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( 13) 1. 1 ( 13) 1. 2 ( 14) 1. 3 ( 19) ( 24) 2. 1 ( 24) 2. 2 ( 24) 2. 2. 1 ( 24) 2. 2. 2 ( 26) 2. 3 ( 31) 2. 4 ( 38) 2. 5 ( 44) 2. 5. 1 ( 44) 2. 5. 2 ( 49) 2. 5. 3 ( 53) 2. 6 Box-W hisker ( 55) 2. 6. 1 Q 1 Q 3 ( 1/ 4 3/ 4 ) ( 55) 2. 6. 2 Box-W hisker ( 59) 2. 7 ( 61) ( 62) 3. 1 ( 62) 3. 1. 1 ( 62) 3. 1. 2 ( 64) 3. 1. 3 ( 67)
3. 2 ( 76) 3. 2. 1 ( 76) * 3. 2. 2 (, F, P ) ( 79) 3. 2. 3 ( 81) 3. 3 ( 83) 3. 4 ( 90) ( 92) 4. 1 ( 92) 4. 2 ( 92) 4. 2. 1 ( 94) 4. 2. 2 ( 104) 4. 3 ( 112) 4. 3. 1 ( 112) 4. 3. 2 ( 120) * 4. 3. 3 ( 122) 4. 4 ( 125) 4. 4. 1 ( 126) 4. 4. 2 ( 129) 4. 4. 3, ( 132) * 4. 5 ( 135) 4. 6 ( 136) * 4. 7 : ( 140) 4. 8 ( 144) () ( 146) 5. 1 ( 146) 5. 2 ( 154) 5. 2. 1 ( 155) 5. 2. 2 ( 164) 5. 2. 3, ( 166) 5. 3 ( 169)
5. 4 ( 174) 5. 4. 1 ( 174) 5. 4. 2 ( 178) 5. 4. 3, ( 183) 5. 5 ( 184) 5. 6 ( 190) 5. 7 : t ( 198) 5. 8 ( 198) () ( 200) 6. 1 ( 200) 6. 2 ( 200) 6. 2. 1 ( 200) 6. 2. 2 ( 205) 6. 3 ( 207) 6. 4 2 ( 215) * 6. 5 ( 219) 6. 6 ( 223) 6. 7 : 2 n F ( k 1, k 2) ( 227) 6. 8 ( 227) ( 229) 7. 1 ( 229) 7. 2 ( 232) 7. 2. 1 ( 232) 7. 2. 2 ( 237) 7. 3 ( 240) 7. 3. 1 ( 240) 7. 3. 2 ( 244) 7. 3. 3 ( 249) 7. 3. 4 ( 252) 7. 4 ( 259)
7. 4. 1 ( 260) 7. 4. 2 ( 268) 7. 4. 3 ( 271) 7. 4. 4 ( 275) 7. 5 ( 279) 7. 5. 1 ( 279) 7. 5. 2 ( 283) 7. 5. 3 ( 286) 7. 6 ( 287) ( 289) 8. 1 ( 289) 8. 2 ( 290) 8. 3 ( 291) 8. 4 ( 295) 8. 5 ( 299) 8. 6 ( 302) ( 303) ( 345) 09 4000 ( 348) n P ( X = k) = k p k ( 1- p ) n - k ( 351) ( 356) t ( 357) F ( 359) 2 ( 366) ( 368) ( 369) ( 370) ( 373) ( 377)
1. 1,,., :, ;,, 60%,.,,,, 30 ;,..,,,.,, ;,,,.,.,,,.,,
14,. 1. 2?,,., (,, 1972) :,,.. :,.,. 1. 1 S, A, A. 1. 1 ( S), ( A), A. 1. 2 01 36 7, 08-31-22-14-27-03-33, : 7 7 ( A)?, A., S.,, S, A. 1. 1, : S, A, A ;, A,
1. 2 15 1. 3,, 100 C( S ), ( A), A S., S, B, B,. : A, ( A ) ;, B, ( B )., ;, =, =. ( 1. 1) : S, S,. 1. 4( ), 1. 1. 5 m,, ( A), S,, : 1/ 1000 s, 10 ( : s) : 1. 010, 1. 010, 1. 010, 1. 010, 1. 010, 1. 010, 1. 010, 1. 010, 1. 010, 1. 010., A. S, 10 ( : s) : 1. 009810350, 1. 009800736, 1. 009810257, 1. 009810072, 1. 009810929, 1. 009810943, 1. 009810704, 1. 009810009, 1. 009810542, 1. 009812001., A. 1. 1 H. L ohninger ( 1999) [ 30].
16, S, A,.,,,. 1. 5( ),. : ( n) ( ) ( ( n) / n) 2048 1061 0. 5180 4040 2048 0. 5069 12000 6019 0. 5016 24000 12012 0. 5005 30000 14994 0. 4998, : n, ( n) / n, 1/ 2. :,, 50%.,. 1. 2 300 ( n) / n. 1. 2 300 ( n) / n 1. 6( ) ( Laplace ) ( 18 ),
1. 2 17 = 0. 5116., 17451784 43, 25 = 0. 5102. 49,,, 22 43 25. 49,,,., 22 43. :,. 1. 7( De Mer e ),. 17 De Mere : 4 6 1 2 ; 24, 6 1 2., P 1 ), P 2 ( :, De Mere P 2 < 1 2 < P 1., ( ), P 2 = 0. 491, P 1 = 0. 518., De Mere,,. ( 1. 5 1, 2) ( n) ( n ). n : ( n) n 1 2 = 0. 5,
18, : 1 2.,,. 1. 2( ) S, A, n, A ( n), n, ( n) n p, p A, P {A } = p. ( 1. 2), Cram r ( 1957) [ 13 ], : ( 1), n, ( n) n, n, ( ),, n, ( n) P {A} n. ( 2), P {} = 1; P {}= 0. ( n) = n, ( n) = 0. ( 3) 1. 2 : A, S,, ( ). ( n), p. :,?,,. :?.,, n( ) ( n), p.
1. 3 19.,,.,, y = sinx ( 0< x < 1), x, y x,. x, x 0, 0. 1, 0. 25, 0. 30, y ; x y : 0, 0. 09983, 0. 24740, 0. 29552, 1. 1. x y 1. 1 x 0 0. 1 0. 25 0. 30 y 0 0. 09983 0. 24740 0. 29552, y.. 1. 1,,. :, y, 4, n= 1, 2, 3, 4. 1. 2.,, y,. 1. 2 n 1 2 3 4 y 3 6 1 3 :,.,, ( ) {1, 2, 3, 4, 5, 6}6 ;
20 X, {01, 02, 03,, 36}36..,,,,,. 0 x,, - 14( C) + 30( C),, 26. 41( C).,, - 14 + 30, [ - 14, + 30]. 2, ( ),, x,.., ;,.,, ( 2000 ), ( ). 2000,.,,,,,.,, ( 500 ), {x 1, x 2,, x 500 },,, {x 1, x 2,, x n}
1. 3 21.,.,,.,..,,.,. 1. 8( ) 1. 3 ( 2002 5 ) 001054. 1. 3 ( 001054 ) 12 14 17 22 27 28 35-18 01 09 11 17 22 25 35-20 02 10 17 23 32 33 34-05 01 06 09 14 20 33 36-13 02 10 14 18 23 27 34-30 02 09 12 16 23 25 26-21 20 24 27 29 30 31 34-22 05 11 12 18 22 28 30-16 10 12 18 22 27 31 35-25 16 17 20 21 22 30 36-03 03 04 06 10 12 18 36-19 01 02 12 22 24 30 32-29 07 08 10 14 16 22 30-31 01 09 11 18 20 27 29-02 03 18 21 22 24 33 35-27 10 16 18 28 29 31 33-30 07 09 11 14 21 27 35-18 06 12 14 17 21 30 33-20 02 06 11 12 16 17 24-34 04 11 14 18 20 22 24-06 04 08 11 15 17 20 21-14 05 11 18 22 24 27 35-26 02 06 10 12 17 26 35-19 01 02 07 15 24 26 33-27 05 06 12 16 33 34 35-21 03 05 11 14 24 34 36-21 03 13 14 20 28 35 36-29 02 03 13 14 15 20 26-33 02 06 11 16 17 28 35-22 02 03 11 20 21 23 30-16 02 07 14 19 20 24 33-26 05 08 14 18 23 27 32-02 06 11 15 22 27 33 35-29 01 05 10 20 24 28 35-34 08 12 14 15 17 19 35-25 03 08 09 17 22 33 34-20 03 10 13 24 25 27 36-08 01 14 15 17 20 29 34-21 08 12 13 15 16 22 35-01 07 15 20 25 27 29 33-21 03 08 14 21 22 25 28-13 04 08 16 26 27 30 36-25 04 12 18 21 25 33 35-24 05 10 16 18 24 26 28-32 12 19 20 23 24 26 28-04 10 19 20 25 32 35 36-12 01 02 22 28 31 32 33-29 11 24 25 26 27 33 34-36 01 11 15 16 21 23 25-24 02 17 21 23 24 28 34-13 03 04 16 25 31 33 35-24 02 11 19 22 27 30 36-26 05 06 07 09 17 31 34-11 04 14 17 26 30 32 33-28 0136, 36, : [ 01, 06], [ 07, 12], [ 13, 18], [ 19, 24], [ 25, 30], [ 31, 36],, 1. 3.
22 1. 9( ), 1. 4. ( ),. 1. 4 1 2 3 4 5 6 1 6 1 6 1 6 1 6 1 6 1 6 1. 10( ) Rut herford, :,. N = 2608, 7. 5 s, 1. 5, k, N k., 1. 6 ( Poisson). 1. 6,, N P k N k, 1. 4.
