1 1.1 (population) (sample) (event) (trial) Ω () 1 1 Ω 1.2 P 1. A A P (A) 0 1 0 P (A) 1 (1) 2. P 1 P 0 1 6 1 1 6 0 3. A B P (A B) = P (A) + P (B) (2) A B A B A 1 B 2 A B 1 2 1 2 1
1 2 2 3 1.3 A B P (A B) = P (A)P (B) (3) A B A B P (B A) A B A B P (A B) = P (B A)P (A) (4) P (B A) = P (A B) P (A) (5) P (A B) P (B A) P (A B) A B P (A B) = P (B A)P (A) = P (A B)P (B) (6) P (B A) = P (A B) P (A) = P (A)P (B) P (A) = P (B) (7) A B Ω Ω = A 1 + A 2 + + A n (8) 2
P (B) = P (B A i )P (A i ) (9) i=1 A i 1.4 Ω Ω = A 1 + A 2 + + A n (10) P (B A i )P (A i ) = P (A i B)P (B) (11) P (A i B) = P (B A i)p (A i ) P (B) = P (B A i )P (A i ) n i=1 P (B A i)p (A i ) (12) P (A i ) B P (A i B) B ( ( ) ) 10 1 5 3 2 2 2 3 5 1 2 2 1 2 2 1 2 P ( ) = P ( )P ( ) P ( )P ( ) + P ( )P ( ) + P ( )P ( ) (13) 3
P ( ) = 2 10, P ( ) = 5 10, P ( ) = 3 10 (14) P ( ) = 6 11, P ( ) = 5 11, P ( ) = 5 11 (15) P ( ) = 3 13 (16) 2 2/10 = 2.6/13 0.4/13 99% (+) 1% 10 (0.001%) P ( +) = P (+ )P ( ) P (+ )P ( ) + P (+ )P ( ) (17) P ( ) = 0.00001, P ( ) = 1 P ( ) = 0.99999 (18) P ( +) = P (+ ) = 0.99, P ( ) = 0.01 (19) 0.99 0.00001 0.001 (20) 0.99 0.00001 + 0.01 0.99999 0.1% 4
2 2.1 1 0 10 X F (x) F (x) = P (X x) (21) X x X a b P (a X b) = F (b) F (a) (22) F X x i f i = P (X = x i ) F (x) = f i (23) x i x x i x x i X X 0 1 l l 0.1 0 ( 0) 0.1 0.1 0 1mm 100g 5
f(x) F (x) f(x) = df (x) dx F (x) f(x) x x X x + x (24) P (x X x + x) = F (x + x) F (x) (25) x x x + x F (x) f(x) F (x + x) = F (x) + f(x) x (26) P (x X x + x) = f(x) x (27) f(x) f(x) a X b F (b) F (a) = b a f(x)dx (28) f(x) F (x) = x f(x)dx (29) 0 f(x) 1 (30) f(x)dx = 1 (31) F (x) 2.2 ( ) X ( )E(X) E(X) = xf(x)dx ( ) (32) E(X) = i x i f i dx ( ) (33) 6
X h(x) E{h(X)} = h(x)f(x)dx ( ) (34) E{h(X)} = i h(x i )f i dx ( ) (35) h(x) = X k (k ) X k E{X k } = x k f(x)dx (36) X 1 X X V (X) 2 1 2 V (X) = E(X 2 ) {E(X)} 2 (37) 2 E{(X E(X)) 2 } = E{X 2 2XE(X)+(E(X)) 2 } = E(X 2 ) 2E{XE(X)}+E{E(X) 2 } (38) E(X) E(X) E(X) E{(X E(X)) 2 } = E(X 2 ) 2E(X)E(X) + {E(X)} 2 = E(X 2 ) {E(X)} 2 (39) 2 S S = V (X) (40) 2.3 (Moment generating function) M(t) = E{e tx } = d k M(t) dt k t=0 = d k dt k etx t=0 f(x)dx = e tx f(x)dx (41) x k f(x)dx = E(X k ) (42) M(t) k k t=0 t t = 0 e x = 1 + x + 1 2 x+ 1 3! x3 + = 7 k=0 x k k! (43)
