (n 1) 1 i p (n i)/2 (1 p) (n+i)/2 = (1 p)q (n 1) 1 i m (n 1) 1+i p (n+i)/2 (1 p) (n i)/2 = P (T 0 = n S 0 = i) = (1 p) 4.13 (4.35a) ( q (n 1) 1 i (1 p)i+1 p i q (n 1) 1+i ( 1 p p ) i q (n 1) 1+i ) 4.3.5 0 n min S 1:n P (min S 1:n = k S 0 = 0) = P (min S 1:n > k 1 S 0 = 0) P (min S 1:n > k S 0 = 0) min S 1:n > k n k n P (min S 1:n > k S 0 = 0) = = = j=k+1 P (S n = j, min S 1:n > k S 0 = 0) P (S n = j, min S 1:n > 0 S 0 = k) j=1 j=1 ( ) k p j+k q (n) j k 1 p q (n) P (min S 1:n = k S 0 = 0) ( ( ) ) k = q (n) p j+k 1 q (n) j k+1 1 p j=1 = q (n) k ( ) k p q (n) 1 k 1 p ( j=1 ( p j+k 1 p q (n) ) ) k q (n) j k

71 4.14 p 0.5 P (min Z 1:n = k Z 0 = 0) = ( ) m (n) k m (n) 1 k 2 n 4.4 Y 0, Y 1, Y 2,..., Y n X n+1 X n E(X n+1 Y 0, Y 1, Y 2,..., Y n ) = X n {X n } {Y n } E( X n ) <, n = 0, 1, 2,... (4.36) X n n Y n n {S n } S n+1 = S n + U n+1 n E( S n ) E( S n 1 ) + E( U n ) E( U i ) < i=1 U n+1 0 (S 0, S 1, S 2,..., )S n S n+1 S n 4.5 {S n } Z n = eθ(s n S 0 ) (M X (θ)) n, n = 0, 1, 2,... (4.37) {Z n } {Z n } {S n } M X (θ) X Z n = eθ(x1+x2+ +Xn) (M X (θ)) n E(Z n+1 S 0, S 1,..., S n ) = eθ(sn S0) (M X (θ)) n E ( e θx n+1 M X (θ) ) = eθ(sn S0) (M X (θ)) n = Z n

72 2 θ M X (θ) = pe θ + (1 p)e θ = 1 e θ = 1 p p (4.38) ( ) X1+X 1 p 2+ +X n Z n = (4.39) p 4.6 p {S n }(S 0 = 0) ( ) Sn 1 p S n =, n = 0, 1, 2,... (4.40) p { S n } { S n } {S n } ( ( ) (1 ) ) Sn +X p n+1 E Sn+1 S 1,..., S n = E S 1,...S n p ( ) ( Sn (1 ) ) Xn+1 1 p p = E p p ( = S 1 p n p + p ) (1 p) = p 1 p S n 4.4.1 optional sampling theorem T stopping time T = n n 4.7 {X n } {Y n } T 1 n < T E ( Xn) 2 A, n = 1, 2,..., T (4.41) A E(X T ) = E(X 0 ) (4.42) Xn 2 E(X T ) = E(X 0 ) {S n } E(Sn) 2 = 1 + 2 E (X i ) + E ( X 2 ) i + 2 E (X i ) E (X k ) = n + 1 i i i<k

73 S 0 = 1 0 T T 1 S 2 n E(S T ) = 0 1 = E(S 0 ) 4.4 A, B N A i A r i T r i = P (S T = 0 S 0 = i) T a i = P (T < S 0 = i) a i 1 a i = pa i+1 + (1 p)a i 1 a i+1 a i = 1 p p (a i a i 1 ) = i = N 1 ( ) N 1 1 p 0 1 a N 1 = (a 1 1) 0 p a N 1 = 1 a N 2 = a N 3 = 1 ( ) i 1 p (a 1 1) (4.43) S T 0 N r i = P (S T = 0 S 0 = i) 4.5 4.6 {S n } ( 1 p S n = p ) Sn S 0 { S n } E( S T ) = E( S 0 ) = 1 ( ) i ( ) N i E( S 1 p 1 p T ) = r i + (1 r i ) = 1 p p p r i = ( 1 p p ( 1 p p ) N i 1 ) i = ) N i ( 1 p p ( 1 p p ) N ( 1 p ( 1 p p p ) i ) N 1 (4.44) Q = 1 r i a i r i = pr i+1 + (1 p)r i 1 = r i+1 + r i 1 2 (4.45) r i = 1 i n Q = 1 r i = 1 i N 4.15 Q = 1 r i = 1 i N