1. 3 23 1. 5 k 0 1 2 3 4 5 6 7 8 9 10 N k 57 203 383 525 532 408 273 139 45 27 16 1. 6 k 0 1 2 3 4 5 6 7 8 9 10 P k 0. 0209 0. 0807 0. 1562 0. 2015 0. 1949 0. 1509 0. 0973 0. 0538 0. 02602 0. 01119 0. 0065 1. 4 1. 9,, 3 4, 15% 20%,.
2. 1.. :,.,,, :, ( 25% ), ( 25% ),,,.,. 2. 2 2. 2. 1,,.,,,,.,,, :. :
2. 2 25 :,,.,,,.,,,, : ( 95000 ), 00001 95000,,,, 5,,, 100, n= 100, 100, 5,,, : 73676, 47150, 01927, 27754, ( 100 )., n,,. 2. 1 1997, 352, : 50,? 352 : 001352, : ( 1), 3, ; ( 2),, 3 ( * 1 * 2 * 3), 3 001352,,, ; ( 3) 50 ( 2. 1) ; ( 4), 2. 1. 2. 1 50 170 286 329 269 005 307 227 118 294 007 148 160 335 173 93 63 75 99 94 76 95 91 93 61 92 85 83 44 315 067 114 080 094 072 024 178 091 346 197 200 087 088 88 68 94 78 88 91 94 85 82 100 90 83 88 94
26 ) 130 273 089 064 032 255 311 117 306 003 020 293 322 247 48 72 85 100 87 96 62 96 63 76 93 91 45 81 097 318 259 214 260 142 036 225 93 30 72 80 92 93 75 81 2. 2 2000 5 1 180, 8, 01 36., 8, 60, :? : N = 180, y, 1y180, 8, x, 1x 8, ( y = 022, x = 7), 22 7., ( y, x ), 60 : ( 083, 3), ( 134, 6), ( 026, 8), ( 116, 7), ( 125, 5), ( 094. 7), ( 173, 4), ( 117, 3), ( 042, 6), ( 155, 7), ( 033, 8), ( 064, 4), ( 110, 5), ( 076, 6), ( 023, 5), ( 036, 6), ( 035, 8), ( 004, 2), ( 016, 6), ( 029, 3), ( 009, 2), ( 124, 3), ( 001, 1), ( 139, 8), ( 130, 7), ( 052, 8), ( 088, 7), ( 147, 5), ( 152, 8), ( 168, 6), ( 160, 5), ( 006, 8), ( 122, 1), ( 166, 3), ( 121, 3), ( 066, 3), ( 046, 4), ( 142, 5), ( 128, 7), ( 061, 2), ( 093, 8), ( 084, 6), ( 126, 6), ( 105, 5), ( 040, 4), ( 179, 4), ( 150, 3), ( 099, 6), ( 065, 7), ( 162, 3), ( 111, 4), ( 109, 5), ( 078, 6), ( 075, 1), ( 171, 6), ( 067, 6), ( 003, 1), ( 055, 2), ( 050, 5), ( 114, 6). 2. 2. 2,.,,,,, ( ) ( ), ( ),.
2. 2 27,,..,. 2. 3 42 : 3, 2, 4, 1, 5, 1, 5, 3, 4, 3, 5, 6, 4, 2, 5, 3, 1, 3, 4, 1, 4, 5, 1, 6, 3, 3, 1, 2, 4, 2, 6, 3, 4, 6, 6, 1, 6, 2, 4, 5, 2, 6.,. :, : 1, 2, 3, 4, 5, 6, 6 ; :, 6, 2. 2 ; 2. 2 k 1 2 3 4 5 6 m k 7 6 8 8 6 7 :, 2. 1. 2. 1
28 2. 1,,,. 2. 4 352, 60 : 63, 76, 83, 91, 45, 81, 93, 30, 72, 80, 82, 83, 81, 76, 67, 84, 72, 58, 83, 64, 93, 63, 75, 99, 74, 76, 95, 91, 83, 61, 82, 85, 83, 44, 88, 72, 66, 94, 68, 78, 88, 71, 94, 85, 82, 79, 100, 90, 83, 88, 84, 48, 72, 80, 85, 80, 87, 76, 62, 96... 2. 3, 6,., : X ma x = 100; X m in = 30., 71, 60,, 32,41,,, 5, ( ), 5., [ X min, X m a x], : 10, 7,, 2. 3. 2. 3,,,, 80., 60, 60 100 5, 2. 4. 2. 3 I k [ 30, 39] [ 40, 49] [ 50, 59] [ 60, 69] [ 70, 79] [ 80, 89] [ 90, 100] m k 1 3 1 8 13 23 11
2. 2 29 2 4 I k [ 30, 59] [ 60, 68] [ 69, 76] [ 77, 84] [ 85, 92] [ 93, 100] m k 5 7 12 18 10 8, ( 2. 2). 2. 2, 80, ( ),. 2. 2., ;,., 6 15, 5 ( ). k, N, : Moore ( 1986) : kcn 2/ 5, C= 13; ( 2. 1) Sturges ( 1928) : k1+ 3. 322( lgn ). ( 2. 2) k. N = 60, Moore, C5. 123, C 13, k= 6 ; St urges, k= 6. 907, k= 6,,.
30 2 5. ( ) ( : mv, 10 ) : 0. 1 1. 5 0. 2 2. 0-2. 3-0. 1 0. 7-0. 4-1. 0-0. 6-1. 0 0. 3 0. 1-0. 4 0. 5 0. 5 1. 2-1. 1 1. 1 2. 3 1. 9 1. 0 0. 7-1. 3 0. 7-0. 1 0. 5 2. 9 0. 0 2. 0-0. 1-1. 3 1. 3-1. 9-2. 1 1. 1 0. 0-1. 1-1. 1-1. 1 0. 0 0. 5 2. 4-0. 5-0. 6 2. 5-0. 5 0. 4 0. 9 1. 2 0. 3-1. 2-0. 5-1. 5-0. 4-2. 6-0. 3 0. 0 1. 7 1. 0-1. 2-3. 4 0. 5-0. 1 2. 4-0. 3-1. 8-0. 4-0. 3 0. 1 0. 0-3. 0-3. 5-1. 1 1. 5 1. 2 0. 2-0. 3 2. 1-0. 5-0. 4-0. 5 0. 4 0. 0 1. 6-0. 8-1. 9 1. 7 0. 7-0. 3 0. 1 1. 9 0. 7 0. 2 0. 6-2. 4-0. 8-1. 5 0. 7-0. 2.,,. :. N = 100, St urges, k = [ 1 + 3. 322( lg100) ] = [ 7. 644] = 7, [ x ] x. :., 0( mv ), : X m a x = 29, X m in = - 35, = X m ax - X mi n k = 29 + 35 7 9. 14., 2. 5 ((,] ). 2. 3. 2. 5 I k ( - 35, - 22. 5] ( - 22. 5, - 13. 5] ( - 13. 5, - 4. 5] ( - 4. 5, 4. 5] ( 4. 5, 13. 5] ( 13. 5, 22. 5] ( 22. 5, 29) m k 6 6 20 32 21 10 5
2. 3 31 2. 3 2. 2,., 85 92., 5, 8592 10,.,,,. f k, k m k N, f k = m k, k = 1, 2,, K. ( 2. 3) N. ( 1), 2. 3, k 1, 2, 3, 4, 5, 6( ), f k 7 42, 6 42, 8 42, 8 42, 6 42, 7, 2. 6. 42, 2. 3, 2. 4.
32 2. 6 k 1 2 3 4 5 6 f k 0. 1667 0. 1428 0. 1905 0. 1905 0. 1428 0. 1667 f k : K f k = 1. ( 2. 4) k= 1, 2. 3 2. 4( a).,, {1, 2, 3, 4, 5, 6}, 2. 4( a ) 2. 4( b). 2. 4( a ) 2. 3 2. 4( b) 2. 3 2. 4( b), {1, 2, 3, 4, 5, 6} {f k},
2. 3 33 = 0. 1667, : 6. ( 2), 2. 4 2. 5,, f k k, p k = f k = k mk, k = 1, 2,, K. ( 2. 5) N k 2. 4, I k 30, 9, 8, 8, 8, 8, N = 60, p k, 2. 7. 2. 4, 2. 5, I k( 2. 4( b) ), I k p k. 2. 7 I k [ 30, 59] [ 60, 68] [ 69, 76] [ 77, 84] [ 85, 92] [ 93, 100] k 30 9 8 8 8 8 m k 5 7 12 18 10 8 f k 0. 0833 0. 1167 0. 2000 0. 3000 0. 1667 0. 1333 p k 0. 00280 0. 01297 0. 02500 0. 03750 0. 02084 0. 01663 2. 5 2. 4 ) :, 1( K p k k = 1. ( 2. 6) k= 1, 2. 5 2. 2.
