Transcription

1 ( /main.tex) ,

2

3 W. (Political Arithmetic) ( ) Statistik 19 Statistik (Blaise Pascal : ) Pierre de Fermat 1607(1608?)-1665, 19 (Pierre-Simon Laplace, ) Leonhard Euler, ) 3

4 ( ) ( ) ( ) ( ) 19 Biometrics*( ) 19 0 * Biostatistics Biometrika 0 W. ( ) Sir A. ( , ) ( ) K. Abraham Wald, John Wilder Tukey, AT&T Bell Laboratories 1965 The future of data analysis 196 Mathematics of Computation 1965 Cooley Software Bit Statistics

5 ISBN GDP* GDP : Gross Domestic Product GDP (i) ( ) estattopportal.do (ii) 5

6 : : : statistics statisticum ( EBM) (iii)

7 URL(Uniform Resource Locator) Web http(hypertext Transfer Protocol) URL R (Win, Mac, Linux) R on Windows: ~takeshou/r/ Wikipedia

8 .1 (set) n > 0 n 1 a A b A a A a A a A b A b / A U U A B A B d a c b A B (a A, a / B) (1) ; {, 4, 6,, 0} () ; {x x = n, n = 1,,, 10}, {n n = 1,,, 10}, {n n 1 10 } U U A B A B A B A B 8

9 U U A B A B A c A A \ B = A B A,B x A x B A B B A A B A B B A A A A A A,B A B, B A A = B (empty set) {} ϕ ( ) ϕ A A, B A B (intersection) A B A, B A B (union) A B U Ω U (Universe set) U A U A (complement) A c A (empty set).1 ( ) : (1) A B = A B () A B = A B. (1) (A B) C = A (B C) () (A B) C = A (B C) (3) A B = B A (4) A B = B A (5) (A B) C = (A C) (B C) 9

10 .1 n,a 1, A,, A n n A i = A i = A 1 A A n i=1 i n A i = A 1 A A n i=1.1 n {A i, i = 1,,, n}. a, b, c {0, 1} A, B, C (i) 1 a A; (ii) a = 1 (1 a) (A) ; (iii) max{a, b} A B; (iv) max{a, b, c} A B C; (v) min{a, b} A B; (vi) min{a, b, c} A B C..3 x i A i i x A i x i A i i x A i [ ] (exist) (all) x i x i A i x A i, i A i x A i, i.4 n x i {1, 0}, i = 1,,, n (i) max{x 1, x,, x n } = 1 (1 x 1 ) (1 x n ) (ii) min{x 1, x,, x n } = x 1 x x n 10

11 . 5 5 =?.3 n n (n 1) (n )... 1 n (n 1) 1 (exclamation, )! n! n n! = n(n 1)(n ) 3 1 (n 1), 0! = 1 11

12 10 3! = 6, 4! = 4, 5! = 10, 6! = 70, 7! = 5040, 8! = 4030, 9! = 36880, 10! = r n 1. : n r = r {}}{ n n n (.1) (permutation) (n) r = r {}}{ n(n 1)(n ) (n (r 1)) =. : n! (n r)! (.) [n] r r! = r {}}{ n(n + 1)(n + ) (n + (r 1)) = r! ( ) ( ) n + r 1 n + r 1 = r n 1 (.3) (combination) (n) r r! = n(n 1)(n ) (n (r 1)) r! = ( ) ( ) n n = r n r r {}}{ (n) r = n(n 1)(n ) (n r + 1) r {}}{ [n] r = n(n + 1)(n + ) (n + r 1) (a 1, a, a 3,, a r ), {a 1, a, a 3,, a r } r! { } = r! { } (a, b, c) {a, b, c} A_ (.4)

13 (i) (ii) (iii) (iv) (v) 5 (i) 5 5! = (ii) 4 4 4! = 48 (iii) 1, 3, ! = 7 5 (i) (ii) (iii) = 7.6 {0, 1,, 3, 5} 5 (4 ) 0 (i) 5 (ii) 5 (iii) 5 (iv) 5 5 (v) 5 10 (i) ! = 96 (ii) ! = ! = = 4 (iii) (i) (ii) 5 5 1, 3, ! = 54 5 (ii) (i) 96 (iv) ! = ! = = ( ) 5 ( 5 ) = 5 4 = 10 10! ( 5 3 ) = = 10 ( 5 3! ) = (5 3 ).8 6 (i) 3 (ii) 13

14 (iii) (i) 6 3 ( 6 3 ) ( 3 ) (6 3 ) (3 ) (ii) (i) 6 ( 6 ) ( 4 ) (6 )(4 ) (iii) (ii) (ii) 45 n, r ( ) ( ) n n = ( r ) n ( r ) n + 1 n = r r ( ) n + r 1 ( ) n r n r 1 n 1 r 1 n 1 r 1 n, r 1 r ( ) π (0! = 1) n = (π) ( ) 3 π (π 1) (π ) 3 = = (3) 5 = 3 3! 6 5 5! 3 (3 1) (3 ) (3 3) (3 4) = 0 5! O = (0, 0) (North) (East) b a + b a O m, n = 0, 1,, (m, n) N(m, n) = ( m+n n ) = (m+n m ) N(0, 3) = 1, N(, 3) = N(3, ) = 10, N(3, 3) = 0 N(m, n) = k k (m, n) k = 15 (m, n) = (, 4), (4, ) k = 5 14

15 (m, n) = (5, 5) N(m, n) + N(m 1, n + 1) = N(m, n + 1) ( m+n n ) + (m+n n+1 ) = (m+n+1 n+1 ) (x + 1) m+n (x + 1) = (x + 1) m+n+1 x n.9 (m, n) N(m, n) = ( m+n n ) = (m+n m ) : a, b n ;(a + b) n (a + b) n = n {}}{ (a + b) (a + b) (a + b) n (a + b) a b a b a ( ) ( r b ) n r a r a r b b r n n = = (n) r = (n) n r r n r r! (n r)! ( ) ( ) ( ) ( ) ( ) n n n n n (a + b) n = a n b 0 + a n 1 b a n r b r + + a 1 b n 1 + a 0 b n 0 1 r n 1 n.4 (event) (probability). : N A #(A) A P(A) P(A) = #(A) N a A, {a} P({a}) = 1, a N n P({a 1, a,, a n }) = n N, a i A : ,,3,4,5,6 6 : Ω( ) #(Ω) < 15

