( )/2 hara/lectures/lectures-j.html 2, {H} {T } S = {H, T } {(H, H), (H, T )} {(H, T ), (T, T )} {(H, H), (T, T )} {1

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( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 1 2011 ( )/2 2 2011 4 1 2 1.1 1 2 1 2 3 4 5 1.1.1 sample space S S = {H, T } H T T H S = {(H, H), (H, T ), (T, H), (T, T )} (T, H) S = {1, 2, 3, 4, 5, 6} i i i = 1, 2,..., 6 S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2),..., (6, 5), (6, 6)} = {(i, j) i, j = 1, 2,..., 6} (i, j) i j S = {,...} 3 1.1.2 1 2.1 a) 1) 2 3

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 2, {H} {T } S = {H, T } {(H, H), (H, T )} {(H, T ), (T, T )} {(H, H), (T, T )} {1, 3, 5} {1, 2, 3, 4} {(1, j) j = 1, 2,..., 6} {(i, j) i + j = } {,..., } { } E, F E F E F E F EF E F E F E F = EF E F E c S\E E E E E = {H}, F = {F } E F = E c = {T } E F = S E = {(H, H), (H, T )}, F = {(H, T )}, G = {(T, H)}, D = {(T, T )} E F = {(H, T )} E G = E G = {(H, H), (H, T ), (T, H)} D c = E G A B = A B 1.2 4 p j P (H) = P (T ) = 1/2 P (j) = 1/6 6 1 P (1) = 1 12, P (6) = 3 12, P (2) = P (3) = P (4) = P (5) = 1 6 S = {e 1, e 2,..., e N } e j p j j = 1, 2,..., N p j 0 1 S 1 4 2.1 a 2) c 0 p j 1, N p j = 1 (1.2.1)

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 3 E = {e 1, e 2, e 3,..., e m } m E = p j (1.2.2) E e 1 e 2... e m S p j (1.2.1) E (1.2.2) p j 1.2.1 ( ) S S P S E P [E] 1. E S 0 P [E] 1 E 2. P (S) = 1 E 3. E 1, E 2 E 1 E 2 = P [ ] E 1 E 2 = P [E1 ] + P [E 2 ] S P P P 1.2.2 P [E c ] = 1 P [E] E c E (1.2.3) E F = P [E] P [F ] (1.2.4) P [E F ] = P [E] + P [F ] P [EF ] (1.2.5) 2 p 1 p q 1 q {(H, H), (H, T ), (T, H), (T, T )} P [ ] = p, P [ ] = q (1.2.6) P [{(H, H), (H, T )}] = p, P [{(H, H), (T, H)}] = q (1.2.7) P [{(T, H), (T, T )}] = 1 p, P [{(H, T ), (T, T )}] = 1 q (1.2.8)

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 4 P [{(H, H)}] = a, P [{(H, T )}] = b, P [{(T, H)}] = c, P [{(T, T )}] = d (1.2.9) a + b = p, a + c = q, c + d = 1 p, b + d = 1 q (1.2.10) P [{(H, H)}] = P [ ] = P [ ] P [ ] = pq (1.2.11) P [{(H, T )}] = P [ ] = P [ ] P [ ] = p(1 q) (1.2.12) P [{(T, H)}] = P [ ] P [ ] = (1 p)q (1.2.13) P [{(T, T )}] = P [ ] P [ ] = (1 p)(1 q) (1.2.14) 1.3 1.3.1 n > 0 ( ) n! := n (n 1) (n 2) 3 2 1 0! = 1 n n! 0 k n := k k!(n k)! ( ) r n n! 0 n i (i = 1, 2,..., r), n i = n := n 1 n 2 n 3 n r n 1! n 2! n 3! n r! 1 n n k Case 1: n k (a 1, a 2,..., a k ) a j j a j 1 n n n n n = n k (1.3.1) Case 2: n k (a 1, a 2,..., a k ) a j a 1 n a 2 a 1 (n 1) n P k n (n 1) (n 2) (n k + 1) = n! (n k)! (1.3.2) Case 3: n k case 2 (a 1, a 2,..., a k ) a j Case 2 k k! n! (n k)! 1 ( ) n k! = = n C k (1.3.3) k Case 3