34 2. 5,. N = 100, p k 2. 8. 2. 5, 2. 6, I k, I k p k., 2. 6, : d 1, d 2, d 3, d 4, d 5, d 6, d 7,, 0, : d 0, d 8, 9 {d 0, d 1, d 2, d 3, d 4, d 5, d 6, d 7, d 8},, ( Frequency Polygon), 2. 7., 1., 0,.,,, 5 ( ). 2. 8 k 1 2 3 4 5 6 7 I k ( - 35, - 22. 5] ( - 22. 5, - 13. 5] ( - 13. 5, - 4. 5] ( - 4. 5, 4. 5] ( 4. 5, 13. 5] ( 13. 5, 22. 5] ( 22. 5, k 14 9 9 9 9 9 6. 5 m k 6 6 20 32 21 10 5 f k 6/ 100 6/ 100 20/ 100 32/ 100 21/ 100 10/ 100 5/ 100 p k 0. 00429 0. 00667 0. 02222 0. 0356 0. 02333 0. 01111 0. 00769 29) 2. 6
2. 3 35 {x 1, x 2, x 3,, x n}, : : k. X 1, X 2,, X k, 2. 3( k = 6), {x i, i = 1, 2,, N } {X k} 5, {X k},. X 1, X 2,, X k, k, {x i, i = 1, 2,, N } {X k} 5 ( 2. 4 ),. k 5, 15. k Moore ( 2. 1) Storges ( 2. 2). ( ), {X k}, I 1, I 2,, I k, k ( 2. 1) ( 2. 2). :.. k, ;,, 2. 4
36 2. 5. : {x 1, x 2,, x N },, x m = min {x i}, x M = max {x i}; 1 i N 1 i N k : = x M - x m k ; ( 2. 7) I j : x m, k : [ x m, x m + ], ( x m +, x m + 2 ], ( x m + 2, x m + 3 ],, I I 1 2 ( x m + ( k - 1), x m + k ] I k, [ ], ( ]. :. {x i, i = 1, 2,, N }, I 3, m1, m2,, mk; ( 2. 3) f j = m j N ( j = 1, 2,, k),., 2. 9. 2. 9 j X 1 X 2 X k k m j m 1 m 2 m k m j = N j = 1 k f j f 1 f 2 f k f j = 1 j = 1 m j 5, ; m1 m k ( 3), x m {x 1, x 2,, x N }, min {x k } k 1kN [ 1, N ] x k ;, x M.
2. 3 37 m1 mk 3. :. 2. 4 2. 5, m j, ( 2. 5) p j = f j / j 2. 10. 2. 10 j 1 2 k I j I 1 I 2 I k m j m 1 m 2 m k m j = N j = 1 f j f 1 f 2 f k f j= 1 j = 1 k p j p 1 p 2 p k p j j = 1 j = 1 :. 2. 9 ( ),, X j, 1, f j, 2. 8. 2. 8 2. 10, I j, p j, 2. 9, 1., 2. 9, x m x M,, I 0 I k+ 1, ( 0 ). d 0, d 1,, d k+ 1,
38 2. 10,, 1. 2. 9 2. 10 2. 4,, :,,,,, ( 2. 7), ( 2. 5),.,, ( stem-leaf plot ),,,..,.
2. 4 39,,. 2. 6 2. 5, 100,., - 3. 5 + 2. 9., ( ) ( ),, ( ),,,,, 2. 5 100, 2. 11. 3-3 4 0 5 4-2 3 1 6 4 16-1 0 0 1 3 3 9 1 1 1 2 5 2 8 1 9 5 24-0 1 4 6 4 1 1 5 6 5 5 4 3 1 3 4 3 3 5 4 5 8 3 8 2 30 0 1 2 7 3 1 5 5 7 7 5 0 0 0 5 4 9 3 0 5 1 0 2 4 0 7 1 7 2 6 7 15 1 5 2 1 9 0 3 1 2 7 0 5 2 6 7 9 8 2 0 3 9 0 4 5 4 1 2. 11 2. 5, 2. 11 90,,,, 2. 12. 2. 11,,.,, [ 0, 1] 30,, [ 0, 1], [ 0. 0, 0. 5) ( 0. 5) [ 0. 5, 1. 0) ( 1. 0),, 2. 12 2. 13. (,,, ( - 1. 5, - 1. 0], ( - 1. 0, - 0. 5], ( - 0. 5, 0. 0). )
40 6 2 7 1 7 2 0 8 4 3 2 8 0 5 1 4 5 5 0 3 3 5 3 9 9 4 4 9 1 3 5 7 8 1 0 6 2 3 0 2 5 4 0 5 2 5 5 0 1 5 7 7 1 6 7 2 1 1 5 5 1 4 9 1 5 3 5 3 1 1 0 4 4 3 4 3 9 0 5 6 1 6 7 1 9 0 1 0 4 2 2 3 4 3 0 1 1 5 0-3 - 2-1 - 0 0 1 2 3 4 16 24 30 15 8 ( 100) 2. 12 2. 5
2. 4 41-3 5 2-3 0 4 1-2 6 3-2 1 3 4 5-1 5 5 8 8 9 11-1 0 0 1 1 1 1 1 2 2 3 3 9-0 5 5 5 5 5 6 6 8 8 15-0 1 1 1 1 2 3 3 3 3 3 4 4 4 4 4 17 0 0 0 0 0 0 0 1 1 1 1 2 2 2 3 3 4 4 13 0 5 5 5 5 5 6 7 7 7 7 7 7 9 8 1 0 0 1 1 2 2 2 3 7 1 5 5 6 7 7 9 9 6 2 0 0 1 3 4 4 2 2 5 9 2. 13 2. 5 2. 13, : ( 1) - 3. 5 + 3. 0( mv ) ; ( 2) ( ) ; ( 3) < 0 > 0 50% ( 47, 48), 0. 0( Md) ; ( 4) 25% ( Q 1) - 1. 0 ( 23) ; ( 5) 75% ( Q 3) 1. 0( 77) ; ( 6) ( - 1. 0, 1. 0) (, 54 ).,, ( Bendat ( 1958) [ 4], ( 1985) [ 48] ). 2. 6, : ( ), ( 20
42, ( ). : 1. 0, 1. 2, 6. 3, 6. 1, 8. 5, 8. 9, 8. 1, 9. 0, 9. 4, 35. 1, 54. 2, 34. 0,, 155. : 30. 28, 32. 79, 33. 76, 35. 01, 30. 17, 33. 41, 34. 44, 32. 14, 31. 33, 31. 56, 32. 89, 33. 35, 33. 65, 32. 88, 35. 27, 34. 79, 31. 68, 32. 27, 33. 42, 33. 85, 32. 90., 3036,,.,,,, 30. 2, 30. 3, 31. 3, 31. 6, 31. 7, 32. 1, 32. 3, 32. 8, 32. 9, 32. 9, 32. 9, 33. 4, 33. 4, 33. 4, 33. 7, 33. 8, 33. 9, 34. 4, 34. 8, 35. 0, 35. 3,, 2. 14. 2 30 2 3 3 31 3 6 7 6 32 1 3 8 9 9 9 6 33 4 4 4 7 8 9 2 34 4 8 2 35 0 3 2. 14.
2. 4 43 2 7 ( 1995 ) ( 1994 ), 20 90 ( 2. 11)., 2. 15. 2. 14,,,. 2. 15 : 2. 11 68 73 76 73 73 73 74 74 71 80 82 79 79 81 81 80 64 67 69 66 65 67 61 59 75 74 75 75 74 73 62 62 68 70 63 67 64 69 72 59 75 76 67 72 69 73 77 59 58 55 62 62 67 60 50 68 62 56 65 66 75 63 54 75 4 5 0 6 9 5 5 8 9 9 3 2 2 2 6 0 1 2 2 3 4 4 9 7 6 5 6 5 6 7 7 7 7 8 8 8 9 9 4 4 3 3 2 1 7 0 2 3 3 3 3 4 4 9 9 7 6 5 5 5 5 5 5 7 6 2 1 1 0 0 8 2. 15
44 90, 5080, 80, 80 ; ( 2) 6569, 75 79 ; ( 3) 67, 67 67, ; ( 4) 74 74 50%, 67 50%., 2. 11, (, ).,. 2. 5 2. 5. 1, :.,,...,, 15, : Z = {72, 81, 90, 85, 76, 90, 80, 83, 78, 75, 63, 73, 30, 82, 90}.,,, 90 3 ;, 76. 5.,, 80, 80.,?