16 1 A 0 P(A) 1 ( ) ϕ P(ϕ) = 0, 3 Ω P(Ω) = 1. : A,B A B = ϕ A B (mutually exclusive) (mutually disjoint) A B (exclusive event) A B A B A B = ϕ P(A B) = P(A) + P(B) A B = ϕ P(A B) = 0 P(A B) + P(A B) = P(A) + P(B) (.5) a b c = a b c {a i } i a i, Π i a i A 1, A,, A n n i=1 A i = i A i = A 1 A A 3 A ni=1 n A i = i A i = A 1 A A 3 A n = A 1 A A 3 A n ( 5 ) = 10 ( 3 ) = 3 (3 )/(5 ) = 3/ ( 1 3 ) = 0 3 A 3 B P(A) = ( 7 3 )/(1 3 ) = 35/0, P(B) = ( 5 3 )/(1 3 ) = 10/0 3 A B A B, A B = P(A B) = P(A) + P(B) = 35/0 + 10/0 = 45/0 = 9/44. ( ) : A A A A A (complement) A c A A P(A) = 1 P(A) (.6) ( 8 3 ) = 56 1 A P(A) 3 P(A) = ( 5 3 )/(8 3 ) = 10/56 P(A) = 1 P(A) = 1 ( 5 3 )/(8 3 ) = 3/8 16

17 .13 A, B, C (1) P(A B) = P(A) + P(B) P(A B) = P(A) + P(B) {P(A B) () P(A B) = 1 P(A B) = 1 P(A B) = 1 P(A B) (3) P(A B C) = P(A) + P(B) + P(C) {P(A B) + P(B C) + P(C A)} + P(A B C) = P(A) + P(B) + P(C) {P(A B) + P(B C) + P(C A)} + P(A B C) (4) P(A B C) = 1 P(A B C) = 1 P(A B C) = 1 P(A B C).5 (conditional probability); A B A B (conditional probability) P(B A) P(A) = 0 P(B A) = P(A B) P(A) (.7) Ω A B A A B A A B B A A B P(Ω) = 1, P(A B) P(B A) = P(A B) P(A) P(A B) = P(A) P(B A) (.8) P(B A) P(A) A B A B B A (independent), P(B A) = P(B A) P(A B) = P(A) P(B) A P(B A) = P(B) A B 17

18 P(A B) = P(A) P(B) (.9) n {A 1, A,, A n } (i), A i A j (ii) P(A 1 A A n ) = P(A 1 )P(A ) P(A n ) (i) (pairwise independent) A 3 B A = {, 4, 6, 8, 10, 1, 14}, B = {3, 6, 9, 1, 15} 3 p p 7 6 A B = {6, 1}. A B A B P(A B) P(B A) = = P(A) 7 {B 1, B,, B n } Ω (partition), (i) B i B j = ϕ (i = j), (ii) Ω = i B i.3 {B 1, B,, B n } Ω P(A) = P(A B 1 )P(B 1 ) + P(A B )P(B ) + + P(A B n )P(B n ) (.10) : 1 A 3 B 4 C A = {, 4, 6}, B = {3, 6}, C = {4, 5, 6}., P(B A) = P(B) A B P(C A) = P(C) A C A, B A B A B A B P(B A) = P(B) n , =

19 : E p n E r ( n r )pr (1 p) n r, (r = 0, 1,,, n) Binom(n, p) = 1 1 ( ) ( ( 4 ) 1 1 ) 5 4 = % 4% A 50 B P(A) = 60 = 0.6, P(A B) = P(A B) = 0.4. P(B A) = = P(A) a b a b 1 a A b B a 4 P(B A) = 4 = 1. P(A B) = P(A) P(B A) = = (1) a 1 b 1 a b () a 1 b 1 a b a A b B (1) A B A A 1 B PA(B) () A A B.6.4 A Ω {B 1, B,, B n } P(B i ),P(A B i ), (i = 1,,, n) P(B i A) = P(A B i) P(B i ) j P(A B j )P(B j ) 19

20 (belief) 1763 Thomas Bayes( ) ( ) n r θ θ (aleatory) (epistemological) ( ) P(B A) = P(B) P(A B) P(B) P(A B), B = {Y = b} A P(A B) = P(A Y = b) b A L(b) L(b) = ( ) P(A Y = b) 0

21 3 3.1 ( ):{H} {T} {1, 0} (random variable) X H (Head),T (Tail) 1 X 4 X X {0, 1, } (1, ) X (H, H) p (H, T) 1 p(1 p) (T, H) 1 p(1 p) (T, T) 0 (1 p) p X = 0 (1 p) X = 1 p(1 p) X = p X 0 1 (1 p) p(1 p) p 1 X x p X (x) (1 p) if x = 0 ( ) p X (x) = p(1 p) if x = 1 p p X (x) = p x (1 p) x, x = 0, 1, if x = x n ( ) n p X (x) = p x (1 p) n x, x = 0, 1,,, n (3.1) x X Binom(n, p) n, p (Binomial distribution) X x 1, x,, x n p i = P(X = x i ), i = 1,,, n (i) 0 p k, (ii) n p i = 1 x i p i i x x 1 x x n P(X = x) p 1 p p n 1,pdf(probability density function) (discrete density) 1

22 X (probability distribution) {X = x k } p X (x k ) = p k, k = 1,,, n {x 1, x,, x n } {X x}, F X (x) = P(X x), < x < (3.) (distribution function) (i), (ii) 0 1, (iii) f X (x) = d dx F X(x) F X (x) = x f X (t) dt (discrete type) P(a X b) = P(X = x i ) = p X (x i ) a x i b a x i b (continuous type) P(a X b) = f X(x) dx = {x : a x b} b a f X (x) dx 3. c X r x 1, x,, x r n nc X 1 + X + + X n c nc (X 1 + X + + X n ) = 0, c c = 1 n (X 1 + X + + X n ) P(X = x i ) = f X (x i ), i = 1,,, r N n (x) x X 1, X,, X n x N n (x) = i 1 {Xi =x} X 1 + X + + X n = r i=1 x in n (x i ), X 1 + X + + X n n = r N n (x x i ) i n i=1 *1 N n (x)/n n f X (x) = P(X = x) (3.3) *1

