統計学のポイント整理

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1 .. September 17, / 55

2 n! = n (n 1) (n 2) 1 0! = 1 10! = = n k np k np k = n! (n k)! (1) P 3 = 5! = = 60 (5 3)! n k n C k nc k = npk k! = n! k!(n k)! (2) 5 3 5C 3 = 5! = 10 3!(5 3)! 2 / 55

3 1 6 X P (X = 1) = 1/6, P (X = 2) = 1/6,, P (X = 6) = 1/ /6 1/6 1/6 1/6 1/6 1/6 X x 3 / 55

4 X X g 3000g 2750g 2750g 2751g g 1 4 / 55

5 P (X = x) P (x) X E(X) E(X) = x xp (x) (3) Var(X) = E[(X E(X)) 2 ] = E(X 2 ) [E(X)] 2 (4) X E(X) = = 3.5 V(X) = (1 3.5) (2 3.5) (6 3.5)2 1 6 = / 55

6 f(x) a b Figure 1 : P (a < X < b) = b a f(x)dx (5) P (x) = 0 6 / 55

7 E(X) = xf(x)dx (6) Var(X) = E[(X E(X)) 2 ] = E(X 2 ) [E(X)] 2 (7) f(x) = 1 (0 < x < 1) f(x) = 0 () Figure 2 : E(X) = E(X 2 ) = [ ] x 2 1 xdx = = [ ] x 2 x 3 1 dx = = Var(X) = E(X 2 ) [E(X)] 2 = 1 3 ( 1 2 ) 2 = / 55

8 X X θ θ X M(θ) = x e θx P (x) (8) e z = 1 + z + z2 + z3 + z4 + 2! 3! 4! M(θ) = ) (1 + θx + θ2 x 2 + θ3 x 3 + θ4 x 4 + P (x) 2! 3! 4! x = P (x) + θ xp (x) + θ2 x 2 P (x) + θ3 x 3 P (x) 2! 3! x x x x + θ4 x 4 P (x) + (9) 4! x 8 / 55

9 θ θ = 0 dm(θ) dθ = xp (x) = E(X) (10) θ=0 x θ 2 θ = 0 d 2 M(θ) dθ 2 = x 2 P (x) = E(X 2 ) (11) θ=0 x θ n θ = 0 X n M(θ) = e θx f(x)dx (12) 9 / 55

10 1 p n x P (x) = n C x p x (1 p) n x = 10 3 P (3) = 10 C = B(n, p) n! x!(n x)! px (1 p) n x (13) (p + q) n = n C 0 p n q 0 + n C 1 p n 1 q 1 + n C 2 p n 2 q 2 + = + n C k p n k q k + + n C n p 0 q n n nc k p n k q k (14) k=1 10 / 55

11 M(θ) = n e θx P (x) = x=0 = n x=0 n x=0 e θx n! x!(n x)! px (1 p) n x n! x!(n x)! (peθ ) x (1 p) n x = (pe θ + (1 p)) n (15) E(X) E(X 2 ) = dm(θ) dθ = d2 M(θ) dθ 2 θ=0 θ=0 = n(pe θ + (1 p)) n 1 pe θ θ=0 = np (16) = npe θ (pe θ + (1 p)) n 1 + npe θ (n 1)(pe θ + (1 p)) n 2 pe θ θ=0 = np + n(n 1)p 2 (17) Var(X) = E(X 2 ) [E(X)] 2 = np(1 p) (18) 11 / 55

12 [ f(x; µ, σ 2 ) = 1 exp 1 ( ) ] 2 x µ 2πσ 2 σ (19) Figure 3 : µ σ 2 X µ σ 2 X N(µ, σ 2 ) (20) 12 / 55

13 0 1 N(0, 1) f(z) = 1 2π e z2 2 (21) X N(µ, σ 2 ) X Z Z = X µ σ N(0, 1) (22) P (Z > z) = α z z α α 100% P (Z > z α ) = P (Z < z α ) = α 13 / 55