74 4.5 4.5.1 10000 R plot(cumsum(sample(c(-1,1),10000,replace=t)),type="l") t S n t 0 t n t = t/n X 1, X 2,... P (X = x) = P (X = x) = 0.5 (4.46) S n = X 1 + X 2 + + X n S(n t) E(S(n t)) = ne(x) = 0 V (S(n t)) = nv (X) = n( x) 2 = ( x)2 t t t 0 S(n t) S(t) ( x) 2 / t σ > 0 ( x) 2 t = σ 2 x = σ t (4.47) x {S(t), t 0} E(S(t)) = 0, V (S(t)) = σ 2 t (4.48) t 1 < t 2 t 3 < t 4 t S(t 2 ) S(t 1 ) S(t 4 ) S(t 3 ) (t 2 t 1 )/ t, (t 4 t 3 )/ t S(n t) E(S(n t)) V (S(n t)) = S(n t) σ t (4.49) S(t) 0 σ 2 t S(t + s) S(s) 0 σ 2 t {S(t), t 0}

75 S(t + s) S(s) 0 σ 2 t S(n t) ( 1 E (exp(θs(n t))) = t 2 eθσ + 1 n t) 2 e θσ t = t/n 1 2 eθσ t/n + 1 2 e θσ t/n = 1 + σ2 2n tθ2 + o(n 1 ) E (exp(θs(n t))) = ) n ( ) (1 + σ2 1 2n tθ2 + o(n 1 ) exp 2 σ2 tθ 2 0 σ 2 t (4.50) 4.5.2 4.1 {X(t), t 0} (1) X(0) = 0 (2) (3) t > 0, s 0 X(t + s) X(s) 0 σ 2 t x ( ) 1 P (X(t + s) X(s) x) = e z2 /(2σ 2 x t) dz = Φ 2πtσ σ (4.51) t 3 s {X(t)} 3 3 [s, s + t] E ( (X(t + h) X(t) h ) 2 ) = 1h 2 E ((X(t + h) X(t)) 2) = σ2 h, (h 0) (4.52) t t σ = 1 {B(t), t 0} {W (t), t 0} W Wiener C(X(t), X(s)) t > s C(X(t), X(s)) = E(X(t)X(s)) = E((X(t) X(s) + X(s))X(s)) = σ 2 s (4.53) t < s C(X(t), X(s)) = σ 2 min {s, t} (4.54)

76 4.16 {B(t), t 0} (1) B(s)B(t + s) (2) B(s) B(t + s) 4.5.3 [0, T ] {B(t), t 0} R(T ) R(T ) > a(> 0) 4.8 {B(t), 0 t T } R(T ) P (R(T ) a) = 2 a 1 2πT e x2 /(2T ) dx (4.55) a > 0 {B(t), 0 t T } B(T ) a B(T ) < a B(T ) a R(T ) a B(T ) = b < a R(T ) a B(t) = a 0 < t < T t t y = a [t, T ] B(t) B(t) [0, t ] B(t) = B(t) B(t) B(t) B(T ) = 2a b > a B(T ) > a B(t) R(T ) a B(T ) < a B(T ) > a R(T ) > a B(T ) > a 2 9 a 4.9 {B(t), t 0} a 0 Y

77 P (Y < t) = 2 a 1 2πt e x2 /(2t) dx (4.56) Y < t t a [0, t] R(t) a 1 P (Y < t) = P (R(t) a) = 2 e x2 /(2t) dx a 2πt 4.6 4.6.1 1 security t 1 + u t 1 + d t u up d down d < 0 2 t (1 + u t) 2 (1 + u t)(1 + d t) (1 + d t) 2 3 n t (1 + u t) i (1 + d t) n i (i = 0, 1, 2,..., n) Y 0 n t Y n R k u t d t R k Y n = (1 + R n )Y n 1 = = n k=1 (1 + R k)y 0 (4.57) P (R k = u t) = p, P (R k = d t) = 1 p (4.58) Y n P ( Y n = Y 0 (1 + u t) k (1 + d t) n k) ( ) n = p k (1 p) n k, k k = 0, 1, 2,..., n (4.59) 2 R k 4.6.2 Y n = (1 + R n )Y n 1 log Y n = log (1 + R n ) + log Y n 1 (4.60)

