96 6

.. 1..3.. 6 11 15 0

Y =1 0 =, L, ) Y Y ( 1 m f (.) Y = f, L, ) Y ( 1 m Y, L, ) ( k < m) ( 1 k ( k + 1, L, m ) Y ) ( 1, L, k, L, ) ( 1 k Y, L, ) ( 1 k Y Y Y Y Y 1

(Flush straght) (Four of a knd) (Full house) (Flush) (Straght) (Three of a knd) (Two pars) (Par) (one) 5 = 598960 5 1:1 : 40 64 5:78 (Almost surely) 15.6 (=78/5) Y Y Ω Y = {,,,,,,,, } 40 : 64 : 3,744 : 5,108 : 10,00 : 54,91 : 13,55 : 1,098,40 : 1,30,540 598960 40 ) 3

40 64 3744 { }) = ; { }) = ; { }) = ; 598960 598960 598960 5108 1000 5491 { }) = ; { }) = ; { }) = ; 598960 598960 598960 1355 109840 130540 { }) = ; { }) = ; { }) =, 598960 598960 598960 { P }) ({ }) 40 ( = ) 59890 9 ) 1 8 9 ) 40 598960 Y A 40 + 64 + 3744 A = {,, } = 598960 A = {,, } Ω Y ΩY φ φ 1. ΩY Y. A Ω Y 3. P [0,1] 4. 4

1. A Ω Y 0. A, B Ω Y A = φ) = 0 P ( A = + 3. A = + A 4. A B 5. Y ) = 1 Ω Y << >> Y = (,, ) () Ω () ω Ω ω) () A A (v) A Y Y Y = (,, ) () ΩY () Ω Y () A A 5

(Senstvty ) (Relablty ) Y = 1, 1, ( Y1, Y ), Y1 = Y = 0, 0, Ω Y = {(0,0), (0,1), (1,0), (1,1)} {( a, b)}), a = 0,1; b = 0,1 Y 00 P ({( 0,0)}) =, ) Y = ( a, b 10 ({( 0,1)}) = 01, {(1,0)}) = {(1,1 )}) = ab P 11 A = {(1,0),(1,1 )} B = {(0,1),(1,1)} 6

10 + 11 01 + 11 = = A B A 10 + 11 Y ( 1,0) (1,1 ) 11 A B B = (B + 11 ) A B A = A B = A B c c c c A B ) B A ) = c A ) 10 11 A Y Ω A A Y = A Y ) ( 1,0) (1,1 A ( 1,0) (1,1 ) A A 1 AB Ω > 0 A Y A B B = A A B B c c B A ) A c c B B A ) A ( ) 7

B Ω = Y B P B ) = P ( B Ω ) A A c Ω ( Y c c = B ( A A )) = B + B A ) A A B A = = = c c c A + A B + B A ) A ) c c c c B A ) = 1 B A ) A ) = 1 c c B B A ) A ) A A Y A B Ω > 0 A c A B A = c c B + B A ) A ) Y 0.99 0.98 1:19 A B c c 敍 B = 0.99, B A ) = 0.98P ( = 0. 05 0.99*0.05 7 A = 0. 0.99*0.05 + 0.0*0.95 0.05 0.7 B A 8

Y = Y 1, Y ) ( Y = I( ), = 1, 0.5 = {(0,0),(0,1),(1,0),(1,1 )} Ω Y ω ΩP ( ω) = 0.5 A = {(0,1), (1,0 )} Y B = {(1,0),(1,1)} = 0.5 = 0. 5 A 0.5 A = 0.5 P ( A = = = 0.5 B A A B A P ( B = P ( A = A B Ω P ( A = Y 9

1 0 7 1 0 1/0 1/ 絶 寛 / 3 1/ / 3 10

敍 ΩY 00 = 01 = 10 = 11 = / 4 ) 1 {, } Ω Y = {(, ),(, ),(, ),(, )} { H, T} Ω = Y {( H, H ),( H, T ),( T, H ),( T, T )} Ω Y = {(0,0),(0,1),(1,0),(1,1 )} ( Y ) = ( Y1, Y ) = Y1 + Y = 0 = 1 = {0,1,,3} Ω Y Y ΩY ΩY ( : Ω R ) Y 敍 Y Y 11