14 X x Y y P (X = x, Y = y) P (x, y) X, Y X\Y y 1 y 2 y m P (x) x 1 p 11 p 12 p 1m p 1 x 2 p 21 p 22 p 2m p 2 x n p n1 p n2 p nm p n P (y) p 1 p 2 p m 1 14 / 55

15 Y X x i P X(X = x i) X P X(X = x i) = m p ij = p i (23) j=1 Y n P Y (Y = y j ) = p ij = p j (24) i=1 15 / 55

16 Y = y j X X Y Y = y j X x i P (X = x i Y = y j ) P (X = x i Y = Y j ) = P (X = xi, Y = yj) P (Y = y j) (25) E(X Y = y j) = n i=1 x i P (x i, y j ) P (y j) (26) Var(X Y = y j) = E[(X E(X)) 2 Y = y j] = n [x i E(X Y = y j )] 2 P (x i, y j) P (y j) i=1 (27) Var(X Y = y j ) = E(X 2 Y = y j ) [E(X Y = y j )] 2 (28) 16 / 55

17 X Y P (x i, y j ) = P (x i )P (y j ) (29) X Y E(XY ) = i E(XY ) = E(X)E(Y ) (30) x i y j P (x i, y i ) j = i = i x i y i P (x i )P (y i ) j x ip (x i) y jp (y j) = E(X)E(Y ) j 17 / 55

18 X Y X Y Cov(X, Y ) = E[(X E(X))(Y E(X))] Corr(X, Y ) = = E(XY ) E(X)E(Y ) (31) Cov(X, Y ) Var(X) Var(Y ) (32) X Y Cov(X, Y ) = 0 X Y E(XY ) = E(X)E(Y ) Cov(X, Y ) = E(XY ) E(X)E(Y ) = 0 18 / 55

19 E(X + Y ) = E(X) + E(Y ) (33) Cov(X, Y ) = 0 Var(X + Y ) = Var(X) + Var(Y ) (34) Var(X Y ) = Var(X) + Var(Y ) 19 / 55

20 P (x, y) f(x, y) X Y X Y f(x, y) = 1 2πσ Xσ Y 1 ρ 2 XY [ ( 1 (x µx ) 2 exp 2(1 ρ 2 XY ) σx 2 + (y µ Y ) 2 σy 2 )] 2ρXY (x µx)(y µy ) σ Xσ Y (35) 20 / 55

21 X f X (x) f X(x) = f(x, y)dy [ 1 exp 1 ( ) ] 2 x µx 2πσX 2 σ X (36) µ X σ 2 X X E(X) = xf(x, y)dxdy xf X(x)dx = µ X (37) Var(X) = (x E(X)) 2 f(x, y)dxdy (x µ X ) 2 f X (x)dx = σ 2 X (38) Y µ Y, σ 2 Y 21 / 55

22 X Y Cov(X, Y ) (x E(X))(y E(Y ))f(x, y)dxdy = ρ XY σ Xσ Y (39) X Y ρ XY Figure 4 : / 55

23 µ σ / 55

24 θ ˆθ ˆθ θ E(ˆθ) = θ (40) ˆθ θ plim n ˆθ = θ (41) ˆθ θ Var(ˆθ) < Var( θ) (42) 24 / 55

25 X 1, X 2,, X n µ σ 2 n X = 1 n X i µ σ 2 /n n i=1 1 X 1, X 2,, X n N(µ, σ 2 ) n X = 1 n X i N(µ, σ 2 /n) n i=1 25 / 55

26 n N(µ, σ 2 /n) 1 n n Figure 5 : 26 / 55

27 S 2 S 2 = 1 n 1 n (X i X) 2 (43) i=1 X 1, X 2,, X n µ σ 2 n S 2 = 1 n (X i X) 2 σ 2 n 1 i=1 S 2 σ n n i=1 (Xi X) 2 27 / 55

28 n (X i µ) 2 = i=1 = n [(X i X) + ( X µ)] 2 i=1 n n (X i X) 2 + n( X µ) 2 + 2( X µ) (X i X) i=1 = (n 1)S 2 + n( X µ) 2 [ n ] [ n ] E (X i µ) = nσ 2 E ( X µ) i=1 nσ 2 = (n 1)E(S 2 ) + σ 2 E(S) = σ 4 i=1 i=1 = σ2 n E(S 2 ) = σ 2 (44) 4 f(x) E[f(X)] = f[e(x)] f(x) 28 / 55