23 X x 1 x x r N n (x)/n N n (x 1 )/n N n (x )/n N n (x r )/n 1 f X (x) = P(X = x) f X (x 1 ) f X (x ) f X (x r ) 1 (3.3) µ = r i=1 x i f X (x i ) c µ > c µ = c X µ = i x i P(X = x i ) = x i p X (x i ) (3.4) i f X (x) = d dx F X(x) = d dx P(X x), ( < x < ) µ = x f X (x) dx (3.5) : X :{p X (x i ), i = 1,, }, :{ f X (x), < x < } µ = E(X) (expectation) (mean) average E(X) = x i p X (x i ) : E(X) = i x f X (x) dx : x m (x m) E[(X m) ] X m X m X (variance) V(X) V(X) = E[(X m) ] = i (x i m) p X (x i ) : V(X) = (x m) f X (x) dx : X (standard deviation) σ(x) σ(x) = V(X) V(X) (X m) = X mx + m V(X) = E[(X m) ] = E[X mx + m ] = E[X ] me(x) + m = E[X ] m V(X) = E[X(X 1)] + m m 3

24 3.1 (1) () 1 X X [0, 1] X (Uniform distribution) X Uni f {1,,, 6}, X Uni f [0, 1] P(X 1 + X = a) = P(X 1 = k)p(x = a k) k 3 ( ) X ( ) 3 X 1 P(X 1 + X = 7) = k P(X 1 = k)p(x = 7 k) = P(X 1 = 1)P(X = 7 1) + P(X 1 = )P(X = 7 ) P(X 1 = 6)P(X = 7 6) = (1/6)(5/36) + (1/6)(4/36) + (1/6)(3/36) + (1/6)(/36) + (1/6)(1/36) = 15/16 7 X 1 = 1,,, 6, X =, 3,, 1 X ( ):X 1 ( ) 1 (1/6) (1/6) 3 (1/6) 4 (1/6) 5 (1/6) 6 (1/6) (1/36) (/36) (3/36) (4/36) (5/36) (6/36) (5/36) (4/36) (3/36) (/36) (1/36) [ ] X a,b Y = ax + b Y (i) E(Y) = ae(x) + b (iii) σ(y) = a σ(x) (ii) V(Y) = a V(X) (b ) n X,Y (X, Y) 4

25 {X = a} {Y = b} (X, Y) = (a, b) f X,Y (a, b) = P(X = a, Y = b) f X (a) = b f X,Y (a, b), f Y (b) = a f X,Y (a, b) P(X 1 + X = a) = f X,Y (x, y) (3.6) {(x,y) x+y=a} x + y = a (x, y) E(X) = i x i p X (x i ) x f X (x)dx E[(X a) ] = i (x i a) p X (x i ) = i xi p X(x i ) a i x i p X (x i ) + a E[(X a) ] = (x a) f X (x)dx = x f X (x)dx a x f X (x)dx + a V(X) a = E(X) i xi p X(x i ) {E(X)} V(X) = x f X (x)dx {E(X)} (3.8) (X, Y) h(x, Y) h(x i, y j )p X,Y (x i, y j ) i j E[h(X, Y)] = h(x, y) f X,Y (x, y) dxdy {p X (x i, y j )} (x i, y j ) {p X (x i, y j )} p X,Y (x i, y j ) = p Y (y j ), p X,Y (x i, y j ) = p X (x i ) i j A i = {X = x i }, B j = {Y = y j } (Marginal distribution) (3.7) (3.9) 3.1 F X,Y (x, y) = P({X x} {Y y}) = P(X x, Y y) Y X x y {X x, Y y} (x, y) Y y (0, 0) x X 3. X Y (covariance) cov(x, Y) = E[(X µ X )(Y µ Y )] = E[XY] µ X µ Y 5

26 µ X, µ Y 3.3 : X a Y b {X = a} {Y = b} P({X = a} {Y = b}) = P(X = a, Y = b) P(X = a) P(Y = b) X Y 3.1 X Y F X,Y (x, y) = F X (x)f Y (y), x, y 3. X, Y, E(X + Y) = E(X) + E(Y) (3.9) h(x, y) = x + y 3.3 X, Y, E(X Y) = E(X) E(Y) X Y = X + 3 [ ] P(X = i) = 1 6, i = 1,,, 6. E(X) = 6 i=1 i 1 6 = ( ) 1 6 = 7 ( ) 7 V(X) = E(X ) = ( ) = = 35 1 Y E(Y) = E(X + 3) = E(X) + 3 = = 10, Y V(Y) = V(X) = = X,Y 3.4 (i) V(X + Y) = V(X) + V(Y) (ii) V(X Y) = V(X) + V(Y),cov(X, Y) = 0 V(X + Y) = V(X) + V(Y) + cov(x, Y) V(X Y) = V(X) + V(Y) cov(x, Y) 3. A,B 1 A

27 1 B A,B X,Y E(X + Y) E(XY) X Y X + Y XY {X = 1} 4/6 = /3 {X = 0} /6 = 1/3 {Y = 1} 1/6 {Y = } 3/6 = 1/ {Y = 3} /6 = 1/3 E(X) = 1 / /3 = /3, V(X) = 1 / /3 (/3) = /9 E(Y) = 1 1/6 + 1/ + 3 1/3 = 13/6, V(Y) = 1 1/6 + 1/ + 3 1/3 (13/6) = 17/36 {XY = 0} /6 = 1/3 {X + Y = } /6 = 1/3 {XY = 1} 1/6 {X + Y = 3} 3/6 = 1/ {XY = } /6 = 1/3 {X + Y = 4} 1/6 {XY = 3} 1/6 E(X + Y) = 1/ / + 4 1/6 = 17/6, V(X + Y) = 1/ / + 4 1/6 (17/6) == 17/36. E(XY) = 0 1/ /6 + 1/ /6 = 4/3, V(XY) = 0 1/ /6 + 1/ /6 (4/3) = 3 16/9 = 11/9. cov(x, Y) = E(XY) E(X)E(Y) = 4/3 /3 13/6 = 1/9 V(X) + V(Y) + cov(x, Y) = /9 + 17/36 + ( 1/9) = 17/36 V(X + Y) 3. X, Y X + Y XY X + Y 3.3, X X {0,1,,3} {X = 1} /6 5/6 3 3(1/6)(5/6) X X (5/6) 3 3(1/6)(5/6) 3(5/6) (1/6) (1/6) A p n A X X q = 1 p 7