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 5 1.3.2 ( ) 1 n (x + y) n = n k=0 ( ) n x k y n k k Case 4. Case 3 n n 1, n 2,..., n r r r n i = n n n 1 n n 1 n 2 n n 1 n 2 n 3 ( n n 1 ) ( n n1 n 2 ) ( ) ( ) n n1 n 2 n! 1 = n 3 n 1! n 2! n 3! n r! = n n 1 n 2 n 3 n r (1.3.4) 1.4 5 1.4.1 ( ) (S, P ) E, F P [E F ] = P [E] P [F ] E F E, F (1.4.1) F E E F E F F E E, F F E 1.4.2 ( ) (S, P ) E, F P [F ] 0 P [ E F ] := P [E F ] P [F ] (1.4.2) F E 1.4.3 E F P [E F ] = P [E] E F F E P [E] P [E F ] P [F ] P [E] = P [E F ] P [F ] + P [E F c ] P [F c ] (1.4.3) P [E] 2.A 1. 2. 5 2.1 b

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 6 2.B 1 2 1. 2. 2.C 1 2 2.D 200 0.9 0.3 200 2.E 2 6 1/6 2 6 1.5 6 P [F E] E F E F 1.5.1 (Bayes ) (S, P ) E, F S P [F E] = P [F E] P [E] = P [E F ] P [F ] P [E F ] P [F ] + P [E F c ] P [F c ] (1.5.1) 3 F i i = 1, 2,..., k F i F j = for k i j F i = S P [F j E] = P [F j E] P [E] = P [E F j] P [F j ] k P [E F i ] P [F i ] (1.5.2) (1.4.3) P [F E] = P [E] = P [F E] P [E] (1.5.3) k P [E F i ] P [F i ] (1.5.4) P [E] (1.5.4) E F i 6 2.1 d

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 7 1.5.2 200 = r 1 I, II = r 2 I = r 0 I, II 0.9 = p 1 I 0.6 = p 2 0.3 = p 0 200 1.5.3 2.D 1.5.2 A A I II II 1.5.4 (1 p) p (1 q) q r p, q, r 1.5.5 m A p P < 1/2 A m 1 1. A 2. A A 3. p, m

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 8 1.5.6 1, 2, 3 1/3 i- p i p i 1 1- i- i = 1, 2, 3 1.5.7 (Laplace) i = 0, 1, 2,..., k (k + 1) i i/k 1. i i = 0, 1, 2,..., k 2. n i i = 0, 1, 2,..., k 3. (n + 1) 4. 2, 3 k 2, 3 1.5.8 1, 2, 3 200m i p i i = 1, 2, 3 i 1. 2. 3. 2 p 1 = 0.2, p 2 = 0.4, p 3 = 0.6 1 1, 2

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 9 2 2.1 7 8 2.1.A X X 1, 2, 3, 4, 5, 6 P [X = 1] = P [X = 2] =... = P [X = 6] = 1 6 (2.1.1) Y 0 10 Y 0, 10 P [Y = 0] = 4 6 = 2 3, P [Y = 10] = 2 6 = 1 3 (2.1.2) 2.1.B Z Z 2, 3, 4,..., 12 P [Z = 2] = 1 36, P [Z = 3] = 2 36 = 1, (2.1.3) 18 X x 1, x 2,..., x n X x i P [X = x i ] i = 1, 2,..., n 2.2 9 P [X = x i ] x i 2.2.1 X x 1, x 2,..., x n n ) P [X = x i ] = p i ( p i = 1 (2.2.1) 7 2.2 a b 8 9 2.2, b

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 10 X n E[X] := X := x i p i (2.2.2) E[X] X X [ (X ) ] 2 Var[X] := E E[X] = E [ X 2] E[X] 2 = X 2 (X X 2 ) 2 = X (2.2.3) σ := Var[X] Var[X] = σ 2 X p i F I[F ] 1 (F ) I[F ] := 0 ( F ) (2.2.4) P [F ] = E[ I[F ] ] = I[F ] (2.2.5) F I[F ] 2.2 c 2.3 10 2.3.A X [0, 1] 2.3.B Y 2.3.C Z 2.3.A X X X = 1 2 P [X = x i ] P [X = x i ] P [a X b] = f(x) 10 2.2 d b a f(x)dx (2.3.1)

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 11 2.3.A f(x) = 1 2.3.B 2.3.C 2.3.1 X f(x) (2.2.2) X E[X] := X := x f(x) dx (2.3.2) X [ (X ) ] 2 Var[X] := E E[X] = E [ X 2] E[X] 2 = X 2 (X X 2 ) 2 = X (2.3.3) X σ := Var[X] Var[X] = σ 2 2.2 e 2.4 11 2 X x 1, x 2,..., x n Y y 1, y 2,..., y m P [X = x i Y = y j ] = p ij (2.4.1) Y X m m P [X = x i ] = P [X = x i Y = y j ] = p ij (2.4.2) X Y P [Y = y j ] = n n P [X = x i Y = y j ] = p ij (2.4.3) 2.4.1 (S, P ) X, Y a > 0 E[X + Y ] = E[X] + E[Y ] (2.4.4) E[aX] = ae[x] (2.4.5) 11 2.3