2. 5 45,., g ( : cm/ s 2 ) : 980. 20, 980. 10, 979. 86, 981. 04, 979. 57, 980. 10, 980. 88, 981. 21, 980. 08, 979. 82., : g 1 ( 980. 20 + 980. 10 + 979. 86 + 981. 04 + 979. 57 + 980. 10 10 + 980. 88 + 981. 21 + 980. 08 + 979. 82) = 980. 286( cm/ s 2 ),.,. 2. 1 x 1, x 2,, x n, X 1 = ( ). 2. 8 n ( x 1 + x 2 + + x n) = 1 n n k= 1 x k ( 2. 8) 2. 7,. {x i}= X, {y i }= Y, X 1 = ( 68 + 73 + 76 + 73 + 73 + 73 + 74 + 74 32 + 64 + 67 + 69 + 66 + 65 + 67 + 68 + 70 + 61 + 59 + 63 + 67 + 64 + 69 + 72 + 59 + 58 + 55 + 62 + 62 + 67 + 60 + 50 + 68) = 66. 125, 1 Y = ( 71 + 80 + 82 + 79 + 79 + 81 + 81 + 80 32 + 75 + 74 + 76 + 75 + 74 + 73 + 75 + 76 + 62 + 62 + 67 + 72 + 69 + 73 + 77 + 59 + 62 + 56 + 65 + 66 + 75 + 63 + 54 + 75)
46 : 2. 7, 20 90 66. 125, 71. 5. 2. 9 2. 5,. X = {x i, i= 1, 2,, 100}= {0. 1, 1. 5,, 0. 7, - 0. 2}, 10 0 X 1 = x i 100 = i= 1 1 ( 0. 1 + 1. 5 + + 0. 7 + ( - 0. 2) ) 100 = 0. 003,.,,. 2. 10,, Z, X 1 = ( 72 + 81 + 90 + 85 + 75 + 63 + 73 + 30 15 + 82 + 76 + 90 + 80 + 83 + 78 + 90) = 76. 5.,,,,. :. Z, 30. X = {x 1, x 2,, x n}, x 1 x 2 x k x n, ( 2. 9) ( X ( 2. 9)
2. 5 47 X = {5, 28, 7, 8, 15, 8, 9, 10, 13, 8, 5, 13, 10}. 5, 5, 7, 8, 8, 8, 9, 10, 10, 13, 13, 15, 28. ( 2. 10) 2. 2 Md X X, : ( 1) n, Md X ( 2. 9), MdX = x ( n+ 1) / 2; ( 2. 11) ( 2) n, Md X ( 2. 9), MdX = 1 2 ( x n / 2 + x n/ 2 + 1). ( 2. 12) ( 2. 10), n= 13, Md X = x ( n+ 1) / 2 = x 7 = 9., ( 2. 9), Md X Md X 2. 11. x. x 2. 7, X = {68, 73, 76, 73, 73, 73, 74, 74, 64, 67, 69, 66, 65, 67, 68, 70, 61, 59, 63, 67, 64, 69, 72, 59, 58, 55, 62, 62, 67, 60, 50, 68}, X = {50, 55, 58, 59, 59, 60, 61, 62, 62, 63, 64, 64, 65, 66, 67, 67, 67, 67, 68, 68, 68, 69, 69, 70, 72, 73, 73, 73, 73, 74, 74, 76}, Md X = 1 2 ( x 16+ x 17) = 1 2 ( 67+ 67) = 67. Y = {71, 80, 82, 79, 79, 81, 81, 80, 75, 74, 76, 75, 74, 73, 75, 76, 62, 62, 67, 72, 69, 73, 77, 59, 62, 56, 65, 66, 75, 63, 54, 75}, Y = {54, 56, 59, 62, 62, 62, 63, 65, 66, 67, 69, 71, 72, 73, 73, 74, 74, 75, 75, 75, 75, 75, 76, 76, 77, 79, 79, 80, 80, 81, 81, 82},
48 Y = 1 2 ( y 16+ y 17) = 1 ( 74+ 74) = 74. 2 ( ) 67 74. 2. 12 Z,. Z= {72, 81, 90, 85, 76, 90, 80, 83, 78, 75, 63, 73, 30, 82, 90} Z = {30, 63, 72, 73, 75, 76, 78, 80, 81, 82, 83, 85, 90, 90, 90}. n= 15, Md Z MdZ = Z ( n+ 1) / 2 = z8 = 80.,., 50% 80, 50% 80, 80. : ( ) ( 50% ) ;,, 2. 12 0 ( 100 ), Md Z, x,.,. n,,.. ( ExcelMat hemat ica TI-89, 92,. ) X = {x 1, x 2,, x n } x M X., 2. 12 Z, 90, x M = 90.., n x M, ;, 2. 11, 4 67 73, ( 4 )., n x M.,
2. 5 49 2 5. 2,, ( )., ( : C) : : 16, 18, 19, 20, 21, 22, 24, 25, 23, 20, 18, 15; : - 20, - 15, 20, 29, 34, 35, 30, 40, 32, 29, 18, 0. X = 20. 08 C, X = 19. 33 C,.,, 20 C,.. X = 20 C ( X ) : - 4, - 2, - 1, 0, 1, 2, 4, 5, 3, 0, - 2, - 5;, ( X = 19) : - 39, - 34, 1, 10, 15, 16, 20, 21, 13, 10, - 1, - 19..,. 2. 3, X = {x 1, x 2,, x n}, X 2 n = 1 n n S 2 n- 1 = 1 n - k = 1 X, ( S)., ( x k - X ) 2 ( 2. 13) n ( x k - 1 k= 1 X ) 2 ( 2. 14) 2 = 8. 75, S 2 = 9. 545; 2 = 371, S 2 = 404. 7;
50 S = 3. 09), = 19. 26 ( S = 20. 11). ( ),. :,?. 2. 3 : ( 1) 2 S 2? ( 2) 1 n n k= 1 x k - X? ( 2. 15),, S, n, S., ( 2. 15) S., ( 2. 15) y,, 2 S 2., 2 1 n n k = 1 x k - X 2, ( 2. 16) ( 2. 15),,. ( coefficient of variation),,. 10 cm,?,.,,,.,, cm, 10 cm.
2. 5 51 X ( S 2 ),, S,, C. X. 2. 13 2. 4 X = {x 1, x 2,, x n} X, S, C= S X ( 2. 17) 2001, ( : ) : 16350, 12480, 7781, 6918, 8811, 7924, 7835, 18531, 10299, 13076, 6989, 10584, 7014, 8772, 6930, 7565, 13823, 7408, 8128, 6974, 7651, 8020, 8323, 7468, 9231, 14976, 7804, 8560, 10050, 8590, 8717.. : ( 1) 2001 X Md S C ; ( 2) X, Md, S C. X, Md, S C ( ( 2. 8), ( 2. 11), ( 2. 12), ( 2. 14) ( 2. 17) ), 2. 12. 2. 12 X M d S C 9853. 579 8128 3441. 994 0. 3493 8863. 667 8441. 5 2093. 449 0. 2362 9470. 387 8323 2992. 584 0. 3160 2002.
52 2. 12,,, C,., 2001 8323 ; ( 8128 ). ( / ) : 2. 14 2001 GDP 20387. 62, 18248. 45, 8270. 65, 5382. 47, 11876. 07, 7450. 44, 9658. 26, 29574. 91, 12791. 88, 14325. 42, 5496. 34, 12268. 42, 5250. 82, 10395. 76, 6098. 77, 7734. 37, 12215. 31, 6928. 71, 6184. 78, 6504. 58, 4970. 35, 5662. 62, 5308. 87, 3070. 04, 4844. 84, 5107. 46, 4195. 36, 5807. 53, 5304. 27, 7706. 38. : GDP X, Md, S C. X, Md, S C, 2. 13. 2. 13 : 2. 13 X S M d C 11081. 03 6214. 9 9658. 26 0. 5608 5316. 57 1190. 53 5304. 27 0. 2239 8967. 39 5696. 11 6716. 65 0. 6352 ( 1) GDP ; ( 2) Md ( 9658. 26 ), ( 5304. 27 ) ; 2002 GDP.
2. 5 53, C,,, C,, ; ( 4) ( 2. 13),, X, Md, GDP, ( ), GDP. 2. 5. 3,,,. X = {x i, i= 1, 2,, N }, Y= {y i, i= 1, 2,, N }, K ( ), Z = {zi = x i + y i, i = 1, 2,, N }, U = {u i = K x i, i = 1, 2,, N }, V = {vi = ( x i - K ), i = 1, 2,, N }, ( 1) Z Z = X + Y,. ( 2) U U = K X,. ( 3) U S U = K S X, 2 U = K 2 2 X. ( 4), 2 Z 2 X + 2 Y, S ZS X + S Y,. ( 5) V, 2 V = 2 X S V = S X,,. {100001. 1, 100000. 0, 100003. 5} {1. 1, 0. 0, 3. 5},, K = 100000. 0.,. ( 6) C, CU = CX, U X,.
54 : X = {x i, i= 1, 2,, N },, X : 2 ( ) = 1 N N i = 1 ( x i - N ) 2 = 1 N i, i= 1 2 i 2. 16., 2 ( ),, 2 ( ). :, 2 ( )? 2. 16 {x i } : = X, 2 ( )., X {x i }., X = {1. 2, 3. 2, - 4. 0, 0. 0, 1. 1}, X = 0. 3. 1) 1= 2, 2 ( 1) = 8. 578. 1= - 0. 8, 2= 1. 2, 3= - 6, 4= - 2, 5= - 0. 9, 2) 2= - 1, 1= 2. 2, 2= 4. 2, 3= - 3. 0, 4= 1. 0, 5= 2. 1, 2 ( 2) = 7. 378. 3) = X, 2 ( X ) = 5. 688., 2 ( X ). 1= 0. 9, 2= 2. 9, 3= - 4. 3, 4= - 0. 3, 5= 0. 8, ( 8) :
2. 6 Box-W hisker 55 ( ), ( ) = 1 N N i = 1 x i -, ( )? : = Md( X ) ( X )., X, Md= 1. 1, 1, 2, X, ( 1) = 1 ( 0. 8 + 1. 2 + 6 + 2 + 0. 9) = 2. 18, 5 ( 2) = 1 ( 2. 2 + 4. 2 + 3. 0 + 1. 0 + 2. 1) = 2. 5, 5 ( X ) = 1 ( 0. 9 + 2. 9 + 4. 3 + 0. 3 + 0. 8) = 1. 84, 5 = Md, ( Md) = 1 5. ( 0. 1+ 2. 1+ 5. 1+ 1. 1+ 0) = 1. 68, 2. 6 Box-Whisker 2. 6. 1 Q 1 Q 3( 1/ 4 3/ 4 ) Q 1 Q3 2. 5 X = {x 1, x 2,, x n}. X Md S C,.,. 2. 13, 2001 9470. 387( / ), ( 8323 ). 2. 2,, 50%, Md 50%. Md, 25%, :
56, 25% ( 75% ), :? 2. 5 X = {x 1, x 2,, x n}, Q( p ) X p ( 0< p < 1), X Q( p ) [ np ], Q( p ) [ ( 1- p ) n], [ z] z. 2. 5, Q ( 0. 5 ) 50%, Md, Q( 0. 25) 25%, Q( 0. 75) 75%, : Q 1 = Q( 0. 25), Q 2 = M d = Q( 0. 50), Q3 = Q( 0. 75). 2001 Q 1, Q 3 ( Q 2= Md= 8323 ) : Q1 = 7565( ), Q 3 = 10299( ). :, 2. 13 ( n= 31) : X = {6918( ), 6930( ), 6974( ), 6989( ), 7014( ), 7408( ), 7468( ), 7565( ), 7651( ), 7781( ), 7804( ), 7835( ), 7924( ), 8020( ), 8128( ), 8323( ), 8560( ), 8590( ), 8717( ), 8772( ), 8811( ), 9231( ), 10050( ), 10299( ), 10584( ), 12480( ), 13076( ), 13823( ), 14976( ), 16350( ), 18531( ) }. 2. 5, 25% p 1 = 0. 25, [ np 1] = [ 7. 75] = 7, Q 1 = 7565(, 8 ) ; 2. 5, 75% p 3 = 0. 75, [ np 3] = [ 23. 25] = 23, Q3= 10299(, 24 ). 7565 : ( 6918) ( 6930) ( 6974) ( 6989) ( 7014) ( 7408) ( 7468), 7 ; 10299 : ( 10584) ( 12480) ( 13076) ( 13823) ( 14976) ( 16350) ( 18531), 7, 31 25%. 2. 17.