n Y = I( ) n Y = 1 = Ω = { 0,1, L, n} =1, L,9 (Flush straght) (Four of a knd) (Full house) (Flush) (Straght) (Three of a knd) (Two pars) (Par) (one) ( ) { 4} = {1,,3,4} { 4}) 4) = {,3, L,1} 7 Ω 7) = 1/36 Ω (Probablty mass functon, p.m.f.) Ω x Ω f = = x) x Ω f = 0 f ( ) p.m.f. 1

x > 0 f Ω f ( ) f (x) = { 1,, L} = k k 1 Ω 1 x Ω, f = Ω ) = 1 x f x=1 1 0.5 1 f = = = 1 x 1 0.5 x= 1 x= 1 Ω Ω p.m.f. f ( ) ) f ( x Ω, f > 0 x Ω, f = 0 f ( x ) = 1 x Ω (Cumulatve densty functon, c.d.f.) F ( ) 13

c.d.f. Ω = { x x R f > 0} ) f ( p.m.f. x R F = x) = f ( y) y Ω, y x F ( ) F ( ) = 1 F ( ) = 0 F f ( ) F ( ) ( ) f ( ) } Ω = {1,,3,4 f 3) = F (3) F () ( f ( ) F ( ) f ( ) F ( ) 10 0.6 f ( ) F ( ) 14

(005) 1 6 3 0.7 0.1 0. (0.7>0.1) }, = { 10,0,40 f Ω ( 10) = 0.7, f (0) = 0. f (40) = 0. 1 1 ε 0.7 + O( ) = 10 1 0. ( ε 1 ε + O ) = 0 0.1 + O( ) = 40( 0.5 1:1 15

+ + + : 1: 1 1 ε O( ) 1 ε O ( ) / ) [0.7 + O( )]*( 10) + [0.1 + O( 1 ε 1 ε )]*(40) 10* f ( 10) + 0* f (0) + 40* f (40) = 3 3 (Expectaton) ) E( Ω = { x x R f > 0} f ( ) p.m.f. E ( ) = x* f x Ω E( ) = 10*0.7 + 15*0. + 40*0.1 = 0 0 E( ) = E( ) ( 1, L, ) 16

1 = 1 ) E( 0000 9999 4 6 1 4 4 4 4 0.0001 4 0.0004 6 0.0006 1 0.001 4 0.004 E( ) = 49950*0.0001 50*0.9999 = 5 Y Z T W 4 6 1 4 E( Y ) = E( Z) = E( T ) = E( W ) = 6 E( ) 17

1 6 3 (1) () (3) 敍, Y, Z ), L, Y, L, Y ) Z, L, Z ), L, ) ( 1 ( 1 ( 1 ( 1 Z, L, Z ) ( 1 (Varance) ) Var( Ω = { x x R f > 0} f ( ) p.m.f. Var ( ) = ( x µ ) * f x Ω µ = E( ) σ ( ) Var( ) = 10 *0.9 + 90 *0.1 = 900 Var( Y ) = 75 Var( Z) = 40 x µ * f ( x x, x j Ω x j ) * f ( x )* f 18 x Ω ( x )

( x µ ) * f = ( x µ x + µ )* f = x * f µ x Ω x Ω x Ω Var( ) = E( ) [ E( )] (Standard devaton) ) ( SD Var( ) SD ( ) = Var( ) σ ( ) (p.m.f) p.m.f. ( ) 19

1,L, 1 ~ D -U (1, ) p.m.f. f =, x = 1, L, 1 > + 1 1 E( ) = Var( ) = 1 p ( 1 p) = 1 = 0 ~ Ber( p) p.m.f f ( 0) = 1 p f (1) p 0 < p < 1 = E( ) = p Var( ) = p(1 p) p ( 1 p) n n ) ~ B( n, p p.m.f f n x n x = p (1 p), x = 0,1, L n x, n 1 0 < p < 1 E( ) = np Var( ) = np(1 p) n 1, L, n = 0 n = 1

p ( 1 p) ) ~ Ge( p p.m.f. f = p(1 p) x 1, x = 1,, L 1 (1 p) 0 < p <1 E( ) = Var( ) = p p p ( 1 p) r x + r 1 r x ) ~ B( r, p p.m.f. f = p (1 p), x = 0,1, L x r(1 p) r(1 p) r 1 0 < p < 1 E( ) = Var( ) = p p r = 1 r, L 1, r = D n n 1

D D ~ H (, D, n) p.m.f. f = /, x n x n max{ 0, n + D} x mn{ n, D}, D, n 1 E ( ) = nd( D)( n) Var( ) = ( 1) nd