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 12 Var[aX] = a 2 Var[X] (2.4.6) X Y Cov(X, Y ) := (X X )(Y Y ) (2.4.7) Var[X + Y ] = Var[X] + Var[Y ] + 2Cov(X, Y ), (2.4.8) X, Y Proof. X x i i = 1, 2,..., N Y y j j = 1, 2,..., M P [X = x i Y = y j ] = p ij E[X + Y ] = ij p ij (x i + y j ) = ij p ij x i + ij p ij y j (2.4.9) M p ij = P [X = x i Y ] = P [X = x i ] N ( M ) N p ij x i = x i p ij = x i P [X = x i ] = E[X] (2.4.10) ij p ij y j = E[Y ] (2.4.11) ij E[X + Y ] = E[X] + E[Y ] E[aX] N N E[aX] = P [X = x i ](ax i ) = a P [X = x i ] x i = a E[X]. (2.4.12) Var[aX] E[(aX) 2 ] = E[a 2 X 2 ] = a 2 E[X 2 ] (2.4.13) Var[aX] = E[(aX) 2 ] ( E[aX] ) 2 = a 2 E[X 2 ] ( ae[x] ) 2 = a 2 E[X 2 ] a 2 ( E[X] ) 2 = a 2 Var[X]. (2.4.14) (2.4.8) X Y A, B R P [X A Y B] = P [X A] P [Y B] (2.4.15) X Y X Y E[XY ] = E[X] E[Y ], Var[X + Y ] = Var[X] + Var[Y ] (2.4.16) 2.4.2 n n N n N n 5 (1, 3, 2, 1, 1) 1, 2, 3 N 5 = 3

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 13 2.4.3 M N 1. 2. M X, Y f(x, y) P [a < X b c < Y d] = b a dx d c dy f(x, y) f X, Y Y X X P [a < X b] = P [a < X b < Y ] = X f 1 (x) = dy f(x, y) b a [ ] dx dy f(x, y) 2.4.1 3 X, Y,..., Z P [X = x i, Y = y j,..., Z = z k ] = P [X = X i ] P [Y = y j ]... P [Z = z k ] (2.4.17) n X 1 + X 2 + + X n = X 1 + X 2 + X n (2.4.18) X j X 1, X 2,... X 1 + X 2 + + X n = n X 1 (2.4.19) n n n Cov Cov Var[X 1 + X 2 + + X n ] = Var[X 1 ] + Var[X 2 ] + + Var[X n ] (2.4.20) n Var[X 1 + X 2 + + X n ] = nvar[x 1 ] (2.4.21) Var[X1 + X 2 + + X n ] = n Var[X 1 ] (2.4.22) n n n n n

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 14 2.5 12 (S, P ) X A R P [a X b] = I[a X b] (2.5.1) I[ ] 1 0 2.5.1 ( ) X a P [X a] X a (2.5.2) 2.5.2 ( ) X µ Var[X] a P [ X µ a] Var[X] a 2 (2.5.3) a (2.5.1) X X I[X a] a I[X a] = a I[X a] = a P [X a]. (2.5.4) Var[X] = X µ 2 X µ 2, I[X a] a 2 I[X a] = a 2 I[X a] = a 2 P [X a]. (2.5.5) a, b > 0 P [ X µ a] X µ n a n (a > 0, n ) (2.5.6) e b X µ P [ X µ a] e ab. (2.5.7) X e bx P [X a] e ab (2.5.8) 2.6 13 µ σ 12 13 2.4 P [a X b] = b a 1 [ exp 1 (x µ ) ] 2 dx (2.6.1) 2π σ 2 σ

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 15 N(µ, σ 2 ) X µ σ 2 σ µ = 0, σ = 1 1 Φ(x) = x Φ(x) := x e y2 /2 2π dy e y2 /2 2π dy (2.6.2) x 0 1 1.645 1.960 2 2.326 2.576 3 4 1 Φ(x) 1 2 0.1587 1 20 1 1 40 0.02275 100 1 200 1.350 10 3 3.167 10 5 X N(µ, σ 2 ) Z := X µ σ (2.6.3) Z X Z [ a µ P [a X b] = P σ Z b µ ] σ (2.6.4)

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 16 3 3.A 1 2 10000 0 10000 5 3.B 10000 4500 p 3.C 10000 1000 1 2 3.1 14 3.A 5000 5000 5000 n X 1, X 2, X 3,... S n := X i X i i X i = 1 X i = 0 S n n X 1, X 2,... X 1, X 2,... X 1 X 2 X 3 X x i p i Y y j q j i = 1, 2,..., n j = 1, 2,..., m X, Y P [ X = x i Y = y j ] = P [X = x i ] P [Y = y j ] X 1, X 2,... X i i i X i 14 2.5 a