2. 6 Box-W hisker 57 2. 17, Q 1, Q 2= Md, Q3 1/ 4, 1/ 2, 3/ 4, X, Q 1 8 ( ), Q 2 12 ( ) Q 3 24 ( ). IQR PIQR Q 1, Q 2, Q 3 2. 17 X = {x 1, x 2,, x n}. IQR ( int erquartile range). Q 1 2. 6 X = {x 1, x 2,, x n}iqr IQR = Q 3 - Q 1., IQR 50% X, Q 3 0. 75, 0. 25, IQ R 50%, IQ R,, IQ R,. 2. 13, IQ R = Q 3 - Q 1 = 10299-7565 = 2734( ), 50% 2734( )., IQR, S 2, 2734( )., 2. 18, A, B, IQR 2734,, A, B
58 ) 2. 18 IQ R, A, B, IQR,, IQR. IQR) : IQR, PIQ R ( percentage of PIQR = IQ R. ( 2. 18) max - min ( 2. 18),. A, B, PIQR( A ) = PIQ R( B) =. PIQR. 2734 18531-6918 = 0. 2354, 2734 = 0. 4206. 9000-2500 2. 15 2. 7, Q 1, Q 2, Q 3, IQ R, 2. 7, n= 32, Q 1= 62, Q 2 = 67, Q 3= 71, min = 50, max= 76, IQ R( ) = 71-62 = 9, PIQR( ) = 9 76-50 = 0. 3461., Q 1= 65. 5, Q 2= 74, Q 3= 76. 5, min= 54, max= 82, IQ R( ) = 76. 5-65. 5 = 11, PIQR( ) = 11 82-54 = 0. 3929.
2. 6 Box-W hisker 59 PIQ R. 2. 6. 2 Box-Whisker Box-W hisker X min max, Q 1, Md, Q 3, X = {x 1, x 2,, x n}. Box-Whisker,, X. Box-Whisker : ( 1) X, min, Q 1, Md, Q 3, max ; x M ; ( 2), x m x M ( x m min, max ), [ x m, x M ], ( 3), Q 1, Md, Q 3, IQ R, Md, ; ( 4) min max, + X, 2. 19 Box- W hisker. 2. 19 Box-W hisker 2. 16 2. 13 Box- W hisker. X = 9470. 39, Q1 = 7565, Md = 8323, Q 3 = 10299, min = 6918, max = 18531., x m = 6800, x M = 19000., 2. 20 ( : ) Box -Wh isker. 2. 20, 31
60 Md max 50%, 50% min Md. Box-Whisker, 2. 13. 2. 20 31 Box-W hisker 2. 17 2. 14 2001 GDP Box-Whisker. 2. 14 X = 8967. 4, min = 3070, Q 1 = 5308. 9, M d = 6716. 65, Q 3 = 11876. 1, max = 29574. 9. x m= 3000, x M = 30000, 2. 21 GDP ( : ) Box-Whisker. 2. 21 2. 20,,. 2. 21 31 GDP Box-W hisker : Box-W hisk er John W. T ukey ( 1977), EDA ( explorat ory dat a analysis),.,,.
2. 7 61 MIN IT A B, S-plus, T exas T i-89 T i-92. 2. 7.,,,, ( ) ;,,, Q 1, Md, Q 3 IQR ; Box -Wh isker. ). : ( 1) ( ( 2). ( ( 2. 1) ( 2. 2) ) ( 2. 5). ( 3).. :,. ( 4) ;. ( 5) Q 1, Md, Q 3 IQR,. Md, 50% ; IQR Md 50%. ( 6) Box-W hisk er.
3. 1 3. 1. 1, ;.,,,,,,.,,,.,,,.,,,,,. :, 6, 1 2 6 ;,, 6,, ;,,.. 3. 1( ),
3. 1 63 ( 1), A 1, A 2,, A N ; ( 2) A i, A i, i 1, 2,, N ; ( 3) A i ( i = 1, 2,, N ),,. {A 1, A 2,, A N }, ( 1), ( 2), ( 3).,, A 1 = {= 1}( 1 ) ; A 2 = {= 2}( 2 ) ; A 6 = {= 6}( 6 ). 3. 1,.,,, B1 B2 B3 =, = 4, =. A i,. B 1 {( A 2 ), ( A 4 ), ( A 6 ) }, B 2 { A 1 A 2 A 3 A 4 }, B 3,, B3 {A i },,. 3. 2( ), {A 1, A 2,, A N }, B, P {B} = B, ( 3. 1) N
64 N B B {A i }. 3. 1, 4., A 1 = {= 1}, A 2 = { = 2},, A 6 = {= 6}, N = 6, B= {4}= {A 1 A 2 A 3 A 4}, B = 4, ( 3. 1), 3. 2 P {B} = B N = 4 6 = 2 3. 36 7 ( ), 7, 500.,? : 36 7 ( ), A = {29, 18, 03, 22, 31, 09, 12}, C k n C k n =, n= 36, k= 7, n! k! ( n - k)! n= 36, k= 7, C 7 36 = 3635 30 76 1 = n( n - 1) ( n - k + 1) k! = 42072307200 5040 = 8347680. N = 8347680, 7 ;,,,,, ( 7 ),. ( B), B, 7, P {B} = 3. 1. 2 1 8347680.,
3. 1 65.,,,. 3. 1 m, n,, m n. 3. 1, ( ), N N = mmm m = m n ( ). n 3. 1 m n 3. 1, ( 1) 8,, {0, 1, 2,, 9}, n = 8, m= 10, 8 m n = 10 8 ( ) (, ). ( 2) 6, {1, 2,, 9},,, N = 9 6. 1 nm,,, N = m( m - 1)( m - 2) ( m - n + 1). ( 3. 2) 3. 1,, m, m- 1,, m- n+ 1.
66 2 n= m, 1 N = m!. 1, n= m, N = m( m - 1)( m - 2) 321 = m!. ( 3. 3) ( 3. 3). 0! = 1. 1 ( 3. 2). 3. 2 n {a 1, a2,, an } r ( 0r n),, ( C 0 n = 1). N = C r n = n! r! ( n - r )! ( 3. 4), ( 3. 4),,., 3. 2. 3. 3 n {a 1, a 2,, a n} k ( 1k n), r 1, r 2,, k r k,, N = n! r 1! r 2! r k!, r i 0 ( i= 1, 2,, k),, 3. 3 : ( 3. 5) k r i = n. ( 3. 6) i= 1 ( 1) k= 2, n, r 1, r 2, r 1+ r 2= n, r 2= n- r 1, ( 3. 5) N = n!. ( 3. 7) r 1! ( n - r 1)! ( 3. 4), ( 3. 7) ( 3. 4). k= 2, 3. 3 N. k ( 2) ( 3. 6) r i = n {ai, i= 1, 2,, n} i = 1
3. 1 67,. 3. 3, : n r 1, ( n- r 1) r 2, ( n- ( r 1+ r 2 ) ) r 3 n- ( r 1+ r 2+ + r k- N = = n! r 1! ( n - r 1)!,, 1 )., ( 3. 4) ( n - r 1)! r 2! [ n - ( r 1 + r 2) ]! [ n - ( r 1 + r 2 + + r k- 1) ]! r k! [ n - ( r 1 + r 2 + + r k) ]! n! r 1! 1 r 2! 1 r k!. [ n- ( r 1+ r 2+ + r k) ]! = 0! = 1. 3. 4 k, n 1 {a 1, a 2,, a n 1 }, n 2 {b1, b2,, bn 2 },, k n k {d 1, d 2,, d n k }, M, m1, m 2,, mk k,, N = C m 1 n 1 C m 2 n 2 C m k n k k, ni mi 0, i = 1. mi = M, M ni. 3. 1 3. 2,. 3. 1. 3,, ( 3. 1). 3. 3 {1, 2,, 9} 9, 6 6 k i= 1
68 ( 1) ( ), ; ( 2),.,.. A i = {6 1 9 }, i = 1, 2,, N. {A i } ( A i ),. N 3. 1, m = 9, n= 6, N = m n = 9 6. ( 1), {A i, i = 1, 2,, N }, 6 ( 6 ), P {B 1} = 1 N = 1 = 0. 000001882. 9 6 ( 2), 6.,., P {B2}= 2 N, 2 3. 2, 2= 99 99= 9 4. P {B2} = 94 9 6 = 1 9 2 = 0. 0123. 3. 2 9 ( ).,.,
3. 1 69 ( 1) 1, 2, 3, 4, 5, 6 6 3,, 1, 2. : A= {1, 2}, B= {3, 4, 5, 6}. 3, A, B ( 3. 4). H, N = C 3 6 = 20, = C 2 2 C 1 4 = 4, P {H } = N = 4 20 = 0. 2. ( 2) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 10, 5 ( ), A = {0, 1, 5, 6} ( ). : A = {0, 1, 5, 6}, B = {2, 3, 4, 7, 8, 9}. H, 5 A, 3 B, N = C 5 1 0 = 252, = C 2 4 C 3 6 = 120, P {H } = N = 120 252 = 0. 4762. ( 3) H A 9,. 3 : A= {0, 1, 5, 6}, B= {9}, C= {2, 3, 4, 7, 8}, N = C 5 10 = 252, = C 2 4 C 1 1 C 2 5 = 6110 = 60, P {H } = N = 60 252 = 0. 2381.,. 3. 4 36 ( ) 7, 1 ( 7 ), 7+ 1., 36 7 ( ).