28 X 0 1 k n ( ) ( ) ( ) ( ) ( ) n n n n n q n p 1 q n 1 p q n p k q n k p n k n (binomial distribution) Binom(n, p) P(X = k) = ( ) n p k q n k, k = 0, 1,,, n (3.10) k X X X q = 1 p ( ) ( ) n n 1 X k = n, k = 1,,, n 1 k k 1 n ( ) n E(X) = k p k q n k n 1 ( ) n 1 = np k p k 1 q n k = np k k=0 k=1 ( ) ( ) n n X X = X(X 1) + X k(k 1) = n(n 1), k =, 3,, n k k 1 X V(X) = n k=0 k(k 1)(n k )pk q n k + np = n(n 1)p n = n(n 1)p + np = npq k= (n k )pk q n k + np 3.5 X q = 1 p X Binom(n, p) = E(X) = np, V(X) = npq ( ) ( ) ( ) Binom(n, p) = ,, 8,, 8, : ( ) n p(k) = p k (1 p) n k, k = 0, 1,,, n k n p 3 (p = 3 10 ), (p = 10 5 = 1 ), (p = 8 10 ) :,p = 1 n = 8, 16, 3 3 B(8, 1 ),B(16, 1 ),B(3, 1 ) 8

29 X X X Binom(n, p), n = 100, p = P(X = k) = ( 100 k ) ( ) 7 k ( 310 ) 100 k, 10 k = 0, 1,,, 100 X E(X) = µx = k k P(X = k) k ( n k ) = n( n 1 k 1 ) µ X = 100 i=0 k (100 k ) ( ) 7 k ( 310 ) 100 k 10 = i=1 ( 99 k 1 ) ( ) 7 k ( 310 ) 100 k ( 10 = ) ( ) 99 = 70. X var(x) var(x) = E(X µ X ) = k (k µ X ) P(X = k) = k k P(X = k) (µ X ) = E(X ) µ X E(X ) = E(X(X 1)) + EX k(k 1) ( n k ) = n(n 1)(n k ) E(X(X 1)) = 100 i=0 k(k 1) (100 k ) ( ) 7 k ( 310 ) 100 k 10 = i= ( 98 k ) ( ) 7 k ( 3 ) 100 k ( = ) ( ) 98 = n(n 1)p = = var(x) = = 1 = np(1 p). : (Poisson) (rare event) p p (n λ p = p n 0, n, np n np n λ e exp(a) = e a = lim n (1 + a n )n a n ( n k )pk (1 p) n k λk k! e λ. Po(λ) X Po(λ) : p X (k) = λk k! e λ, k = 0, 1,, (3.11) np np(1 p) np λ, λ np(1 p) = np (np) p λ λ 0 = λ 3.4 X Po(λ) E(X) = λ, var(x) = E(X ) {E(X)} = λ ; n Z N(0, 1) ϕ(z) = 1 ( exp 1 π z) (3.1) 9

30 Z, f Z (z) ϕ(z) ( ) N(0, 1) 1 X X y = f (x) (1) () y = f (x) x 1 b (3) a x b X f (x)dx a f (x) X X P(X = x) = lim h 0 P(x X x + h) = x+h lim h 0 f (x)dx = 0 P x P(X = a) = P(X = b) = 0 P(a X < b), P(a < X b), P(a < X < b), P(a X b), X µ, σ f X (x) = 1 (x µ) e σ = 1 (x µ) exp{ πσ πσ σ } (3.13) X N(µ, σ ) X µ σ 1 1 πσ πσ µ σ σ X N(µ, σ ) E[X] = x f X (x)dx = µ, var[x] = E(X ) E(X) = x f X (x)dx µ = σ X (1) x = µ y x = µ () x (3) σ x = µ 30

31 X N(µ, σ ) Z = X µ N(0, 1) σ σ Z N(0, 1) ϕ(z) = 1 π e z < z < (standard normal distribution) = π Φ(x) = P(Z x) = x ϕ(z)dz = x 1 π e z dz x Φ(x) X N(µ, σ ) Z = X µ σ X = σz + µ Z N(0, 1) (i) P ( 0.5 Z 0.5) = (ii) P ( 1 Z 1) = (iii) P ( Z 1.645) = (iv) P ( 1.96 Z 1.96) = (v) P ( Z ) = (vi) P ( 3 Z 3) = ( z F(z) = F X (z) =NORMSDIST(z) *F(z) -1 P z X µ ) ( z = F(z) 1. p p = P z σ p/ X µ ) z σ p/ = ) ) P ( z p/ Z z p/ = P ( Z z p/ z p/, =NORMSINVDIST((1+p)/) z p/ (Normal distribution table) : Φ(z) = F Z (z) = P(Z z) = 1 P(Z > z) = 1 P( Z < z) = 1 P(Z z) = 1 F Z ( z) = 1 Φ( z) 3.5 Z, (1) P(Z <.5) () P(1 Z <.5) (3) P( 0.8 < Z) (4) P( 0.8 < Z < 1.5) (1) P(Z <.5) = P(Z.5) = Φ(.5) = () P(1 Z <.5) = P(1 Z.5) = P({Z.5} {1 > Z}) = P(Z.5) P(Z 1) = Φ(.5) Φ(1) = = (3) P( 0.8 < Z) = P(Z > 0.8) = 1 P(Z 0.8) = (4) P( 0.8 < Z < 1.5) = P(Z 1.5) P(Z 0.8) = Φ(1.5) (1 Φ(0.8)) = = X N(3, 4 ) P(1 X 7) 31

32 X N(µ, σ ) Z = X µ N(0, 1) σ σ X N(3, 4 ) Z = X 3 N(0, 1) 4 P(1 X 7) = P( 1 3 Z 7 3 ) = P( 0.5 Z 1) = P(Z 4 4 1) P(Z > 0.5) = Φ(1) (1 Φ(0.5)) = = :central limit theorem CLT... µ σ n N(µ, σ /n). ( ).. µ σ n x n µ σ /n n N(µ, σ ) n N(µ, σ /n) n σ /n x µ : Wikipedia (009/0 4/5 1:17 UTC ) 3

33 .. Abraham De Moivre( ) 1730 Miscellanea Analytica 1733 Document of Chance X Bin(n, p) ( ) n P(X = x) = p x q n x x x = 0, 1,,, n npq ( n x ) 1 ( ) x np p x q n x 1 e npq π q = 1 p π e e π n n n! Bin(5, 1/),Bin(0, 1/) : 二項正規 二項正規 3.6 X 1, X,, µ 0 < σ < n i=1 X i nµ nσ N(0, 1) (n ) t ( n ) P i=1 X i nµ t 1 t e x / dx nσ π