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 17 Theorem 3.1.1 ( ) X 1, X 2,... S n := µ X i Var[X 1 ] Var[X 1 ] < n X i X i [ lim P Sn ] n n µ = 0 (3.1.1) ϵ > 0 [ S ] n P n µ > ϵ Var[X 1] n ϵ 2 (3.1.2) S n /n n µ S n n S n n X 1, X 2,... S n 3.1.2 (S, P ) X, Y a > 0 X, Y E[X + Y ] = E[X] + E[Y ], E[aX] = ae[x] (3.1.3) Var[aX] = a 2 Var[X] (3.1.4) Var[X + Y ] = Var[X] + Var[Y ]. (3.1.5) n E[S n ] = E[X i ] = nµ, Var[S n ] = n Var[X i ] = nvar[x 1 ], [ S ] n P n µ > ϵ Var [ Sn E [ Sn ] = µ (3.1.6) n ] = 1 n n 2 Var[S n] = 1 n Var[X 1] (3.1.7) 1 [ ϵ 2 VarVar Sn ] = Var[X 1] n n ϵ 2 (3.1.8) S n n S n /n 1/n n S n /n 3.1.2 X x i i = 1, 2,..., N Y y j j = 1, 2,..., M P [X = x i Y = y j ] = p ij E[X + Y ] = ij p ij (x i + y j ) = ij p ij x i + ij p ij y j (3.1.9)

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 18 M p ij = P [X = x i Y ] = P [X = x i ] N ( M ) N p ij x i = x i p ij = x i P [X = x i ] = E[X] (3.1.10) ij p ij y j = E[Y ] (3.1.11) ij E[X + Y ] = E[X] + E[Y ] E[aX] N N E[aX] = P [X = x i ](ax i ) = a P [X = x i ] x i = a E[X]. (3.1.12) Var[aX] E[(aX) 2 ] = E[a 2 X 2 ] = a 2 E[X 2 ] Var[aX] = E[(aX) 2 ] ( E[aX] ) 2 = a 2 E[X 2 ] ( ae[x] ) 2 = a 2 E[X 2 ] a 2 ( E[X] ) 2 = a 2 Var[X]. (3.1.13) (3.1.5) E[X i ] = 1 2, Var[X i] = 1 4 [ S n P n 1 ] > ϵ 2 1 4n ϵ 2 (3.1.14) 3.1.3 i X i X i n S n := X i S n n n S n n 3.1.4 X X N S N S N N i X i S N = X i n S n n

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 19 3.2 15 X 1, X 2,... X i µ N [ 1 ] S N := X i lim P N N S N µ = 0 S N N 3.2.1 X i i = 1, 2,... S N := N a < b µ := E[X i ], σ := Var[X i ] (3.2.1) X i, Z N := 1 σ N N ( Xi µ ) = S N S N σ N (3.2.2) [ ] b lim P e x2 /2 a Z N b = dx (3.2.3) N a 2π S N S N Nµ N N S N Nµ N N S N Nµ N Z N N Z N N p i 1 0 X i S N = 1 N N X i N N m P [ S N = m ] = ( ) N p m (1 p) N m, m ( ) N N! := N C m := m m! (N m)! p (3.2.4) q := 1 p X 1 = 1 p + 0 (1 p) = p, VarX 1 = (1 p) 2 p + (0 p) 2 (1 p) = p(1 p) = pq (3.2.5) 15 2.5 b

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 20 Z N Z N := S N Np pqn (3.2.6) N Z N N 3.2.2 3.1.3 N 49% 51% N = 100, 1000, 10000 3.2.3 3.2.2 49% 51% N 49% 51% 0.95 N 3.1.3. X i 1 6 1 6 X i = 1 6 (1 + 2 + 3 + 4 + 5 + 6) = 7 2, (Xi ) 2 = 1 6 (12 + 2 2 + 3 2 + 4 2 + 5 2 + 6 2 ) = 91 6, (3.2.7) X i Sn n Var[X i ] = 91 ( 7 ) 2 6 35 35 = 2 12, σ = 12 = X 1 = 7 [ 2, Var Sn ] n = 1 n Var[X 1] = 1 35 n 12 (3.2.8) (3.2.9) Sn n 7 2 [ S n P n 7 ] > ϵ 2 35 12 1 ϵ 2 n (3.2.10) 3.1.4. 1 4 i i = 0, 1, 2, 3 ( ) 3 (1 i ( 3 ) 3 i P [X = i] = (3.2.11) i 4) 4 X = 3 4, Var[X] = 9 16, σ = 3 4 (3.2.12) X S N = N X = 3 4 N, Var[S N ] = N Var[X] = 9 N. (3.2.13) 16 3 4 3.2.2. Z N i X i = 1 X i = 0 S N = N X i X i = 1 2, VarX i = 1 4, σ = 1 2 (3.2.14)