70 7 6 ( ) ( ), ; ( 2) 5, 1,.,,?, : A i = { 36 7 }, i = 1, 2,, N., N N = C 7 36 = 8347680., : 7+ 1( ), A i ( i= 1, 2,, N ), ( ), ( 1) ( 2),. ( 1) B 1, P {B 1} = B 1 N, N = C 7 3 6, B 1 : {A i, i= 1, 2,, N }? B 1 : 36, A= {7 }, B= {1 }, C= {A, B 28 }, 7, 6 A, C, B, P {B 1} = B 1 = C 6 7 C 0 1 C 1 2 8 = 196, 196 8347680 = 2. 34810-5. ( 2) B 2, P {B 2}= B 2 N, N = C 7 36. B 2 : 5 A, B, C, P {B 2} = B 2 = C 5 7 C 1 1 C 1 2 8 = 588, 588 8347680 = 7. 04410-5.. 3. 5, 8, 10,
3. 1 71. 8, 10,. : ( 1) 8 ; ( 2). 10, 8 N = 86= 48, 8 1, 2,, 48, 20 3. 1. 48 ( ) 20, ( ), ( ) N = 48 20. 3. 1 1 2 3 4 19 20 ( ) 13 4 26 18 13 41 ( 1) B1= {8 }, P {B1} = B 1 N, B 1 48 120,, 1 B 1 = 484746 29, P {B 1} = B 1 48 47 46 29 = = 0. 009659. N 48 2 0,. 20 48,,. ( 2) B2 = { }, P {B2} = B 2 N,
72 B 48 120, ( 48 ), 18 47,, B 2 = C 2 20 C 1 48 ( 47464530), P {B2} = B 2 N = 1904847 30 48 20 = 0. 0633. ( 1) 3. 5, : 1, 2,, m, n, : 1 2, 2,, n k ( km)?, 3. 5, / N. :, 3. 2,,,., m, n ( ),. n m. 3. 1, N = n m,. 3. 2 1 2 3 m 5 n 5 4. : 1).,,. 2).,,.,,,. 3).. 0 1 r,,
3. 1 73 4). m, n,, m n... 5).,. 6). m, m 365( ) 7).,. 8).,. ( 2),., Maxw ell-bolt zmann ;,,, Bose-Einst ein, ( Feller ( 1961) [ 19] ). m, n ( ),,?, 3. 3. 3. 3 m, n : 3. 3 ( ) 1, n- 1 1 ( ) ; 0, m 0. m+ ( n - 1) 01., m 0n- 1 1 ;,. m+ ( n- n- 1 1, N = C n- 1 ( n- 1) + m = C m n+ m- 1. 1)
74 3. 6 ( m > n),,,.,, n, m- n n, N = C m - n ( n- 1) + ( m- n) = C m - n m - 1. : 5, 10,,.,. ( 1) 12 : A 1: 5500 ( ) ; A 2: 5410 ( ) ; A 3: 5320 ( ) ; A 4: 5311 ( ) ; A 5: 5221 ( ) ; A 6: 4420 ( ) ; A 7: 4411 ( ) ; A 8: 4330 ( A 7) ; A 9: 4321 ( 2 ) ; A 10: 4222 ( ) ; A 11: 3331 ( 2 ) ; A 12: 3322 ( 5 ). A 3, A 5, A 6 ( 2).,.,,. A 1 : 20 10, N = C 10 20 = 184756. 5500, 2 5, 1= C 2 4 1, C 2 4 4 2, 1,
3. 1 75 P {A 1}= C2 4 C 1 0 2 0 = = 0. 000032. 6 184756 A 2 : N = C 1 0 2 0, 5410, 4 5, 4, 1, 0, 4! ;, 5410, C 5 5C 4 5C 1 5C 0 5 P {A 2} =, 2= 4! 55= 600, 600 184756 A 3 A 2, = 0. 003248. N = C 1 0 2 0, 3 = 2410 2, P {A 3} = 0. 01299. A 4 : N = C 1 0 2 0, 5311 11 4, C 2 4, 5, 3 3, 5, 2C 2 4,, 5311 5, C 5 5C 3 5C 1 5C 1 5, 4= 2611055 = 3000, P {A 4} = 3000 184756 = 0. 016237. A 5, 5 = 6000, P {A5}= 0. 032474., A 6, A 7,, A 1 2 P {A 6} = 0. 016237, P {A 7} = 0. 020296, P {A 8} = 0. 032474, P {A 9} = 0. 324751, P {A10} = 0. 108251, P {A11} = 0. 108251, P {A12} = 0. 324751. A 1, A 2,, A 12 12 P {A k} = 0. 999999. k= 1, 3322( A 12) 4321( A 9) 3331( A 11), 5 2,
76 P }= P {A 12} + P {A 1 1} + P {A 9} = 0. 324751 + 0. 108251 + 0. 324751 = 0. 757753.., 3322,. 4321( A 9) 4330( A 8) ; 0. 324751, 0. 032474.,,,. 3. 2 3. 2. 1 3. 1,.,. ( 1) ( ) : A, B, C A B ( ), C A, B, C= A B( C= A+ B)., 1, 2, 3,, 10 10,, A = { 2, 4 }, B= { 6, 8, 10 }, C= { }. C= AB( C= A+ B), 2, 4, 6, 8, 10. ( 2) ( ) : A, B, C, C A B ( ), C, A, B, C= A B( C. A 2 4 ; B 6 8 10.
3. 2 77 A B).,, A = { 7}, B= { 6, 8, 10 }, C= { 6 }. C = A B ( C = A B). A, B 6. ( 3) : A, B, C, C A B A B, C= A\ B.,, A = { 1, 2, 3, 4, 5 }, B = { }, C= { 1, 3, 5 }., A ( 5) B, 5, C, C= A\ B. ( 4) :, A, A = \ A A ( A ).,, = { 1, 2, 3,, 10}, A= {}, \ A {}, A A -. =, = ( ). ( 5) : A, B, A B, A B ( B A ), A B. ( 6) : A, B, A B B A, A B, A = B., A= { 1, 3, 5, 7, 9 }, B= { }, A B, B A, A= B. ( 7) ( ) : A, B, A B= ( AB= ), A B, A, B ( ). Venn Venn :,,,, A, B, A + B, AB, ( A+ B) 3. 4.
78 V enn V enn, : ( a) A + B= A+ ( B\ A) = A + ( B\ AB) ; ( b) A= ( A \ B) + AB; ( c) A+ B= B+ A= B+ ( A \ AB) ; ( d) AB= BA ; ( e) ( A+ B) + C= A+ ( B+ C) ; ( f) ( AB) C= A( BC) ; ( g) C( A + B) = CA+ CB; ( h) ( A + C) ( B+ C) = AB+ C. ( a) ( h),,, ( h ) ( A\ B) + BA. n n A 1 + A 2 + + A n = A k, ( 3. 8) k = 1 A 1A 2A n = n A k, ( 3. 9) k= 1, ( De Morgan) : n A k k = 1 n = A k= 1 k, ( 3. 10)
3. 2 79 n A k k = 1 n = A k= 1 k. ( 3. 11) * 3. 2. 2 (, F, P ),. Betrand ( ( 1976) [ 46 ] ). :? ( Kolmogorov) 20 30,,,.,,.,, : ( 1),, 1= { 1}, 2= { 2},, 6 = {6}.,, = { 1, 2,, 6} = { k, k = 1, 2,, 6}. ( 3. 12) ( 2) F. 3. 1,, B ( 3. 12) i,. B, B,.,, F, ( +, ), F = {{ i, i = 1, 2,, 6}, { i + { i + j, i, j = 1, 2,, 6, i j }, j + k, i, j, k = 1, 2,, 6, i j k},, { 1 + 2 + + 6} =, }. ( 3. 13) B= F.,, = F.,
80 n, F, C 1 n + C 2 n + + C n n + C 0 n = ( 1 + 1) n = 2 n ( 3. 14). ( 3. 13) 2 6 = 64. ( 3) P., y = f ( x ),. x, y ; x, y., : B ( 3. 13), P {B} = 1 6, B, P {B }= 2 6,,.,, P : F B( B F ), P {B}. : x, B, F. P 1. (, F, P ). (, F, P ), : :,,. F : F ( )., F. F : ( 1) F ; ( 2) AF, A F ; ( 3) A kf ( k= 1, 2,, n, ), k A k F, ( 3. 15) ( 3. 15),.