78 log Y n Z n log(1 + R n ) X n Y 0 = 1 Z 0 = log Y 0 = 0 n Z n = log(1 + R k ) = Z n 1 + X n = X 1 + X 2 + + X n, n = 1, 2,... k=1 {Z n, n = 0, 1, 2,...} 1 t P (X i = log(1 + u t)) = p, P (X i = log(1 + d t)) = 1 p t 0 log(1 + u t), log(1 + d t) t log(1 + u t) = σ t 1 + u t = e σ t (4.61) log(1 + u t) log(1 + d t) = 1 1 + d t = e σ t 1 + R n r t E(1 + R n ) = e r t (4.62) E(1 + R n ) = 1 + pu t + (1 p)d t = e r t p = er t e σ t e σ t e σ t (4.63) t p = er t e σ t e σ t e σ t 1 + r t (1 σ t + σ 2 t/2) 2σ t = 1 2 + 1 2σ ) t (r σ2 (4.64) 2 e σ t p = 1 2 + 1 2σ (r σ2 2 ) t (4.65) e σ t 1 1 p {Y n } 4.6.3 X k E(X k ) = E(log(1 + R k )) = pσ t (1 p)σ t = ) (r σ2 t (4.66) 2 V (X k ) = ( x) 2 E(X k ) 2 = σ 2 t + o( t) (4.67)

79 t = n t Z (n) (t) = Z n t 0 ( ) Z(t) Z(t) r σ2 2 t σ 2 t Z(t) Y (t) = e Z(t) {Y (t), t 0} Y (t) e rt E(e Z(t) ) (= e rt ) e E(Z(t)) (= e (r σ2 /2)t ) Z(t) log Y (t) = Z(t) 4.17 µ σ 2 X Y = e X Y X M X (θ) M X (1) ( ) Z(t) r σ2 2 t σ 2 t Y (t) = e Z(t) e rt 4.18 µ σ 2 X e X 1 2πσx e (log x µ)2 /(2σ 2 ) (4.68) M X (1) M X (θ) X 4.6.4 Y (t) t Y (t) = Y (t + t) Y (t) Y (t) t Z(t) = Z(t + t) Z(t) ( ) Y (t) = Y (t + t) Y (t) = e Z(t+ t) e Z(t) = e Z(t) e Z(t) 1 Y (t) Y (t) = e Z(t) 1 Z(t) e h 1 = h + h2 2 ( Z(t) r σ2 2 t W (t) + o(h) e Z(t) 1 Z(t) + 1 2 ( Z(t))2 (4.69) ) t σ 2 t 0 Z(t) = (r σ2 2 ) t + σ W (t) ( Z(t)) 2 σ 2 t Z(t) + 1 2 ( Z(t))2 r t + σ W (t) Y (t) Y (t) r t + σ W (t)

80 t 0 t, Y, W dt, dy, dw dy (t) Y (t) = rdt + σdw (t) (4.70) dt, dy, dw o( t) t 4.1 (1) {x 1, x 2,..., x N } (x n+1 x n )/x n (2) t = 1/N X(n t) = x n (1) µ t, σ t X(t + t) X(t) X(t) = µ t + σ(b(t + t) B(t)) B(t) 4.2 p = 0.5(1 + µ t) t 1 p t t t 0 µ 4.3 X(n t) t p = 0.5(1 + µ σ t) e σ t X(n t) 1 p e σ t X(n t) n = 1, 2,... t = n t t 0 log X(t) µt σ 2 t 4.4 µ σ {X(t), t 0} X(t), X(t + s) 4.5 a, b > 0 b a b a A b < i < a X 0 = i A r i r i r i 1, r i+1 4.6 a > 0 a