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 21 Z N Z N = S N N 2 2S N N = (3.2.15) 1 4 N N 49% 51% 0.49 S N N 0.51 S N N 1 1 2 100 Z N N 50 (3.2.16) [ P 0.49 S ] [ N N 0.51 = P Z N N ] N 50 50 N 50 e z2 /2 dz (3.2.17) 2π N 50 N 1/5 N = 100 e z2 /2 dz 0.1585, 1/5 2π (3.2.18) 10/5 N = 1000 e z2 /2 dz 0.4729, 2π (3.2.19) N = 10000 10/5 2 2 e z2 /2 dz 0.9545, (3.2.20) 2π 3.2.3. N 50 N 50 e z2 /2 dz 0.95 (3.2.21) 2π N Φ Φ(x) = x e z2 /2 (3.2.21) Φ b e z2 /2 a dz = 2π b e z2 /2 a < 0 ae z2 /2 dz = 2π N 50 N 50 e z2 /2 (3.2.21) ( N 2Φ 50 dz ( N = Φ 2π 50 dz 2π e z2 /2 a dz (3.2.22) 2π a e z2 /2 dz = Φ(b) Φ(a) (3.2.23) 2π dz = 1 Φ( a) (3.2.24) 2π ) N {1 Φ( )} = 2Φ( 50 ) 1 0.95 1 Φ ( N ) 0.025 = 1 50 40 N ) 1 (3.2.25) 50 (3.2.26) N 50 1.960 = N (50 1.960)2 = 9604 (3.2.27) N 9600

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 22 4 4.1 4.1 10 10 4.2 10 10 p p p p 4.2 4.1 4.1 10 10 10 10 10 ( 1 ) 10 9.77 10 4 10 3 (4.2.1) 2 5.1 a. 10 3 b. 10 a

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 23 a b 1. 9.77 10 4 2. 3. 9.77 10 4 4. 16 α 0.05 0.01 5% 1% 17 18 10 6 4 6 4 ( ) 10 (1 6 ( 1 ) 4 210 = 0.2051 (4.2.2) 6 2) 2 210 0.2 16 17 18

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 24 10 9.77 10 4 10 2 fair 10 fair 10 5 4.3 N N N 5% 1% 4.4 10 n n 5% 1% 4.2.1 4.3 4.4 N N 2 N 0.05 N 5 5% N 7 1% 4.4 X P [X = 0] = P [X = 10] = 1 10 45, P [X = 1] = P [X = 9] =, P [X = 2] = P [X = 8] = 1024 1024 1024, (4.2.3) P [X = 3] = P [X = 7] = 120 210 252, P [X = 4] = P [X = 6] =, P [X = 5] = 1024 1024 1024 (4.2.4)

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 25 4.3 n = 9 9 1% n = 9 n = 10 P [X 9] = 11 0.0107 (4.2.5) 1024 0.01 1% n = 0, 10 5% n = 0, 1, 9, 10 n = 2, 8 9 9 10000 10000 5000 0.00798 1% 4.3 4.2.2 4.4 1000 4.6 10000 n n 5% 1% 10000 n n = 5000 n < 5000 P [X n] < 0.01 (4.2.6) n σ = 1/2 Z N = 2 ( n N ) = n 5000 N 2 50 (4.2.7) Z N N = 10000 P [X n] = (n 5000)/50 e z2 /2 e z2 /2 (5000 n)/50 dz 2π dz = 1 Φ ( 5000 n). (4.2.8) 2π 50

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 26 1 Φ(x) 1% 5000 n 50 2.326 = n 4880 (4.2.9) n = 5000 n 5120 5% 5000 n 50 n 5080 1.645 = n 4920 (4.2.10) 4.7. 10000 2000 4.3 4.2 p = 1 2 p 10 p p 10 p 0.5 0.6 0.6309574 0.7 0.7411345 0.8 0.9 1.0 10 0.000977 0.00605 0.010000025 0.0282 0.050000034 0.107 0.349 1.0 p 10 p = 1 10 p = 1 2 p 10 p 0.01 0.01 p 0.6309574 10 0.01 0.01 10 10 p 0.6309574 p 1% p 0.6309574 5% p 0.7411345 10 p 10