{ } (tx) k M(t) = E = k! k=0 k=0 1 k! E(Xk ) = 1 + E(X) + 1 2 E(X2 ) + 1 3! E(X3 ) + (44) (characteristic function) Ψ(t) = E{e itx } = e itx f(x)dx (45) i (i 2 = 1) X d k Ψ(t) dt k t=0 = i k E(X k ) (46) ( ) Ψ(t) = M(it) (47) Ψ(it) = M( t) (48) ( ) X k (cumulant)c k G(t) log M(t) = k=1 1 k! C k (49) k = 1 G(t) M(t) t log(1 + x) = x 1 2 x2 + 1 3 x3 (50) C 1 = E(X) (51) C 2 = E(X 2 ) {E(X)} 2 = V (X) (52) k k 1 2 8
3 3.1 DU(n) X n x 1, x 2,..., x n f i = P (X = x i ) = 1 n DU(2) DU(6) X = 1, 2, 3,..., n E(X) = 1 n E{X 2 } = 1 n k=1 k 2 = k=1 k = n + 1 2 (n + 1)(2n + 1) 6 V (X) = E{X 2 } {E(X)} 2 = n2 1 12 (53) (54) (55) (56) 3.2 B N (n, p) p 1 p ( ) n X X = x( n x) f(x) = n C x p x (1 p) n x (57) n C x n x nc x = x! n!(n x)! (58) f(x) (a + b) n = nc m a m b n m (59) 1 = {p + (1 p)} n ( n ) m=0 1 = {p + (1 p)} n = nc x p x (1 p) n x = f(x) (60) x=0 x=0 9
E(X) = x n C x p x (1 p) n x = n x=0 n 1C x 1 p x (1 p) n x (61) x=1 n 1 = np n 1C x p x (1 p) n x 1 = np{p + (1 p)} n 1 = np (62) x=0 E(X 2 ) E{X(X 1)} E{X(X 1)} = x(x 1) n C x p x (1 p) n x = n(n 1) x=0 n 2C x 2 p x (1 p) n x (63) n 2 = n(n 1)p 2 n 2C x p x (1 p) n x 2 = n(n 1)p 2 {p + (1 p)} n 1 = n(n 1)p 2 (64) x=0 V (X) = E(X 2 ) {E(X)} 2 = E{X(X 1)} + E(X) {E(X)} 2 = np(1 p) (65) n x f(x) = n C x ( 1 2 x=2 ) x ( ) n x 1 (66) 2 E(X) = np = n 2 (67) V (X) = np(1 p) = n 2 (68) 10 1.6 3.3 P 0 (λ) (1 10 23 ) ( ) 10
X λ f(x) = λx x! e λ (69) Ψ(t) = x=0 λ x x! e λ e itx = e λ E(X) = 1 i V (X) 2 x=0 1 ( λe it ) x { ( = exp λ e it 1 )} (70) x! dψ(t) dt = λ (71) t=0 V (X) = d2 log Ψ(t) dt 2 t=0 = λ (72) 3.4 U(α, β) X α X β f(x) = { 1 β α (α x β) 0 ( ) (73) E(X) = β α E(X 2 ) = xf(x)dx = 1 β xdx = α + β β α α 2 β α x 2 f(x)dx == 1 β 3 α 3 β α 3 V (X) = (β α)2 12 (74) (75) (76) 4 4.1 N(µ, σ 2 ) µ σ 2 f(x) = 1 2πσ e (x µ)2 2σ 2 (77) 11
4.2 N(0, 1) ϕ(x) ϕ(x) = 1 2π e x2 /2 (78) N(µ, σ 2 ) z Z = X µ σ ϕ(z) σ (79) ϕ(x)dx = 1 (80) e x2 /2 dx = 2π (81) I = e x2 /2 dx = e y2 /2 dy (82) ( ) ( ) I 2 = e x2 /2 dx e y2 /2 dy = dx dy e (x2 +y 2 )/2 (83) xy x 2 + y 2 (x, y) 2 r 2 = x 2 + y 2 (84) dx dy (85) r r + r r 2πr r r dx 12 dy = 0 2πrdr (86)
I 2 = 0 2πre r2 /2 dr (87) d 2 dr e r /2 = re r2 /2 I 2 = 2π 0 (88) ( ) d 2 dr e r /2 dr (89) I 2 = 2π [e ] r2 /2 = 2π (90) 0 4.3 E(X) = 1 2π xe x2 /2 dx = 0 (91) xe x2 /2 0 ( ) dj(λ) dλ = λ=0 J(λ) = e λx2 /2 dx (92) ( ) d ( ) 2 dλ e λx /2 x 2 dx = 2 λ=0 2 e λx /2 dx = frac12 x 2 e x2 /2 dx λ=0 (93) λ 2 2π e λx /2 dx (94) N(0, 1/ λ) 1 J(λ) = 2π λ (95) ( E(X 2 ) = 2 1 dj(λ) = 2 1 ) d 2π 2π dλ λ=0 2π dλ λ = 1 (96) λ=0 13