34 X X X Binom(800, 1 ). X Z = X = X Z N(0, 1) P(X 380) = P(Z 10 ) = P(Z ) = P(Z 1.414) = 1 Φ(1.414) = P(0 X 380) = 380 k=0 (800 k ) 800 =

35 3.5 n n (Pierre Simon de Laplace), Polya(190) 3.7 (Central Limit Theorem) X 1, X,, µ σ S n = n i=1 X i, n, S n nµ nσ N(0, 1) X n = 1 n n i=1 X i X n nµ σ/ n N(0, 1) 1730 De Moivre Miscellanea Analytica 1733 The Doctrine of Chances 1738 The Doctrine of Chances Analytical Theory of Probabilities (181) ( ) n, p X Bin(n, p), n P(X = x) = p x (1 p) x x ( ) P(X = x) 1 exp π x np 1 / np(1 p) np(1 p) ( ) ( ) 35

36 0.4 ϕ(x) x ϕ(x) = 1 ( exp 1 ) π x = 1 e 1/x π npqp(x = x) ϕ (u), u = x np npq, q = 1 p n = 10, p = n n = 30, p = m! = πmm m e m log(1 + x) = x x + O(x3 ) ( ) n npq p x q n x 1 u x π e pq pq x/n p = u, (n x)/n q = u n n, log log x/n p (n x)/n q ( ) q q = log 1 + u = u np np u q np ( ) p p = log 1 u = u nq nq u p nq n Pierr Simon Laplace( ) ψ(t) = E(e itx ),n 36

37 1 (n) n 1 1 (Karl Friedrich Gauss), d ϕ(x) = xϕ(x) dx log ϕ(x) = ( x)dx + const. = {x } = cx = 1 x n M i = z + v i, i = 1,, n n M 1 + M + + M n = z ϕ(v)dv n v i = M i z P = ϕ(v 1 )dv 1 ϕ(v )dv ϕ(v n )dv n P log P log ϕ(v 1 ) + log ϕ(v ) + + log ϕ(v n ) ϕ(v) z P z = 1 ϕ(v 1 ) + 1 ϕ(v ) ϕ(v n ) = 0 ϕ(v 1 ) z 1 ϕ(v ) z ϕ(v n ) z n = ϕ (v) v z ϕ (v) = ψ(v) ϕ(v) ψ(v 1 ) v 1 z 1 + ψ(v ) v z + + ψ(v n ) v n z n = 0 i v i = M i z v i z i = 1 ψ(v 1 ) + ψ(v ) + + ψ(v n ) = 0 (a) v 1 + v + + v n = 0 (b) v 1 + v + + v n = (M 1 z) + (M z) + (M n z) = M 1 + M + + M n nz = 0 (a), (b) ψ(x) = cv c ϕ (v) ϕ(v) = cv log ϕ(v) = c v + c (c, c : const) c, c ϕ(v) ϕ(v) = ke h v (k, h : const) k = 1 π, h = 1 37

38 (Law of Large Numbers): p, 1 p X 1, X,, X n Σ i X i X p (Central Limit Theorem): X 1, X,, X n, µ, σ n X n = Σ n i=1 X i X n µ 0 X n µ σ /n X n n a, b ( lim P a X ) n µ b n σ /n b 1 exp( x a π ) dx a, b n(x n µ) ( N(0, σ ) N(µ, σ ) µ σ 1 (x µ) exp( πσ σ ) < x < 1 πσ 1 πσ x = µ x = µ ± σ p 38

39 X n, p np, np(1 p) Y Z ( (k 0.5) np P(X = k) P(k 0.5 < Y < k + 0.5) = P < Z < np(1 p) P(i X j) P(i 0.5 < Y < j + 0.5) = P ) (k + 0.5) np, np(1 p) ( ) (i 0.5) np (j + 0.5) np < Z < np(1 p) np(1 p) Z x Φ(x) = P(Z x) P(Z x) = P(Z < x) x Φ(x) ϕ(x i ) = a i, i = 1, a 1 < α < a c = x x 1 a a 1 (α a 1 ) + x 1 (1)P(0 < Z < 1.) = P(0 < Z 1.) = P(0 Z < 1.) = P(0 Z 1.) ()P( 1. < Z < 0) = P(0 < Z < 1.) (3)P( 1. < Z < 1) = P(0 < Z < 1.) + P(0 < Z < 1) = 0.76 (4)P(Z < 1.) = P(Z < 0) P( 1. < Z < 0) = 0.5 P(0 < Z < 1.) = X 10, 4 (1) P(X 13) () P(X > 11) (3) P(9 < X < 1) (4) P(X < c) = 0.1 c , 0.( 0. = 0.04)mm 0.5mm 39

40 3.11 P(X = k + 1) P(X = k) (k = 0, 1,, )

41 0 面体のサイコロでこれを乱数サイとよんでいます 3 個をころがして その結果を記録するのですが い まはコンピュータのシュミレーションがふつうに行われます 41

42 4 4.1 : kg kg ±1.5kg 3.0kg : : 1 : 0 4

43 ( )Average 1 n {x 1, x,, x n } x r i (i = 1,,, r) a i f i n n = f 1 + f + + f r a a 1 a a r x 1 + x + + x n = a 1 f 1 + a f + + a r f r = i a i f i x f f 1 f f r n (I) (II) X = 1 n n k=1 x k X = 1 n i a i f i c x i c x i y i = x i c c 4.1 X Y 4.. (, ) Median {x 1, x,, x n }, {x (1) x () x (n) } Me n n/ (n + 1)/ Me = x ( n ) + x ( n+1 ) Me = x ( n+1 ) 50% 50% 4..3 ( ) Mode Mo 43

44 L Mo Me X L Box Whisker Chart) 1 (1) (min), () 5% ( :Q 1 ), (3) (average), (4) 50% ( ;Me = Q ), (5) 75% ( :Q 3 ),(6) (max) Range R = max i x i min i x i 5% 50% 5% (box chart) 4.3. Mean Deviation x x i, i = 1,,, n n x i X (MD:mean deviation) X = MD = = i (x i X) = 0 44