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 27 4.4 4.4. 400 220 180 p 400 220 p 220 220 400 400 X 1, X 2, X 3,... X j j N X 1, X 2,..., X N µ = E[X 1 ] σ = Var[X 1 ] N 2 X N = 1 N N X j (4.4.1) N V N = 1 N 1 N ( Xj X ) 2 N (4.4.2) XN = X1 = µ, VN = Var[X1 ] = σ 2 (4.4.3) X N V N N µ σ 2 N µ X N µ N X 1, X 2,..., X N Z N = 1 σ N N Z N N ( Xj µ ) (4.4.4) b lim P [ a < Z e z2 /2 N < b ] = dz (4.4.5) N a 2π µ σ P [ a < Z N < b ] b a e z2 /2 2π dz (4.4.6) Z N N X 1, X 2,..., X N Z N σ NZ N = N ( Xj µ ) = N X N ( N Nµ = Z N = XN µ ) (4.4.7) σ

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 28 Z N P [ a < Z N < b ] 1 α a, b 1 α = b a e z2 /2 2π dz (4.4.8) a, b a, b a = b 1 α = b b e z2 /2 2π dz (4.4.9) b Z N < b X N µ N σ XN µ < b X N µ < b σ N (4.4.10) σ µ σ t- N V N N σ 2 σ VN µ X N µ < b VN N (4.4.11) 4.5 5.5 100 170 cm t- X i X i µ σ 2 Z N := 1 σ N N (X i µ) (4.5.1) N Z N 1 N N Y N := (X i µ) = ( N VN V X N N µ), XN := X i, VN := 1 N ( Xi N N 1 X ) 2 N (4.5.2) X i Y N (N 1) t- t- t- (4.5.1) µ σ µ σ (4.5.2) Y N µ σ V N X N 100 t- σ X i N Y N t-

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 29 5 5.1 1 3.2 5.1.1 ) 20 n 20 n n n 20 n n n n 6/30 n n 1 1 n n 1 n N n n n n x j j = 1, 2,..., n µ = := 1 n n x j, V = := 1 n n (x j µ) 2, σ = := V (5.1.1) X S µ µ σ 2 /n n n n X j nσ 2 n µ nσ 2 /n 2

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 30 σ n σ σ Z := µ µ σ/ n = n µ µ σ Z 0.95 (5.1.2) Z < 1.96 (5.1.3) Z < 1.96 = n µ µ < 1.96 = µ µ < σ 1.96 σ n (5.1.4) 0.95 µ µ, σ µ 95% n 0.95 µ 0.05 0.95 99% 1.96 2.576 5.1.2 ) n µ 175cm 175cm µ 0 H 0 : µ = µ 0 (5.1.5) 5% Z := µ µ 0 σ/ n = n µ µ 0 σ (5.1.6) (a) H 1 : µ µ 0 Z 0 0.05 H 0 Z > 1.96 µ 0 µ > 1.96 σ n (5.1.7) H 0 µ µ 0 µ = µ 0 H 0 5% (b) H 1 : µ > µ 0 Z 0.05 H 0 H 0 µ > µ 0 (c) H 1 : µ < µ 0 Z > 1.65 µ µ 0 > 1.65 σ n (5.1.8) Z 1.65 σ < 1.65 µ µ 0 < (5.1.9) n H 0 µ < µ 0

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 31 5.1.3 p n m p n m p n m p p := m n (5.1.10) p p n X ( ) n P [X = m] = p m (1 p) n m (5.1.11) m j 1, 0 X j X = n X j X j n Z = X pn np(1 p) (5.1.12) p p 95% Z < 1.96 m pn np(1 p) < 1.96 (5.1.13) p m n p p p p < 1.96 (5.1.14) p(1 p) n n p p n p p < 1.96 n p(1 p) (5.1.15) p(1 p) 1/2 1/ n p p n p p < 1.96 p(1 p) n (5.1.16) 5.1.4 p = 0.5 H 0 : p = p 0 p 0

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 32 H 0 Z = n (a) H 1 : p p 0 Z > 1.96 X np 0 np0 (1 p 0 ) = p p 0 p0 (1 p 0 ) n (5.1.17) H 0 p p 0 (b) H 1 : p > p 0 p p 0 p0 (1 p 0 ) n > 1.96 (5.1.18) Z > 1.65 p p 0 p0 (1 p 0 ) n > 1.65 (5.1.19) H 0 p > p 0 (c) H 1 : p < p 0 Z < 1.65 p p 0 p0 (1 p 0 ) n < 1.65 (5.1.20) H 0 p < p 0 5.1.5 n j n j j = 1, 2,..., 6 j p j p 0 j 2 3 H 0 : j p j p 0 j j = 1, 2,..., 6 j X j X j np j n j np j n j p j H 0 6 X 2 (n j np 0 j := )2 (5.1.21) (6 1) = 5 χ 2-2.4b n j X 2 n j np 0 j X 2 > χ 2 6 1(0.05) (5.1.22) 5% H 0 χ 2 - np 0 j 5.2 2 3.3 2