V (X) = E(X 2 ) {E(X)} 2 = 1 (97) 0 1 N(µ, σ 2 ) Z X = µ + σz (98) Z E(X) = E(µ + σz) = µ + σe(z) = µ (99) V (X) = V (µ + σz) = E{(µ + σz) 2 } {E(µ + σz)} 2 = σ 2 (100) µ σ 2 ( σ ) 4.4 M(t) = E{e tx } = 1 e tx (x µ)2 /2σ 2 dx = exp (µt + σ2 t 2 ) 2πσ 2 (101) G(t) = log M(t) = µt + σ2 t 2 (102) 2 t 2 3 0 3 0 4.5 Z 0 Z z Z P ( σ X µ σ) 0.682 P ( 2σ X µ 2σ) 0.954 14
P ( 3σ X µ 3σ) 0.997 2 3 5 2 5.1 X Y X Y F (x, y) = P (X x, Y y) (103) X Y 1 F 1 (x) = P (X x) = F (x, ) (104) F 2 (y) = P (Y y) = F (, y) (105) f(x, y) = 2 F (x, y) (106) x y XY x y dx f(x, y) x y (107) f(x, y) 0 (108) X Y f 1 (x) = df 1(x) dx f 2 (x) = df 2(y) dy X Y dyf(x, y) = 1 (109) = = f(x, y)dy (110) f(x, y)dx (111) f(x, y) = f 1 (x)f 2 (y) (112) 15
5.2 2 h(x, Y ) E{h(X, Y )} = E(X) = µ 1, E(Y ) = µ 2 (113) V (X) = σ 2 1, V (Y ) = σ 2 2 (114) dx dyh(x, y)f(x, y) (115) h(x, Y ) X Y X Y h(x, Y ) = h 1 (X)h 2 (Y ) (116) E{h(X, Y )} = E{h 1 (X)}E{h 2 (Y )} (117) X Y 2 2 Cov(X, Y ) = E{(X µ 1 )(Y µ 2 )} = E(XY ) E(X)E(Y ) = E(XY ) µ 1 µ 2 (118) X Y Z Z 1 = X µ 1 σ 1, Z 2 = Y µ 2 σ 2 (119) ρ ρ Corr(X, Y ) Cov(Z 1, Z 2 ) = Cov(X, Y ) V (X)V (Y ) (120) ρ = ±1 1 ρ 1 (121) Z 2 = ±Z 1 (122) Y = µ 2 ± σ 2 σ 1 (X µ 1 ) (123) Y X 16
X Y E(XY ) = E(X)E(Y ) 0 0 0 2 ρ > 0 X Y ρ < 0 X Y 5.3 2 X Y U = ax + by (124) E(U) = E(aX + by ) = ae(x) + be(y ) = aµ 1 + bµ 2 (125) V (U) = E{(aX + by ) 2 } {E(aX + by )} 2 = a 2 V (X) + b 2 V (Y ) + 2ab Cov(X, Y ) (126) U X Y 5.4 2 2 N 2 (µ, Σ) µ = ( µ1 µ 2 ) (127) Σ = ( V (X) Cov(X, Y ) Cov(X, Y ) V (Y ) ) ( σ 2 = 1 ρσ 1 σ 2 ρσ 1 σ 2 σ2 2 ) (128) [ { 1 f(x, y) = 2πσ 1 σ 2 (1 ρ2 ) exp 1 (x µ1 ) 2 2(1 ρ 2 ) σ 2 1 2 ρ(x µ 1)(y µ 2 ) σ 1 σ 2 + (y µ 2) 2 X Y z [ ] 1 f(z 1, z 2 ) = 2π (1 ρ 2 ) exp 1 2(1 ρ 2 ) (z2 1 2ρz 1 z 2 + z2) 2 σ 1 σ 2 2 }] (129) (130) ρ = 0 [ 1 f(x, y) = exp 1 ( ) ] [ 2 x µ1 1 exp 1 ( ) ] 2 y µ2 = f 1 (x)f 2 (y) (131) 2πσ1 2 2πσ2 2 17 σ 2
2 0 2 x + y f(x, y) = [ 1 exp 1 2πσ 1 σ 2 2 ( ) ] 2 x µ1 σ 1 [ 1 exp 1 2πσ2 2 f(z 1, z 2 ) ( ) ] 2 y µ2 = f 1 (x)f 2 (y) (132) u = z 1 + z 2, v = z 1 z 2 (133) σ 2 1 2(1 ρ 2 ) (z2 1 2ρz 1 z 2 + z2) 2 = 1 ( ) u 2 2 1 + ρ + v2 1 ρ (134) U = Z 1 + Z 2 V = Z 1 Z 2 2 Z 1 Z 2 (ρ = 0) U f(u) = 1 2π e u2 /2 (135) 6 6.1 X 1, X 2,..., X n µ σ 2 ( ) X n = 1 n (X 1 + X 2 + + X n ) (136) n E( X n ) = µ (137) V ( X n ) = 1 n 2 [ E{(X1 + + X n ) 2 } {E(X 1 + + X n )} 2] = σ2 n (138) 1/ n 18