29 χ 2 Z 1, Z 2,, Z n U = Z1 2 + Z Zn 2 (45) n χ 2 U χ 2 (n) X 1, X 2,, X n N(µ, σ 2 ) n ( ) 2 Xi µ χ 2 (n) (46) i=1 σ S 2 (n 1)S 2 σ 2 = n ( ) 2 Xi X χ 2 (n 1) (47) i=1 σ 29 / 55

30 t t Z U k χ 2 Z U T = Z U/k (48) k t T t(k) 2 X 1, X 2,, X n µ σ 2 n X S 2 T = X µ S/ n t(n 1) (49) σ 2 σ S n 1 t 30 / 55

31 t T = X µ S/ n = X µ σ/ n S2 /σ 2 = X µ σ/ (n 1)S2 N(0, 1) n σ 2 T n 1 t X µ σ/ n (n 1)S 2 σ 2 /(n 1) χ 2 (n 1) (50) t 31 / 55

32 F F U m χ 2 V n χ 2 U V F = U/m V /n (51) m, n F F F (m, n) n 1, n 2 S 2 1, S 2 2 S 2 1 S 2 2 F (n 1 1, n 2 1) (52) 32 / 55

33 µ α 1 α 33 / 55

34 σ 2 n X µ σ 2 /n Z = X µ σ/ n (53) 5 ( P z α/2 < X ) µ σ/ n < z α/2 = 1 α (54) 5 34 / 55

35 (54) ( ) σ σ P X z α/2 n < µ < X + z α/2 n = 1 α (55) X ± z α/2 σ n µ 1 α X x µ (1 α) 100% ( x z α/2 σ n, x + z α/2 σ n ) (56) Table 1 : α z α/ / 55

36 σ 2 σ 2 S 2 S 2 = 1 n 1 (53) σ S n (X i X) 2 (57) i=1 T = X µ S/ n (58) n 1 t P ( t (n 1) α/2 < X ) µ S/ n < t(n 1) α/2 = 1 α (59) 36 / 55

37 (59) ( ) P X t (n 1) S α/2 < µ < X + t (n 1) S n α/2 = 1 α (60) n x s µ 95% ( ) x t (n 1) s α/2, x + t (n 1) s (61) n n α/2 t n t α/2 z α/2 n n 37 / 55

38 X 0 1 P (X = 1) = p P (X = 0) = 1 p q X 1, X 2,, X n n X i B(n, p) (62) i=1 n B(n, p) N(np, npq) X = 1 n X i N(p, pq/n) n i=1 Z = X p pq/n N(0, 1) (63) 38 / 55

39 P ( z α/2 < X p pq/n < z α/2 ) = 1 α (64) ( ) P X z α/2 pq/n < p < X + z α/2 pq/n = 1 α (65) p, q ˆp = x, ˆq = 1 x ( ) P X z α/2 ˆpˆq/n < p < X + z α/2 ˆpˆq/n = 1 α (66) p (1 α) 100% 6 ( x z α/2 ˆpˆq/n, x + zα/2 ˆpˆq/n ) (67) 6 (64) ( X p) 2 < z 2 α/2 p(1 p) n (n + z 2 α/2 )p2 (2n X + z 2 α/2 )p + n X 2 < 0 p p 39 / 55

40 X 1, X 2,, X n N(µ, σ 2 ) P (χ 21 α/2(n 1) < (n 1)S 2 σ 2 χ 2 (n 1) (68) ) (n 1)σ2 S 2 < χ 2 α/2(n 1) = 1 α (69) ( (n 1)S 2 P χ 2 α/2 (n 1) < σ2 < ) (n 1)S2 (n 1) χ 2 1 α/2 (70) (1 α) 100% 7 7 χ 2 α/2% 40 / 55