81 T T n 4.7 a, b > 0 b a ab i b a t i t i t i+1.t i 1 t b = t a = 0 t a 1 = t b+1, t a 2 = t b+2,... 4.8 B(t) P (B(2) > 2 B(0) = 0) 4.9 B(t) c > 0 B(ct)/ c B(ct)/ c 0 t 4.10 B(t) E(B(t)B(t + s)) s > 0 4.11 σ B(t) B(1) = σ B(2) > 0 4.12 σ B(t) B(2) = σ B(1) > 0 4.13 σ B(t) a(> 0) T a T a > 1 B(1) a T a > 1 4.14 σ B(t) [0, 1] a a a 4.15 σ t, T, a > 0 t a t + T 0 4.16 t + T 4.17 σ t, T, a > 0 [t, t + T ] a t B(t) = x > a 1 x < a

82... 4.18 σ b(> 0) b a(> 0) c(> 0) a c(> 0) 4.19 100 120 90 1 105 10 0

83 5 5.1 {U 1, U 2,...} X n = X n 1 + U n, n = 1, 2,... X 0 = 0 n 1 X n 1 n X n n 1 n 1 {X(t), t 0} t, s > 0, k, j, {x(u), 0 u s} P (X(t + s) = k X(s) = j, X(u) = x(u)(0 u < s)) = P (X(t + s) = k X(s) = j) s P ( ) = P ( ) {X n, n = 0, 1,...} n > 0, k, j, i,..., h P (X n = k X n 1 = j, X n 2 = i,..., X 0 = h) = P (X n = k X n 1 = j) t n 1 P (X n+1 = k X n 1 = j, X n 2 = i,..., X 0 = h) = u P (X n+1 = k, X n = u X n 1 = j,...) = u = u P (X n+1 = k X n = u, X n 1 = j,...)p (X n = u X n 1 = j,...) P (X n+1 = k X n = u)p (X n = u X n 1 = j) {U n } U n X n 1 5.1 P ( ) = P ( )P ( )

84 5.1.1 3 5.2 2 0.8 2 0.2 0.6 5.2 {X n, n = 0, 1, 2,...} n k, j, i,..., h P (X n+1 = k X n = j, X n 1 = i,..., X 0 = h) = P (X n+1 = k X n = j) p jk p jk = P (X n+1 = k X n = j)

85 j k n 5.1 {S n, n = 0, 1, 2,...} S n = S n 1 + X n P (X n = 1) = 1 P (X n = 0) = p X n S n 2, S n 3,... p jk = P (S n = k S n 1 = j) = P (X n = k j) k j 0, 1 0 p j,j+1 = 1 p j,j = P (X n = 1) 5.2 1 1 n S n n X n S n = X 1 + X 2 + + X n = S n 1 + X n, n = 1, 2,... S 0 = 0 S n P (X n = 1) = 1 P (X n = 1) = p S n 1 {S n, n = 0, 1, 2,...} 5.2.1 i k p ik = P (X n+1 = k X n = i) P = (p ik ) 1 1 p ik 0, k p ik = 1 X 0 = i n k n p (n) ik = P (X n = k X 0 = i)

86 n i k p (n) ik n P (n) ( P (n) = n p (n) ik ) 5.3 p ( ) p (n) n ik = p k i (1 p) n (k i), k i k = i, i + 1,..., i + n 5.2.2 5.1 n, m 0, i, k S P (X n+m = k X 0 = i) = j P (X n+m = k X n = j)p (X n = j X 0 = i) n + m n n P (n+m) = P (n) P (m) P (1) = P n 1 n P (n) = P (n 1) P = P (n 2) P 2 =... = P n n 5.3 P P r r r n P n 5.2.3 n P λi = 0 P x k = λ k x k

87 P λ 1, λ 2,... λ k x k (x 1, x 2,..., x m ) X λ 1 O λ 2 P (x 1, x 2,..., x m ) = (x 1, x 2,..., x m )... = XΛ P = XΛX 1 O λ m P n = XΛ n X 1 = X λ n 1 O λ n 2... O λ n m X 1 5.4 2 2 ( 1 a a P = b 1 b ) a b 0 0 a 2 + b 2 > 0 P 1, 1 a b ( ) 1 a a P = b 1 b = 1 ( ) ( 1 a 1 0 a + b 1 b 0 1 a b P n = 1 ( 1 a a + b 1 b p (n) 00 = b a + b + p (n) 10 = b a + b ) ( b a 1 1 ) ) ( 1 0 ) ( b a 0 (1 a b) n 1 1 a (1 a b)n a + b b (1 a b)n a + b 5.4 2 ) ( 1 a a P = b 1 b = 1 ( 1 a a + b 1 b ) ) ( 1 0 0 1 a b ) ( b a 1 1 ) 5.5 2 ( ) 1 p p P = p 1 p n p