45 4.3.3 Variance Standard Deviation [ ]: (x i X), i = 1,,, n (variance) s s = 1 n i = 1 n i x i X = n i x i ( i x i ) n, x 1 + x + + x n = a 1 f 1 + a f + + a r f r = i a i f i s = 1 n i (a i X) f i = 1 n i a i f i X n i (x i X) = 0 n n 1 (unbiased variance) u = 1 n 1 i (x i X) = 1 ( n ) (x i x i ) i<j 1 i<j n {(i, j) ; 1 i < j n} ( n ) (x i x i ), i, j = 1,,, n cm cm u u = u u x (standard deviation) [ ]: u = 1 n 1 n i=1 x i n n 1 X u = u n xi ( x i ) = n(n 1) X AVERAGE( ) :{x 1, x,, x n } ( )Me ( )Mo MEDIAN( ) MODE( ) Q QUARTILE( ) 0: 1: 1 (5%), : (50%)= 3: 3 (75%) 4: u STDEV( ) ( - ) = (n-1) s STDEVP( ) n u VAR( ) (n-1) s VARP( ) n 45

46 4.1 ; X 61.0 s 6.04 (SD) u 5.38 u 8.93 (Me) (Mo) Non available 4 (Q 0 ) (Q 1 ) (Q ) (Q 3 ) (Q 4 ) Score {x 1, x,, x n } x i z i x a, b y = ax + b b = 0 y i = x i X, i = 1,,, n, : a = 1 u u, b = X u (4.1) z i = 10y i % (quantile) 10% 10 (decitile) 100 (percentile) (range) (quantile range) 3/4(75%) 1/4(5%),skewness) (,kurtosis) kg cm x kg y cm 46

47 x, y n {(x i, y i ); i = 1,,, n} x X, y Y x s x = 1 n n i=1 (x i X) = 1 n n i=1 x i X y s y = 1 n n i=1 (y i Y) = 1 n n i=1 y i Y s xy = 1 n n i=1 (x i X)(y i Y) = 1 n n i=1 x iy i XY x y (x i X) (y i Y) x y x y (x, y) x, y ( ) s x,s y x i X, y i Y, i = 1,,, n s X s Y x, y r 4.1 x y r r = s XY s X s Y (4.) r r 1 r 1 r 1 r 0 ( kg cm r = CORREL(, ) = CORREL(, ) = CORREL(, ) =

48 Corrado Gini Wikipedia (Gini coefficient) 100 Gini Index Japan 4.9 United Kingdom 36.0 Sweden 5.0 Iran 43.0 Germany 8.3 United States 46.6 France 3.7 Argentina 5. Pakistan 33.0 Mexico 54.6 Canada 33.1 South Africa 57.8 Switzerland 33.1 Namibia 70.7 ( Bob Sutcliffe (007), Postscript to the article World inequality and globalization (Oxford Review of Economic Policy, Spring 004), Retrieved on The World BANK : Working for a World Free of Poverty Jonhson Kots Balakrishnan, Continuous Univariate Distribution,Vol.1 48

49 4. X, X 1, X F X (t) = P(X t) p X (t) FX 1(t) = inf x{x; F X (x) t} 1. (Gini mean difference); γ(x) = E[ X 1 X ]. (Lorenz curve); L(p) = 1 p F 1 E(X) X 3. (Gini concentration index); C(X) = 0 (t)dt {p L(p)}dp = 1 {L(p)}dp ( ) n 1 (1) g = X i X j γ(x) (i < j ) i<j () L(F X (x)) = E[X X x] F X(x) F E[X] X (x) (3) L(p) p, (0 p 1), L(0) = 0, L(1) = 1. (4) C(X) = 1 E[X 1 X 1 X ] = 1 γ(x) E[X] E[X] (5) γ(x) = E[X 1 X 1 X ] E[X 1 X 1 X ] ( ) E[X X x]f X (x) = E[X ; X x], P(X 1 X ) = P(X 1 X ) = 1 E[X] = E[X X 1 X ]P(X 1 X ) + E[X X 1 X ]P(X 1 X ) = 1 {E[X X 1 X ] + E[X X 1 X ]} γ(x) = E[ X 1 X ] = E[X 1 X X 1 X ]P(X 1 X ) + E[ (X 1 X ) X 1 X ]P(X 1 X ) = 1 E[X 1 X 1 X ] + 1 E[X X 1 X ] 1 E[X 1 X 1 X ] 1 E[X X 1 X ] = E[X 1 X 1 X ] E[X 1 X 1 X ] x = F 1 X 1 0 L(p) dp = 1 1 p dp E(X) X 0 0 = 1 p X (y) dy E(X) = 1 E[X] (t), p X(x)dx = dt, y = FX 1(p), p = F X(y), dp = p X (y)dy 0 0 F 1 y 0 1 (t) dt = E(X) 0 x p X (x) dx = 1 E[X] FX (y) p X (y) dy E[X X y] F X (y) df X (y) = 1 E[X] = 1 E[X] E[X 1 X 1 X ] P(X 1 X ) 0 0 FX 1 (t) dt E[X ; X y] df X (y) { x } t p X (t) dt p X (x)dx (4) E[X 1 X 1 X ] = E[X] 1 γ(x) G = 1 0 L(x) L(x)dx = 1/ 1 0 L(x)dx 1/ or = 1 L(x i ) (i.e.l(x) ) i 49

50 y G y = x L(x)dx y = L(x) (1) i = 1,,, n 0 1 x x i x 1 x x n p i = f (x i ) p 1 p p n F i F 1 = p 1 F = p + F 1 F n = p n + F n 1 = 1 L i = i j=1 x j f (x j ) / L L = i+1 + L i i (F i+1 F i ) L = n j=1 x j f (x j ) (i = 1,,, n) L 0 = 0 L 1 L L n 1 L n = () < x < x f (x) F(x) = x f (t)dt L(x) = x / t f (t)dt L (0 x 1) L = t f (t)dt (i) L(0) = 0 L(x) L(y) L(1) = 1, 0 < x < y < 1 (ii) 0 < L(x) x, 0 < x < n = 3 i = 1 i = i = = = = (L i ) 00/1000 = /1000 = /1000 = = = (F i ) 4/10= /10 = /10=1.0 50

51 (F i, L i )i = 0, 1,,, n (F 0, L 0 ) = (0, 0), (F n, L n ) = (1, 1) (F i ) (L i ) L(F i ) F i = i n 1 i=0 L i+1 + L i (F i+1 F i ) L (1, 1) (0.8, 0.6) (0.4, 0.) (0, 0) F i L(F i ) F i = ( ) + ( ) + ( ) = = 0.36 G = 1 i L(F i ) F i = = 0.8 ( ) Frank Cowell, Gini,Deprivation and Complaints 005, Discussion Paper. (aggregation) (Gini coefficient, Gini s coefficient) p