41 µ µ = 6.0 µ < 6.0 µ = 6.0 X / 55

42 Step 1 H 0 H 1 H 0 : µ = 6.0 H 1 : µ < / 55

43 Step 2 α α α % % % 43 / 55

44 Step X N(6, /20) Z = X 6 0.5/ Z 20 X x = 5.75 Z z z = / 20 = / 55

45 Step 4. P (Z < 1.645) = 0.05 Z % Figure 6 : z % z z = / 55

46 Figure 7 : H 0 : µ = µ o H 1 : µ < µ o H 1 : µ > µ o H 1 : µ µ o 46 / 55

47 σ 2 σ 2 σ 2 H 0 : µ = 6.0 H 1 : µ < x = 5.75 s 2 = T = X µ S/ n 1 t n 47 / 55

48 t = % t = / 20 = t n 48 / 55

49 N(µ 1, σ 2 1) N(µ 2, σ 2 2) µ 1 µ 2 H 0 : µ 1 = µ 2 H 1 : (1)µ 1 > µ 2, (2)µ 2 > µ 1, (3)µ 1 µ 2 n 1, n 2 X 1 X 2 σ 2 1, σ 2 2 X 1 X 2 X 1 N(µ 1, σ 2 1/n) X 2 N(µ 2, σ 2 2/n) (71) 49 / 55

50 X 1 X 2 E( X 1 X 2 ) = E( X 1 ) E( X 2 ) = µ 1 µ 2 (72) Var( X 1 X 2 ) = Var( X 1 ) + Var( X 2 ) = σ σ 2 2 (73) 8 X 1 X 2 N(µ 1 µ 2, σ 2 1/n 1 + σ 2 2/n 2 ) (74) ( X 1 X 2) (µ 1 µ 2) σ 2 1 /n 1 + σ 2 2/n 2 N(0, 1) (75) µ 1 µ 2 = 0 Z = ( X 1 X 2) σ 2 1 /n 1 + σ 2 2/n 2 N(0, 1) (76) 8 50 / 55

51 α (1)Z > z α, (2)Z < z α, (3) Z > z α/2 σ 2 1, σ 2 2 S 2 1, S 2 2 Z = ( X 1 X 2 ) S 2 1 /n 1 + S 2 2/n 2 (77) n 1, n 2 51 / 55

52 p 1, p 2 H 0 : p 1 = p 2 H 1 : (1)p 1 > p 2, (2)p 2 > p 1, (3)p 1 p 2 n 1, n 2 X 1 X 2 X 1 p 1 p 1 (1 p 1 )/n 1 X 2 p 2 p 2 (1 p 2 )/n 2 ˆp 1 ˆp 2 E( X 1 X 2 ) = p 1 p 2 (78) Var( X 1 X 2 ) = p 1(1 p 1 ) + p 2(1 p 2 ) n 1 n 2 (79) 52 / 55

53 n 1, n 2 X 1 X 2 ˆp 1 ˆp 2 p 1 = p 2 p Z = p(1 p) X 1 X 2 ( 1 n n 2 ) N(0, 1) (80) p ˆp = n1 i=1 X 1i + n 2 i=1 X 2i n 1 + n 2 α (1)Z > z α, (2)Z < z α, (3) Z > z α/2 53 / 55

54 N(µ 1, σ 2 1) N(µ 2, σ 2 2) σ 2 1 σ 2 2 H 0 : σ 1 = σ 2 H 1 : σ 1 > σ 2 n 1, n 2 X 1 X 2 n 1 ( ) 2 X1i X 1 (81) i=1 n 2 σ 1 ( ) 2 X2i X 2 (82) i=1 σ 2 n 1 n 2 χ 2 54 / 55

55 n 1 n 2 F 1 n 1 ( ) 2 X1i X 1 F = n 1 1 σ 1 i=1 1 n 2 ( ) 2 X2i X 2 n 2 1 F (n 1, n 2 ) (83) i=1 σ 2 F = F > F (n 1 1,n 2 1) α 1 n n 2 1 n 1 (X 1i X 1) 2 i=1 n 2 (84) (X 2i X 2 ) 2 i=1 55 / 55

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