88 5.2.4 π k (n) = P (X n = k) π k (n) k π(n) 0 π(0) 0 π k (0) = P (X 0 = k) n n π k (n) = P (X n = k) = j P (X n = k X 0 = k)p (X 0 = k) = j π 0 (j)p (n) jk π(n) = π(0)p n {X n, n =..., 2, 1, 0, 1, 2,...} 5.2.5 p jk > 0 j, k p jk k N k (0) 0 k N k (0)p kj j k, j N k (1) 1 k N j (1) = k N k (0)p kj (5.1) N n k N k (n)/n k k i k 5.5

89 5.6 M i j l ij = 1 l ij = 0 L i = j l ij N n i N i (n) q 1 q i j q/l j + (1 q)/m N i (n + 1) = ( q N j (n) + 1 q ) L j j M = 1 q M + q j N j (n) L j (5.2) 5.7 0 N(> 0) n 0 N 0 N N N 0 5.8 p i,i+1 i 1 p i,i+1

90 i 5.6 5.7 {0.1, 0.2, 0.3, 0.4} 0.1 0.1

91 5.3 5.3.1 2 ( 1 a a P = b 1 b ) n p (n) 00 = b a + b + p (n) 10 = b a + b a (1 a b)n a + b b (1 a b)n a + b 0 < a + b < 2 n p (n) 00, p(n) 10 0 {X n, n = 0, 1, 2,...} lim n P (X n = k X 0 = j) = π k π k k π k 2 a + b = 2 a = 0 p (n) 00, p(n) 10 a + b = 0 a = 0 0 1 0 p (n) 00 = 1, p(n) 10 b = 0 = 1 (1 b)n a = b = 0 a + b = 2 a b 5.3.2 i k k i k i i k i, k

92 i S(i) S(i) S S(i) S(i) j S(i) k S(i) c k j i k S(i) S 5.3.3 S(i) S(i) p jk > 0 j S(i) k S(i) c S(i) S(i) S(i) S(i) transient S(i) P 1 S(1) = {1, 2} P 1 = 0 1 2 3 1 0 0 0 0.1 0.4 0.5 0 0 0.3 0.4 0.3 0 0 0 1 S(i) S(i) S(i) j j recurrent i S(i) i S(i) i P 2 S(2) = {2, 3} 0 0.1 0.8 0.1 0 1 P 2 = 0.6 0.4 0 0 2 0 0 0 1 3 0 0 0.4 0.6 S(i) S(i) j S(i) S(i) j j S(i) S(i) = {i} i absorbing

93 5.3.4 S P 3 P 4 0 0.1 0.9 0 0 0 0.1 0.9 0 0 1 P 3 = 0.2 0 0.8 0 2 0.4 0 0 0.6, P 1 4 = 0.6 0.4 0 0 2 0 0 0 1 3 1 0 0 0 3 0 0 0.4 0.6 P 4 0, 1 2, 3 p (n) jk π k (n ) j, k 0 P 1 P 2 {0, 1} P 2 {2, 3} 5.3.5 m cyclic m 1 i m k i k m i k m m m P 5 0 2 2 0 0 0 0.3 0.7 1 P 5 = 0 0 0.8 0.2 2 0.4 0.6 0 0 3 1 0 0 0 0 P 5 2 3

94 0 1 5.3.6 k X 0 = k k 1 k recurrence time T (k) T (k) = {n; X 1 k, X 2 k,..., X n 1 k, X n = k X 0 = k} T (k) 1 k P (T (k) < ) = 1 k P (T (k) < ) = R k < 1 R k k k 1 R k 5.9 p 01 = a, p 10 = b P (T (0) = n) = a(1 b) n 2 b, n = 2, 3,... P (T (0) = 1) = 1 a P (T (0) < ) = 1 b > 0 0 a > 0, b = 0 5.8 0 P = 0 p 1 p 1 0 0 1 0 0 5.3.7 T (k) r n = P (T (k) = n) (n = 1, 2,...)