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 33 5.2.1 3.3b 2 10 100 n 1 n 1g n 2 n 2g Π 1, Π 2 2 j = 1, 2 Π j n j n jg p j 19 2 n 1 n 2 1 2 1 m 1 2 m 2 0 p 1 p 2 95% 100% 2 2 1 n 1g n 1b n 1 = n 1g + n 1b 2 n 2g n 2b n 2 = n 2g + n 2b m g = n 1g + n 2g m b = n 1b + n 2b n = n 1 + n 2 p j p j := n jg n j (j = 1, 2) (5.2.1) j = 1, 2 ( N p j, p j(1 p j ) ) n j p 1 p 2 ( N p 1 p 2, p 1 (1 p 1 ) + p 2(1 p 2 ) ) n 1 n 2 (5.2.2) (5.2.3) Z = ( p 1 p 2 ) (p 1 p 2 ) p1 (1 p 1 ) n 1 + p 2(1 p 2 ) n 2 (5.2.4) p 1 p 2 95% Z Z = 0 0.95 z < 1.96 ( p 1 p 2 ) (p 1 p 2 ) p1 (1 p 1 ) n 1 + p 2(1 p 2 ) n 2 < 1.96 (5.2.5) 19

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 34 1 p 1 p 2 p j p j ( p 1 p 2 ) (p 1 p 2 ) p1 (1 p 1 ) n 1 + p 2(1 p 2 ) n 2 (p 1 p 2 ) = ( p 1 p 2 ) ± 1.96 < 1.96 (5.2.6) p 1 (1 p 1 ) n 1 + p 2(1 p 2 ) n 2 (5.2.7) p 1 p 2 95% 20 5.2.2 3.3b H 0 : p 1 = p 2 H 0 H 0 p j p ( p 1 p 2 ) Z = ( 1 p(1 p) + 1 ) (5.2.8) n 1 n 2 p p 2 p = n 1g + n 2g n 1 + n 2 = m g n (5.2.9) Z ( p 1 p 2 ) = ( 1 p(1 p) + 1 ) (5.2.10) n 1 n 2 H 1 : p 1 p 2 Z > 1.96 H 0 p 1 p 2 ( 1 p(1 p) + 1 ) > 1.96 ( p 1 p 2 ) 2 ( 1 p(1 p) + 1 ) > (1.96) 2 3.84 (5.2.11) n 1 n 2 n 1 n 2 H 0 n j, n jg (3.23) H 1 : p 1 > p 2 p 1 p 2 ( 1 p(1 p) + 1 ) > 1.65 (5.2.12) n 1 n 2 H 0 H 1 : p 1 < p 2 p 1 p 2 ( 1 p(1 p) + 1 ) < 1.65 (5.2.13) n 1 n 2 H 0 20 X = a ± b a b X a + b

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 35 5.2.3 3.3a 5.3 200 500 5.3.1 X j j = 1, 2,..., n N(µ j, σj 2) Y = X 1 + X 2 + + X n N(µ, σ 2 ) µ = µ 1 + µ 2 +... + µ n σ 2 = σ1 2 + σ2 2 +... + σn 2 X j j = 1, 2,..., n Y = X1 2 + X2 2 +... + Xn 2 n χ 2 - Y f(y) = 1 ( y ) n/2 1 e y/2 2Γ(n/2) 2 (5.3.1) y 0 Y n χ 2 P [Y > C] = α C χ 2 n 1(α) 100α% p.49 X Y k χ 2 - X, Y T := X Y/k (5.3.2) k t- ( k + 1 ) Γ f(x) = 2 ( k ) πk Γ 2 ) (1 + x2 (k+1)/2 (5.3.3) k P [ T > C] = α C t k (α)

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 36 0.99 0.975 0.95 0.05 0.025 0.01 1 0.00 0.00 0.00 3.84 5.02 6.63 2 0.02 0.05 0.10 5.99 7.38 9.21 3 0.12 0.22 0.35 7.81 9.35 11.34 4 0.30 0.48 0.71 9.49 11.14 13.28 5 0.55 0.83 1.15 11.07 12.83 15.09 6 0.87 1.24 1.64 12.59 14.45 16.81 7 1.24 1.69 2.17 14.07 16.01 18.48 8 1.65 2.18 2.73 15.51 17.53 20.09 9 2.09 2.70 3.33 16.92 19.02 21.67 10 2.56 3.25 3.94 18.31 20 48 23.21 1: χ 2 n(α) n α X 1, X 2,..., X n N(µ, σ 2 ) µ n σ T := n 1 ( µ µ) σ (5.3.4) (n 1) t- X 1, X 2,..., X n N(µ, σ 2 ) µ n V σ χ 2 := σ2 σ 2 = V V V = σ 2 (5.3.5) (n 1) χ 2 - V := 1 n 1 n (x j µ) 2 (5.3.6) (n 1) T T := ( µ µ) n n 1 = ( µ µ) (5.3.7) σ V χ 2 χ 2 := σ2 σ 2 = V V = (n 1)V V (5.3.8) χ 2 n 1(0.05) χ 2 n 1(0.01) 5.3.1 5.3.2 1