52 5 10 ( ) OECD ( regression analysis) 1 y ax b Y X (regression line of Y on X) y = ax + b x: y: X X Y 5

53 (1) () coefficient of reversion coefficient of regression 1888 co-relation r (x) (y) (x) (y) a b (x 1 ) (y 1 ) y = ax + b a b ( ) Fitting the Regression Line n (x i, y i ) : i = 1,,, n (x i, y) : y = ax i + b d i = y i y = y i (ax i + b) : i = 1,,, n i d i Galton, F. (1886) x, y 53

54 y = ax + b y (x n, y n ) y = ax + b a n d i (x i, y i ) (x i, ax i + b) (x 3, y 3 ) (x, y ) i (x 1, y 1 ) x (x 1, y 1 ), (x, y ), (x i, y i ),, (x n, y n ) ( ) y = ax + b ( ) a b x y y i (ax i + b) = δ i, i = 1,, δi = (y i ax i b) i δ i δ i i δ i a b a b i δ i 0 a b 0 a (y i ax i b) = ( ) (y i ax i b)x i = 0 b (y i ax i b) = ( ) (y i ax i b) = 0 a b { ( x i )a + ( x i )b = x i y i ( x i )a + nb = y i (4.3) a b a b = n x iy i x i y i n x i ( x i ) = x i y i x i x i y i n x i ( x i ) = σ xy σ x = (σ x + x ) y x (σ xy + xy) σ x = y x σ xy σ x (4.4) y x y y = σ xy (x x) σ x a b x i y i δ i δ i = (y i ax i b) = y i b y i + nb + a x i + ab x i a x i y i 54

55 a b σ a σ b σa = n/(n xi ( x i ) ) δi /(n ) (4.5) σb = x i /(n x i ( x i ) ) δi /(n ) r a r b σ a σ b y x y = y i n x = x i y = ax + b y = ax + b n y σ y = 1 n (y i y) ax + b ŷ i = ax i + b, i = 1,,, n ax + b σ ŷ = 1 n (ŷ i y) = 1 n (ax i + b ax b) = a n (x i x) = a σ x a (x i x) (y i y) a = n x iy i x i y i n x i ( x i ) a (x i x) (y i y) = a n ( ) x i ( x i ) n y i ( y i) = n x i y i x i y i n x i ( x i ) n xi ( x i ) n y i ( y i) = (n x i y i x i y i ) { n x i ( x i ) } { n y i ( y i) } = r r = n x i y i x i y i n x i ( x i ) n y i ( y i) = σ xy σxx σyy = σ xy σ x σ y r r r a r a = σ y σ x r ( ) 55

56 5 Microsoft Excel, OpenOffice calc 5.1 1,,3, A,B,C, C 3 C3 A B C D E A B3 35, [ ] ; SUM [ ] ; {+,-,*,/,%,^} = X1 Y1 =X1+Y1 X1 Y1 =X1-Y1 X1 Y1 =X1*Y1 X1 Y1 =X1/Y1 X1 Y1 =X1%Y1 X1 Y1 =X1^Y1 5. x y 1 X1 X1 56

57 X1 =SQRT(X1) X1 =ABS(X1) X1 =INT(X1) X1 =ROUND(X1) X1 =ROUNDUP(X1) X1 =ROUNDDOWN(X1) X1 X X3 Xn X1 X X3 Xn Y1 Y Y3 Yn (X1:Yn) (X1;Yn) : Excel Openoffice =SUM(X1:Xn) =AVERAGE(X1:Xn) =MEDIANE(X1:Xn) =MODE(X1:Xn) 5.3 =if(, 1, ) =and( 1,,...) =or( 1,,...) =countif(, ) (true) 1 ; =IF(X1>Y1," ","OK") X1 Y1 ( ) (OK) 1 =if(and( 1,,...), 1, ) 1 =if(or( 1,,...), 1, ) =IF(X1>=100,SUM(Y1:Yn),""), X1 100 Y1:Yn ( ) =COUNTIF(X1:Xn," ") COUNTIFS 1 3 (Histgram) (count) (frequency) countif Frequency Ctrl Shift Enter 57

58 (cross tabulation) 3 1 ( ) 1 countifs MS pivot Microsoft Office Excel [ ] [ ] / Microsoft Office Excel 007 [ ] [ ] [ ] [ ] [ ] Tally mark ; ***** ::: *** ; ; ; *************** ; ; :::: ************** ; ::: ******** ; ; :: ************ ::: *** (%) : / : [ ] A1:A10 9 ;{ 150, 00, 50, 30, 330, 360, 380, 40, 480} (1) (4) 58

59 A : (1) 00 () (3) (4) / < [ ] 1. (A1:A10). [ ]-[ ] 3. [Excel ] [ ] [ ] 4. [ ] $A$1:$A$10 [ ] 5. [ ] 6. [ ] [ ] 7. [ ] 8. [ ] 9. [OK] 10. [ ] D1 11. [ ] 1. D -[ ] 13. [ ] [ ] [ ] 100 [OK] frequency [ ] (1) =max{ } =min{ } () =count{ } (3) (4) D13 D17 [ ] (5) ( E13 E17 (6) =frequecy (7) FREQUENCY(, ) help (8) A A10 (9) D13 D17 (10) OK CTRL+SHITFT OK 5.4 [ ][ ] 5.0kg 55.0kg

60 (1) ( ) () D, (3) 1,,3, (4) 1,,3,, OK (5) (6) (0%) OK JPEG A B C D D B C = B * C 3 B11 SUM 60

61 3 E E B B11 = B - B11 4 F G9 F = (B - $B$11)^ * C F3 F8 $ $ 5 6 B1 B13 A B C D E F G A1 A A1 A A1 A 4 A B C D E F G A B C D E F G

62 5.5 pseudorandom numbers (Mersenne twister) [0, 1] U i, i = 1,,, 1 X = i=1 U i 6 X N(0, 1) = rand()