95 n P (X n = k X 0 = k) = p (n) kk = n m=1 n P (X n = k T (k) = m)p (T (k) = m X 0 = k) m=1 r m p (n m) kk G(z) = n=0 p (n) kk zn, H(z) = r n z n n=1 z n n = 1, 2,... n=1 p (n) kk zn = n n=1 m=1 ( (r m z m ) p (n m) kk z n m) G(z) 1 = G(z)H(z) G(z) = 1 1 H(z) k T (k) 1 H(1) = 1 H(1) = 1 G(1) = n=0 p (n) kk = n 5.3.8 k k 3 1 1 k 3 1 k 3 1 n k 1 0 Z n Z n = { 1, if Xn = k X 0 = k 0, if X n k X 0 = k N n=0 Z n N k N k E(T (k)) m k m k = E(T (k)) N N n=0 Z n E(Z n ) = P (X n = k X 0 = k) = p (n) kk

96 n=0 p (n) kk ( ) = E Z n = n=0 p (n) kk π k N n=0 p(n) kk N π k π k m k 5.9 5.8 5.10 5.11 k T (k) P (X n = k X 0 = k) = n P (X n m = k X 0 = k)p (T (k) = m X 0 = k) m=1 5.12 n=0 p(n) kk k 5.3.9 0 p (2n) 00 2n n n p (2n) 00 = ( 2n n ) ( 1 2 ) 2n n! n n e n 2πn 00 (2n)2n e 2n 4πn 1 n 2n e 2n 2πn 2 2n = 1 nπ p (2n) p (2n) 00 p (2n) 00 0 5.13 n 1 p (2n) 00 (nπ) 0.5

97 5.3.10 k π k = lim n π k(n) = lim n P (X n = k X 0 = j) k 9 5.14 5.15 p jk = 1 j M π k = 1 M, k = 1, 2,..., M 5.16 0, 1, 2, 3, 4 p 1 p X n n {X n, n 0} (1) (2)

98 5.4 5.4.1 p (n+1) ik = j p (n) ij p jk n π k = j π j p jk (π k ) 1 (stationary distribution) π(0) π(1) π(2)... steady state 5.17 0 0.3 0.5 0.2 0 0 p 1 p P 1 = 1 1 0 0, P 2 = 1 1 0 0 2 0.5 0.5 0 2 1 0 0 5.4.2 0 (0.5, 0.5) ( ) ( ) ( ) 0 1 0.5 0.5 = 0.5 0.5 1 0 5.4.3 lim n p(n) kk = π 1 k lim m m m n=0 p (n) kk = π k m n=0 p kk(n) m k m m π k = 1 E(T (k))

99 5 1 5.4.4 k N k k N k i N ip ik N k i N i p ik N k = N k N i p ik = N i i π i p ik = Nπ k N k = N k = N k A A c A A N i p ik = N π i p ik i A k A c i A k A c N i p ik = N π i p ik i A c k A i A c k A i A k A c π i p ik = π i p ik i A c k A 5.18 Excel 0.99 0.01 0 0 0 0 0.6 0.4 P 1 = 0 0.99 0.01 0 0 0 0.99 0.01, P 2 = 0 0 0.2 0.8 0.3 0.7 0 0 0.01 0 0 0.99 0.69 0.3 0 0.01 5.19 N 0.7 0.2 0.1 0.2 0.6 0.2 0.1 0.4 0.5 N

100 5.20 4 3 5 5.21 2 3 2 1 5.22 A π i p ik = π i p ik i A k A c i A c k A

101 5.5 5.5.1 (absorbing Markov chain) 0 5.5.2 (first-step analysis) P (A X 0 = k) = j P (A X 1 = j)p (X 1 = j X 0 = k) = j p kj P (A X 1 = j) E(Z X 0 = k) = j p kj E(Z X 1 = j) A Z P (A X 0 = k), E(Z X 0 = k) A P (A X 0 = k) = P (A X 1 = k) Z E(Z X 1 = k, X 0 = i) = 1 + E(Z X 0 = k) 5.10 P = 0 ( ) 1 p p 1 0 1 T E(T X 0 = 0) = 1 + (1 p)e(t X 0 = 0) E(T X 0 = 0) = p 1 5.11 p n X n A P (A X 0 = k) = p P (A X 1 = k + 1) + (1 p)p (A X 1 = k 1) Q k = pq k+1 + (1 p)q k 1 5.23 1 0 0 1 0 0 P = 1 q 1 p q p 2 0 0 1