3. 2 81 P : P ( 1) AF, P {A}0; ( 2) P { }= 1; F, : ( 3) A kf ( k= 1, 2,, n, ),, k j, A ka j =, P A k = k k P {A k}, ( 3. 16) ( 3. 16),. (, F, P ).,. 3. 2. 3 ( 3. 16),,.. 3. 5 A, BF, A B, P {B\ A} = P {B} - P {A }. ( 3. 17) 3. 2. 1 ( a) A + B = A + ( B\ A). ( 3. 18) A B, A + B= B, ( 3. 18) B = A + ( B\ A). ( 3. 19) ( B\ A ) A, A ( B\ A ) =, ( 3. 16) ( 3. 19), P {B} = P {A} + P {B\ A }, P {B\ A}= P {B}- P {A}. 3. 6 A, B F, P {A + B} = P {A} + P {B} - P {AB}. ( 3. 20) 3. 2. 1 ( a)
82 A ( B\ AB) =, A B = A + ( B\ AB), P {A + B} = P {A } + P {B\ AB}. ( 3. 21) AB B, 3. 5, ( 3. 21) P {A + B} = P {A} + P {B} - P {AB}. 1 A F, P {A }= 1- P {A}. 3. 5 B=, A B, P {A } = 2 P { \ A} = P { } - P {A } = 1 - P {A}. A 1, A 2,, A nf, P {A 1 + + A n} P {A 1} + + P {A n}. n= 2, ( 3. 20) P {A + B} = P {A} + P {B} - P {AB}. P {AB}0, P {A+ B}P {A }+ P {B}., n3,.. 3. 7 12, 4, 8. 10, 1, : 10,? R, W 1, P {R} = 8 12 = 2 3, P {W} = 4 12 = 1 3. : A 1= {10, R 1 }, A 2= {10, W 1 }, A = {10, W, R 1 }., A= A 1A 2, 3. 6 1 P {A } = 1 - P {A } = 1 - P {( A 1 A 2) }, ( 3. 22) De Morgan ( 3. 11), ( 3. 22) P {A 1 ( 3. 20), ( 3. 23) A 2} = P {A 1 + A 2}, ( 3. 23)
3. 3 83 P A 1 A 2} = P {A 1} + P {A 2} - P {A 1A 2}. ( 3. 24) A 1A 2 10,, P {A 1 A 2} = P {A 1} + P {A 2}. ( 3. 22), P {A}= 1- ( P {A 1}+ P {A 2}) = 1-2 3 10 + 1 3 = 1- ( 0. 0173+ 1. 6910-5 ) = 0. 9826. 10 3. 3,., 12 11,,,., A B. 3. 3 A, B, P {A }> 0, P {AB} P {A} A, B, P {B A}. ( 3. 25) ( 3. 25), : 52,,, 10., 1 13. ( 3. 25) : A = { }, B= { 10 }, P {A} = 13 52 = 1 4, P {AB} = 1 52,
84 P B A} = P {AB} P {A} =. 1 52 1 4 = 1 13. P {B A}= P {AB}/ P {A} ( 3. 26). 3. 8 P {AB} = P {A} P {B A}. ( 3. 26) 10, 4 6, 2, 2, 1 1 2. A = {, 1 }, B= {, 2 }, AB. ( 3. 26), P {A} = C1 4 C 1 6 C 2 10 = 24 45, P {B A} = C0 3 C 2 5 C 2 8 P {AB} = P {A} P {B A} = 24 45 10 = 0. 1905. 28 = 10 28,., :, ( 3. 25),.,. A, B, A 1, A 2,, A n, P {A }> 0, ( 1) 0P {B A}1; ( 3. 27) ( 2) P { A }= 1; ( 3. 28) ( 3) A ka j = ( kj ), P n A k k = 1 n A = k= 1 P {A k A}; ( 3. 29) ( 4) P {B A }= 1- P {B A}; ( 3. 30)
3. 3 85 P {A 1+ A 2 A}= P {A 1 A }+ P {A 2 A}- P {A 1A 2 A}; ( 3. 31) ( 6) P {B }= P {B}. ( 3. 32) ( 3. 26). 0, 3. 7( ) A 1, A 2,, A nf, P {A 1A 2 A n }> P {A 1A 2A n}= P {A 1} P {A 2 A 1} P {A 3 A 1A 2} P {A n A 1A n - 1}. ( 3. 33),,,.. 3. 9. 10, T 100 ; 10, T 50. 1/ 5. :,? T, A= { }, B= { }, C= { }, P {ABC}. ( 3. 33), P {ABC} = P {A}P {B A}P {C AB}. ( 3. 34) 10 P {A}= 1-100 = 0. 9, P {B A}= 0. 2, P {C AB}= 1- P {C A B}., T 1/ 5, ( 100-10) 1 5 = 18( ). P {C AB}= P {C AB}= 1- ( 3. 34) 10 50+ 18 = 10 68, 10 68 = 58 68. P {ABC} = 0. 90. 20 58 68 = 0. 1535.
86,,,. 3. 8( ) B 1, A 1, A 2,, A n F, { A k} ( k= 1, 2,, n) : ( 1) A ka j = ( kj, k, j = 1, 2,, n) ; n ( 2) A k= ; k= 1 ( 3) P {A k}> 0 ( k= 1, 2,, n), P {B} = n P {A k} P {B A k}. ( 3. 35) k= 1, V enn :, A 1, A 2, A 3, A 4 ; B ( B ), V enn B A 1B, A 2B, A 3B, A 4B. ( 3. 35).. 3. 5 Venn B= B = B A k n k= 1
3. 3 87 n k= 1 ( BA k) ( 3. 2. 1 ( g) ). BA k A k, {A k}, {BA k},, n n P {B} = P ( BA k) = k = 1 k= 1 P {A k}> 0, ( 3. 26). P {B} = n P {A k}p {B A k}. k= 1 P {BA k}. ( 3. 36). 3. 10( ( Polya) ) 3, 1 3 7, 2 4 6, 3 8 2.,,. A k = { k }, k= 1, 2, 3; B = { }, P {A k}= 1/ 3( k= 1, 2, 3),, 3. 11 P {B}= P {A 1}P {B A 1} + + P {A 3}P {B A 3} P {A 2}P {B A 2} = 1 3 3 10 + 1 3 4 10 + 1 3 8 10 = 1 30 ( 15) = 1 2., 1 200, 2 500, 3 300 ; 40, 25, 10.,?,, {A k},,. 3. 10,
88 P B}= P {A 1}P {B A 1} + P {A 2}P {B A 2} + = = = P {A 3}P {B A 3} 200 200 + 500 + 300 40 200 + 500 1000 25 500 + 300 1000 10 300 1 ( 40 + 25 + 10) 1000 75 1000 = 0. 075.. 3. 12, 3. 6. {s1, s2, s3}., si n 1, n 2, n 3, P {n j si}= p ij ( i, j = 1, 2, 3) 3. 3., {si } P {si }= qi ( i = 1, 2, 3), 3 qi = 1, i = 1 3 p i k = 1 ( i = 1, 2, 3). k= 1 n 1, n 2, n 3 P {ni } ( i = 1, 2, 3). 3. 6 i j 3. 3 n 1 n 2 n 3 s 1 p 11 p 12 p 13 s 2 p 21 p 22 p 23 s 3 p 31 p 32 p 33
3. 3 89 3. 6, n 1, s1, s2 s3, P {n 1}= P {s1}p {n 1 s1} + P {s2}p {n 1 s2} + P {s3}p {n 1 s3}, = q1p 1 1 + q2p 21 + q3p 31, ( Ba yes) 3 P {n j }= qkp kj ( j = 1, 2, 3). k= 1,. {A k} B., B, A k 3. 9. 3. 9. B, A 1, A 2,, A nf, { A k}, ( 1) A ka j = ( kj, k, j = 1, 2,, n) ; n ( 2) A k= ; k= 1 ( 3) P {A k}> 0 ( k= 1, 2,, n), P {B}> 0, k ( 1kn), P {A k B} = P {A kb} = P {A k}p {B A k} n. ( 3. 37) P {A j }P {B A j } j = 1 P {B}P {A k B} = P {A k}p {B A k}, P {A k B} = P {B} ( 3. 35) ( 3. 37). P {A k}p {B A k}. ( 3. 38) P {B}. 3. 13, A 10%, B 40%, C 50%. T : A 30%, B
90 C? 10%. T, : A 1 = { A }, A 2 = { B }, A 3= {C }. D = { }, P {A 1 D }, P {A 2 D }, P {A 3 D},. P {A k D } = P {A k}p {D A k} 3 j = 1 P {A j }P {D A j } ( k = 1, 2, 3), ( 3. 39) 3 P {A j }P {D A j }= 0. 10. 3 + 0. 40. 25 j = 1 P {A 1 D } = P {A 2 D } = P {A 3 D } = + 0. 50. 1 = 0. 18, 0. 10. 3 0. 18 0. 40. 25 0. 18 0. 50. 1 0. 18 = 0. 16, = 0. 56, = 0. 28. B. 3. 4. ;,. : ( 1). P {B}= B/ N, N B
3. 4 91 3. 1. 2). B.,. ( 2) A, 1 - P ( A ),. ( 3) Venn,. ( 4),,.,. ( 5).
4. 1 1. 3,,,. :,,.,,..,,..,. 4. 2.,,, {x 1, x 2, x 3,, x n, x n+ 1, },., {x k}..