6.2 (law of large numbers) n X n µ X n µ E( X n ) = µ X n n X n µ 1 ( )µ 6.3 (central limit theorem) n n X n N(µ, σ 2 ) ( ) ( n ) n N(µ, σ2 n ) : n i 1 0 X i (i = 1, 2,..., n) M M = X 1 + X 2 + + X n (139) M B N (n, 1/2) n n M N(n/2, n/4) 7 7.1 n µ σ 2 X 1, X 2,..., X n ( 19
) X = 1 n (X 1 + X 2 + + X n ) (140) n N(µ, σ2 n ) SX 2 = 1 (X i n X) 2 = 1 Xi 2 n X 2 (141) i 1 V ( X) 2 2 i 1 E( X) = µ, V ( X) = E{ X 2 } {E( X)} 2 = σ2 n (142) E( X 2 ) = σ2 n + µ2 (143) X i µ σ 2 E(X 2 i ) = σ 2 + µ 2 (144) E(SX) 2 = σ 2 + µ 2 σ2 n µ2 = σ 2 σ2 n (145) σ2 n (bias) (unbiased) ˆσ 2 x = 1 n 1 (X i X) 2 (146) i 1 x 1, x 2,..., x n ( ) x = 1 n (x 1 + x 2 + + x n ) (147) n N(µ, σ2 n ) s 2 x = 1 n (x i x) 2 = 1 n i 1 20 x 2 i x 2 (148) i 1
2 ˆσ 2 x = 1 n 1 (x i x) 2 (149) i 1 x ˆσ 2 x 7.2 x x 100 X B N (100, 1/2) N(50, 25) z Z = X 50 25 = X 5 10 (150) Z N(0, 1) 3 Z 3 99% (99.7%) Z z 3 X 5 10 3 (151) 35 X 65 (152) 35 65 99.7% 2 Z 2 95.4% 40 60 100 p X B N (100, p) 21
N(100p, 100p(1 p)) Z Z = X 100p 100p(1 p) (153) Z N(0, 1) Z 99.7% 3 Z 3 Z 100p 3 10 p(1 p) X 100p + 3 10 p(1 p) (154) X 100p X 100 0.3 p(1 p) p X 100 + 0.3 p(1 p) (155) p p p p X 100 p X 100 0.3 X 100 ( 1 X ) p X 100 100 + 0.3 X 100 ( 1 X ) 100 (156) p X p 99.7% 99.7% 99% 95.4% 2 3 95% 99% 2.575 Z 2.575 95% 1.96 Z 1.96 100 40 99.7% p 0.25 p 0.55 95.4% 0.3 p 0.5 p = 0.5 ( ) 99% 99% 99% ( ) 99% p p N(100p, 100p(1 p)) 99% ( 40 ) 7.3 ( ) σ 2 n ( n ) X Z n( X µ) Z = (157) σ 22
N(0, 1) 95.4% µ 2 Z 2 (158) X 2 σ n µ X + 2 σ n (159) N(0, 1) X zα Z zα 1 α zα α z 0.05 = 1.96 µ 1 α X zα σ µ X + z σ n α (160) n 7.4 ( ) σ 2 ˆσ n 2 σ 2 ˆσ n 2 Z T T = n( X µ) ˆσ n (161) N(0, 1) T n 1 t n t t n 1 t α t n 1,α µ 1 α X t ˆσ n n 1,α µ X + t ˆσ n n n 1,α (162) n 7.5 χ 2 Z 1, Z 2,, Z n N(0, 1) n 2 S 2 = Z1 2 + Z2 2 + + Zn 2 (163) n χ 2 χ 2 n n = 1 N(0, 1) Z 2 S = Z 2 P (0 S s) = P ( Z s) = 2 s e x2 /2 dx (164) 2π 23 0
2 s 2π 0 y = x 2 (165) dy dx = 2x = 2 y (166) e x2 /2 dx = 1 2π s 0 1 y e y/2 dy (167) S f(s) = dp (0 S s) ds = 1 2π 1 y e y/2 (168) 8 100 60 40 (null hypothesis) 100 60:40 24
1/2 B(100, 1/2) 100 n P (n) = 1 2 100 100 C n (169) 60 1% 1 60 60 p(n 60) = 1 2 100 100 100C n (170) n=60 B N (100, 1/2) N(50, 25) Z Z = X 50 25 = X 5 10 (171) N(0, 1) X = 60 Z = 2 P (Z 2) P (Z 2) = 0.023 (172) 2 Z 2 0.954 Z 2 P (Z 2) = 1 P (Z 0) P (0 Z 2) = 1 0.5 P ( 2 Z 2)/2 (173) 100 60 2.3% 100 1000 23 60 p p 2.3% 1 100 2 100 2 100 20 25
20 1 0.05 60 0.05 ( ) 1. 2. 3. 4. 1 4 2 3 2 3 ( ) () p p 26
p 0.05 27