102 5.5.3 ( ) Q R P = O I I 0 0 R 0 Q 1 5.5.4 T A j T, k A j k ( P (X = k X 0 = j) = + ) p ji P (X = k X 1 = i) i T i A = p ji P (X = k X 0 = i) + p jk i T P (X = k X 0 = j) = a jk a jk = i T p ji a ik + p jk A = (a jk ) A = QA + R A = (I Q) 1 R A 5.5.5 I Q (I Q) 1 I Q A = QA + R A = Q(QA + R = Q(Q(QA + R) + R) + R =... = (I + Q + Q 2 +...)R (I Q)(I + Q + Q 2 +...) = lim n (I Qn ) = I r < 1 r n 0 A = (I Q) 1 R

103 5.5.6 T ( E(T X 0 = j) = + ) p ji E(T X 1 = i) i T i A = 1 + p ji E(T X 0 = i) i T t j = E(T X 0 = j) t 1 1 t = 1 + Qt t = (I Q) 1 1 5.24 Q m I k n P n ( ) Q R P = 0 I 5.25 Q I + Q + Q 2 + Q 3 +... (I Q) 1 5.26 p A (0) (1) (2) (3) (4) p (5) p 5.27 0, 1,..., N (1) j k (2) j 0 N x j x j, j = 0, 1,..., N

104 (3) k kp jk = j, j = 1, 2,..., N 1 (2) x k = k/n 5.28 n 0 N n 0 q n 0 X n n {X n ; n = 0, 1, 2,...} X 0 = 1

105 5.6 branching process tree Y jk (j = 1, 2,...; k = 1, 2,...) P (Y jk = m) = p m (m = 0, 1, 2,...) {X n, n = 0, 1, 2,...} X 0 = 1 X n+1 = Y n1 + Y n2 + + Y n,xn (n = 0, 1, 2,...) {X n, n = 0, 1, 2,...} branching process n X n n X k k Y ki k i X 0 = 1 n n n X n 0 X n = 0 Y {p 0, p 1, p 2,...} p 0 = 0 p 0 + p 1 = 1 1 0 1 p 0 > 0 p 0 + p 1 < 1 5.6.1 {X n, n = 0, 1, 2,...} 0 2 n X n 0

106 n X n E(X n ) = E(E(X n X n 1 )) = E(X n 1 )E(Y ) n X n Y m = E(Y ) E(X n ) = m n (n = 0, 1, 2,...) 1 0 1 0 1 P (X n = 0) 1 5.29 E(X n ) = E(Y ) n P (Y = 0) = p, P (Y = 1) = 0.5, P (Y = 2) = 0.5 p(0 < p < 0.5) p E(X 5 ), E(X 10 ), E(X 20 ) 5.6.2 0 n α n = P (X n = 0 X 0 = 1) X 0 = k P (X n = 0 X 0 = k) = α k n α 1 = p 0 2 1 1

107 α 2 = P (X 2 = 0 Y 11 = k, X 0 = 1)P (Y 11 = k) k=0 = P (Y 21 + Y 22 + + Y 2k = 0)P (Y 11 = k) k=0 = p k 0p k = α1p k k k=0 k=0 Y G Y (z) = p n z n n=0 G Y (α 1 ) α n = P (X n = 0 Y 11 = k, X 0 = 1)P (Y 11 = k) k=0 = p k P (X n 1 = 0 X 0 = 1) k = p k (α n 1 ) k = G Y (α n 1 ) k=0 1 α 1 = p 0 5.30 P (Y = 0) = 0.25, P (Y = 2) = 0.75 n = 1, 2, 3, 4 n {α n } 1 k=0 α n = P (X n = 0 X 0 = 1) = = α n 1 + P (X n = 0 X n 1 = k)p (X n 1 = k X 0 = 1) k=0 p k 0P (X n 1 = k X 0 = 1) α n 1 k=1 α α = lim n α n α n = G Y (α n 1 ) n α α = G Y (α) α G Y (z) α = 1 G Y (z) G Y (z)(0 z 1) G Y (z) (1) G Y (z) > 0 (2) G Y (z) > 0 (3)G Y (0) = p 0 > 0, G Y (1) = 1