63 6 6.1 [ ] ; [ ] ; Garbage in, garbage out 6. 63

64 6.1 µ σ σ n X i Z = c 1 X 1 + c X + + c n X n? {c i } 1. E(Z) = (c 1 + c + + c n )µ. V(Z) = (c 1 + c + + c n)σ [ ] ; 6. [ ]: X ; 1. E(X) = µ. V(X) = σ n µ σ n X n Z = X µ σ/ n Z N(0, 1) n 6.1 µ = 10 σ = 16 n = 5 X (1) P(X < ), () P( X 10 > 3 5 ) 6. (i) X, Y N(0, σ ) X + Y X Y X (ii) X = i +X j i<j, u = ( n ) 1 (X i X j ) i<j (iii) X i, i = 1,,, n 6.3 m m σ( σ ) ( ) n {X 1, X,, X n } X = 1 n i X i n X N(m, σ n ) Z = X m σ/ n N(0, 1) P( Z c) = P( c Z c) = c 64

65 1.96 { Z c } = { X m c σ } { = X c n m σ n m X + c [ X c σ } n σ n m X + c σ n ] 95% c = 1.96 c.58 95% 99% [ ]: σ n X m 95% X 1.96 σ n m X σ n 99% p.58 σ n p p +.58 σ n σ σ s = 1 n i X i X s = s n > cm 6cm 1 m 95% : A p n A X X X m = np σ = np(1 p) n p = X n Z = { Z c } = X np np(1 p) = p p p(1 p)/n N(0, 1) { } { } X np c np(1 p) = p c p(1 p)/n p p + c p(1 p)/n p 1/n [ ] p(1 p) p(1 p) p c p p + c n n 6.4 ( ) n p p 95% 99% p 1.96 p.58 p(1 p) n p(1 p) n p p p p p(1 p) n p(1 p) n

66 A A 95% 6.4 H 0 H 1 1 ( 1 1 (1) () (3) (4) (5) 1. N(µ, σ ) σ = 15 = x = 38.0 H 0 : µ = 40 H 0 : µ = 40 α =

67 σ = 15 = 5 z = x µ 0 σ /n = x 40 = 1.33, α = /100 α/ = 0.05 =.5%, z α/ = z = z > z = x = 38.0 N(40, 5) 9 o C o C 5% H 0 : µ = 1455 H 1 : µ = 1455 x 1, x,, x 9 n = 9, i x i = 1986, i = ( 1400 α = 0.05, ν = 9 1 t 0.05 (8) =.306 t >.306 (.306 < t <.306) t = x µ = = 1.78 u /n 416.1/9 H 0 ( Kg, 3.Kg 6.8Kg,.5Kg ( 5% 1% µ 1 µ H 0 : µ 1 µ = 0 H 1 : µ 1 µ > 0 n 1 = n = 36 z = x 1 x s 1 + s n 1 n = =.51

68 α = 0.05 ( z 0.05 = {z : z > 1.645} z =.51 > ( 68

69 7, 69

70 1. (i) H 0 : µ = µ 0 H 1 : µ = µ 0 (ii) H 0 : µ = µ 0 H 1 : µ > µ 0 (iii) H 0 : µ = µ 0 N(µ, σ ), σ z = x µ 0 σ/ n (n 30), σ u ( α = 0.05, 0.01 ) N(0, 1) (i) z > z α/ (ii) z > z α/ (iii) z < z α/ H 1 : µ < µ 0 z = x µ 0 u/ n. ( ) (i) H 0 : µ = µ 0 H 1 : µ = µ 0 (ii) H 0 : µ = µ 0 H 1 : µ > µ 0 (iii) H 0 : µ = µ 0 N(µ, σ ), σ t = x µ 0 u/ n t ( ν = n 1 ) (i) t > t α (ν) (ii) t > t α (ν) (iii) t < t α (ν) H 1 : µ < µ 0 3. (i) H 0 : σ = σ0 H 1 : σ = σ0 (ii) H 0 : σ = σ0 H 1 : σ > σ0 N(µ, σ ) ν = n 1 χ = u σ /ν = s σ /n ( ν = n 1 ) (i) χ < χ 1 α (ν), χ > χ α (ν) (ii) χ > χ α(ν) 4. ( (i) H 0 : p = p 0 H 1 : p = p 0 (ii) H 0 : p = p 0 H 1 : p > p 0 B(p), (n > 30) (iii) H 0 : p = p 0 H 1 : p < p 0 np, n(1 p) > 5 z = p p 0 p 0 (1 p 0 ) n p N(0, 1) (i) z > z α/ (ii) z > z α/ (iii) z < z α/ 5. ( ) (i) H 0 : µ 1 = µ H 1 : µ 1 = µ (ii) H 0 : µ 1 = µ N(µ 1, σ1 ), H 1 : µ 1 > µ N(µ, σ ), (iii) H 0 : µ 1 = µ n 1, n H 1 : µ 1 < µ σ1, σ z = x 1 x u u = σ 1 /n 1 + σ /n N(0, 1) (i) z > z α (ii) z > z α (iii) z < z α 70

71 7.1 F Excel ; Uni f [0, 1] ; Binom(n, p) ; N(µ, σ ) ; N(0, 1) ; χ ( f ) ; F( f 1, f ) RAND() 0 1, [a, b], RAND() (b a) + a BINOMDIST(k, n, p, T/F) k:, n:, p:, T/F:TRUE, FALSE NORMDIST(x, µ, σ, T/F) x:, µ:, σ:, T/F: TRUE FALSE NORMINV(p,µ,σ) NORMDIST, p:,µ:,σ:, NORMDIST(x,µ, σ, TRUE) = p x NORMSDIST(z) z:, P(Z z) Z N(0, 1), NORMSINV(p) p:, NORMSDIST CHIDIST(x, f ) x;, f :, ( ) :CHIDIST= P(X > x) CHIINV(p, f ) p:, f :, CHIDIST, p =CHIDIST(x, f ), CHIINV(p, f ) = x FDIST(x, f 1, f ), x:, ( f 1, f ): FINV(p, f 1, f ) FDIST, FDIST(x, f 1, f ) = p, FINV(p, f 1, f ) = x, p:, ( f 1, f ): t ; t( f ) TDIST(x, f, B) x:, f :, B:, 1 TDIST(x, f, 1) = P(X > x) TDIST(x, f, ) = P( X > x) = P(X > x or X < x) x < 0, TDIST( x, f, 1) = 1 TDIST(x, f, 1) = P(X > x) TDIST( x, f, ) =TDIST(x, f, ) = P( X > x) t TINV(p, f ) TDIST, p:, f :, TINV P( X > t) = p t, P( X > t) = P(X < torx > t) t p p = 0.05 f = 10 TINV(0.05, 10) =.8139 TINV( 0.05, 10) =