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 37 1 µ σ n µ σ T T := n 1 ( µ µ) σ = µ µ σ/ n 1 (5.3.9) n 1 t- T < t n 1 (0.05) T µ 95% µ µ < t n 1 (0.05) σ n 1 (5.3.10) µ 95% t n 1 (0.05) 2 1 µ = µ 0 µ 0 H 0 : µ = µ 0 T = µ µ 0 σ/ n 1 (5.3.11) H 0 µ µ 0 T H 0 (n 1) t- H 1 : µ µ 0 T > t n 1 (0.05), µ µ 0 σ/ n 1 > t n 1(0.05) H 0 (5.3.12) H 1 : µ > µ 0 5% 10% T > t n 1 (0.10), µ µ 0 σ/ n 1 > t n 1(0.10) H 0 (5.3.13) H 1 : µ < µ 0 T < t n 1 (0.10), µ µ 0 σ/ n 1 < t n 1(0.10) H 0 (5.3.14) t- 5.3.3 1 1 µ σ n µ σ χ 2-5.3.1 χ 2 := σ2 σ 2 = V V (5.3.15)

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 38 (n 1) χ 2 - χ 2 V σ (n 1) χ 2 0.95 V = σ 2 V 95% χ 2 n 1(0.975) < V V < χ2 n 1(0.025) (5.3.16) V 95% χ 2 0.025 χ 2-5.3.4 2 N(µ 1, σ 2 ) N(µ 2, σ 2 ) µ 1, µ 2, σ 2 1 n 1 µ 1 σ 1 2 n 2 µ 2 σ 2 µ 1 µ 2 t- µ 1 N(µ 1, σ 2 /n 1 ) µ 2 N(µ 2, σ 2 /n 2 ) µ 1 mu 2 N(µ 1 µ 2, σ 2 /n 1 + σ 2 /n 2 ) Z = ( µ 1 µ 2 ) (µ 1 µ 2 ) 1 n 1 + 1 n 2 σ (5.3.17) µ 1 µ 2 σ σ 2 V = n 1 σ 2 1 + n 2 σ 2 2 n 1 + n 2 2 T = ( µ 1 µ 2 ) (µ 1 µ 2 ) 1 n 1 + 1 n 2 V (5.3.18) (5.3.19) T µ 1 µ 2 T T (n 1 + n 2 2) t- p.88 t- µ 1 = µ 2 t-

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 39 6 6.1 1 V = IR n (V 1, I 1 ), (V 2, I 2 ),..., (V n, I n ) R 1 (x, y) y = ax + b a, b n (x 1, y 1 ), (x 2, y 2 ),..., (x n, y n ) a, b 1 b = 0 n n x 1, x 2,..., x n n N(0, σx) 2 n y 1, y 2,..., y n n N(0, σy) 2 σ x, σ y a, b n y j = ax j + b j y j (ax j + b) = 0 (j = 1, 2,..., n) (6.1.1) ϵ j ϵ j = y j (ax j + b) (6.1.2) y j, x j y j (ax j + b) ϵ j N(0, σ 2 ) a, b a, b y = ax + b y = a x + b ϵ j := y j (a x j + b ) (6.1.3) a, b ϵ j = y j (a x j + b ) = y j (ax j + b) (a a)x j (b b) = ϵ j (a a)x j (b b) (6.1.4)

( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 40 ϵ j ϵ j (a a)x j (b b) ϵ j N(0, σ 2 ) n ( ) 2 n (ϵj ) 2 ϵj = = nσ 2 (6.1.5) ϵ j n ( ) ϵ 2 j = n (ϵ ) 2 = j n {ϵj (a a)x j (b b) } 2 (6.1.6) x j x 0 j = n (ϵj ) 2 + n {(a a)x j + (b b) } 2 = nσ 2 + n { (a a)x j + (b b) } 2 (6.1.7) nσ 2 2 2 a = a, b = b 0 n ( ) ϵ 2 j a, b a, b f(a, b) := n { yj (ax j + b) } 2 (6.1.8) a, b a, b 2 σ 2 x := 1 n a = σ xy σ x 2, b = ȳ a x (6.1.9) n (x j x) 2, σ xy := 1 n x j x, y n (x j x)(y j ȳ), (6.1.10)

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