4. 2 93 4. 1, {x 0, x 1, x 2,, x n, }, x k 2, ), ( 1) p k0 ( k= 0, 1, 2, ) ; ( 2) p k= 1, k, p k, P { = x k}= p k( k= 0, 1, x 0 x 1 x n p 0 p 1 p n ( ),. x 0 x 1 x n p 0 p 1 p n ( 4. 1) : {x 0, x 1,, x n}, ( 4. 1) P { = x n+ 1 }= P { = x n + 2}= = P { = x n + k}0( k1), ( 4. 1) x 0 x 1 x n p 0 p 1 p n 4. 1. 4. 1,. 4. 1( ),0,, p = 1/ 2, 4. 2. ( 4. 2), 1 1 0 p 1 - p. ( 4. 3),, 1, 0, 1984 ( 4. 3), P { = 1}= p = 0. 51> 1/ 2; 2000 p = 0. 492< 1/ 2. 4. 3,,, P { = k}= 1/ 6( k= 1, 2,, 6),
94.. 4. 2. 1 1 6 2 3 4 5 6 1 6 1 6, : A, B, P {A}> 0, 1 6 1 6 P {B A} = P {AB} P {A} A B., P {B A} P {B}, ( 4. 4) A B, B,. A, B, ( 4. 4), A B,. ( 4. 4), P {B A } P {AB}= P {A }P {B}.. 4. 2( ) A, B, P {AB} = P {A } P {B} ( 4. 5), A, B. 1 ( 4. 5),,, A, B. ( 4. 4), P {B A }= P {B},, A B, A B,, 4. 2. 2, A, B ;, ( 4. 5).,, 1 6
4. 2 95.,,. A, B.,, A B, ( 4. 5).,. (, ), 1,0, (, ) {( 1, 1), ( 1, 0), ( 0, 1), ( 0, 0) }. ( 4. 6),, P {= 1, = 1} = 1 4. ( 4. 7) P { = 1} 1, ( 4. 6), P {= 1} = 2 4 = 1 2. ( 4. 8), P {= 1} = 2 4 = 1 2, ( 4. 9) ( 4. 8), ( 4. 9) ( 4. 7) 1 4 = P {= 1, = 1} = P {= 1}P { = 1} = 1 2 1 2. ( 4. 10) 4. 2, A= { = 1}B= { = 1}. ( 4. 6),, P {= k, = j } = P {= k}p {= j }, k, j = 1, 0.,. 3 ( 4. 5) : ( 4. 11). ( 1) A=, B, B= B, P {B} = P {B} = 1 P {B} = P {}P {B}. ( 4. 12)
96 ( 2) A=, B=, P {}= 0, P {B} = P {}P {B}. ( 4. 13)., P {}= 0, ( 3. 25), ( 4. 4), ( 4. 13).,,. 4. 4, 1, 0, 5, : 5 2 1 ( )?. 5 1, P {= 2} = 2 N, N 5 1 0, 2 2 1. 3. 1, N = 2 5. 2 5, 2 1, 0 : 2= C 2 5. ( ), P {= 2} = 2 N = C2 5 2 5 = 0. 3125. ( 4. 14), 5 1 k (0 5- k ), k= 0, 1, 2, 3, 4, 5, ( 4. 14) P {= k} = C k 5 1 2 5, k = 0, 1,, 5. ( 4. 15) ( 4. 15),, ( 4. 15) P {= k} = C k 5 1 2 k 1-1 2 5- k, k = 0, 1,, 5, ( 4. 16)
4. 2 97 1p = 1/ 2( ), q= 1-1/ 2, 0. ( 4. 16) : 5, 1 k,0 5- k., 1 0, p = q= 1/ 2.,, :, n, k p = q= 1/ 2. P {= k} = C k np k q n- k, k = 0, 1, 2,, n, ( 4. 17),, p 1/ 2, 0< p < 1, ( 4. 17) ( 4. 14), ( ( 1985) [ 4 8] ), ( 4. 17),. 4. 1( ), 1 0 ; 1 p, 0< p < 1. n, 1 k P {= k} = C k np k q n- k, k = 0, 1, 2,, n; q = 1 - p. ( 4. 18),. ( 4. 18), : ( 1). ( 2), P { = 1}= p q= 1- p, ( 3) n. 1 0 p q. ( 4. 19) ( 4) : n, 1 k. n, p, q, k ( 4. 18). : k 1, 0, ( 4. 18)..
98 4. 5, n ( n10), T e - T,. 1,, : T., T ( 0 ) q= e - T, ( 1 ) p = 1 - e - T, 1 0. ( 4. 20) 1 - e - T e - T n, 1 ( ), P { > 1}. P { 1}= P {= 0} + P {= 1} = C 0 n( 1 - e - T ) 0 ( e - = ( e - T ) n + n( 1 - e - T ) ( e - T ) n + C 1 n( 1 - e - T ) 1 ( e - T ) n- 1, P {> 1}= 1 - P { 1} = 1 - [ ( e - T ) n + n( 1 - e - T ) ( e - T ) n- 1 ]. T ) n- 1 ( 4. 21), T =,,, n. ( 4. 21) P n( ), 4. 1 n= 5, 3, 10, P n( ), 4. 1 4. 1. 4. 1 0. 05 0. 10 0. 15 0. 20 0. 30 0. 40 0. 50 0. 60 0. 70 0. 80 0. 90 1. 00 P 5 ( ) 0. 022 0. 076 0. 148 0. 229 0. 392 0. 538 0. 658 0. 752 0. 822 0. 868 0. 907 0. 935 P 3 ( ) 0. 007 0. 026 0. 053 0. 088 0. 170 0. 259 0. 348 0. 433 0. 511 0. 574 0. 637 0. 693 P 10 ( ) 0. 084 0. 250 0. 422 0. 572 0. 782 0. 895 0. 952 0. 978 0. 990 0. 995 0. 998 0. 999 4. 1 : n,,, ;,, = 0, n.
4. 2 99 n 4. 1 : n, P n ( ) ; 0, P n( 0) n?? 4. 2 = 0. 4, P n( 0. 4) n, 4. 2 P n( 0. 4). 4. 2, 0= 0. 4, n 12,. 4. 2 ( = T = 0. 4) n 2 3 4 5 6 7 8 9 10 12 15 20 P n ( 0 ) 0. 108 0. 240 0. 407 0. 527 0. 650 0. 730 0. 794 0. 850 0. 890 0. 940 0. 979 0. 996 4. 2 P n ( 0. 4) n De Mer e 17 De Mere B. P ascal( 16231662) :, 4 6 1/ 2, 24
100 1/ 2. P 1, P 2, De Mere : P 2 < 1 2 < P 1. ( 4. 22)? De Mere, P 1, P 2. ( 1), 6, 1 0 1 5. 6 6, 4, 1 1, P 1 = P { 1}= 1 - P {= 0} = 1 - C 0 4 = 1 - > 1 2. 5 6 4 1 6 0 5 6 4-0 = 0. 517747 ( 4. 23) ( 2), 6, 1 0 1 35. 36 36 24, 1 1, P 2 = P { 1}= 1-24 35 36 = 0. 491404 < 1 2. ( 4. 24) ( 4. 23) ( 4. 24), P 2 = 0. 491404 < 1 2 < 0. 517747 = P 1. ( 4. 25) De Mere., De Mere, ( 4. 25) J. L.,.., 1987.
4. 2 101 0. 491.. Pascal P 2 = Uspensky U spensky :, 1, 0. 25. 100,?, p n k, ( 4. 18).. 4. 1 *,. ( 4. 18),. 4. 1, : A, B, P {AB} = P {A } P {B}. (, ), ( 4. 11), k, j = 0, 1, P {= k, = j } = P {= k} P { = j }, k, j = 0, 1. n : i 1 0 p q, q = ( 1 - p ), i = 1, 2,, n, ( 4. 26) { 1, 2,, n}, {i, j,, k}, ( i, j,, k= 0 1). P { 1 = i, 2 = j,, n = k} = P { 1 = i}p { 2 = j }P { n = k} ( 4. 27) ( 4. 27)., n,1 k ( 4. 27) :
102 A = { 1 = 1, 2 = 1, 3 = 0,, n = 1} ( k 1 ), A 2 = { 1 = 0, 2 = 1, 3 = 1,, n = 0} ( k 1 ), A m = { 1 = 0, 2 = 1, 3 = 0,, n = 1} ( k 1 ). m= C k n, ( 4. 27) ( 4. 28) P {A 1}= P { 1 = 1} P { 2 = 1} P { 3 = 0} P { n = 1} ( k 1 ) = p k q n - k = P {A l}, l = 1, 2,, m. m P {= k} = P {A l} = C k np k q n - k, k = 0, 1, 2,, n. l = 1 P {= k} = C k np k q n- k, k = 0, 1,, n ( 4. 29), {0, 1, 2,, n}, k, P { = k}. p = 0. 4, n= 10, 4. 3 P { = k}, 4. 3( a) P ( k) = P { = k}. 4. 3 ( n= 10) k 0 1 2 3 4 5 6 7 8 9 10 P ( k) 0. 006 0. 040 0. 121 0. 215 0. 251 0. 201 0. 111 0. 042 0. 011 0. 002 0. 000 4. 3( a ) P ( k) ( n= 10, p = 0. 4)
4. 2 103 4. 4 p = 0. 4, n= 3, 5, 7 P ( k) ; n, P ( k) k,. 4. 3( b) 4. 4 n= 3, 5, 7 p = 0. 4. 4. 4 p = 0. 4, n= 3, 5, 7 P ( k) k 0 1 2 3 4 5 6 7 n= 3 0. 216 0. 432 0. 2880 0. 0640 n= 5 0. 0776 0. 2592 0. 3456 0. 2304 0. 0768 0. 0102 n= 7 0. 0279 0. 1306 0. 2613 0. 2903 0. 1935 0. 0774 0. 0172 0. 0016 4. 3( b) p = 0. 4, n= 3, 5, 7 P ( k) 4. 5 n, k, P k( p ) p, n= 10, k= 4, p 0. 1 0. 9. 4. 5 ( k= 4, n= 10) p 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 P k ( p ) 0. 011 0. 088 0. 200 0. 251 0. 205 0. 111 0. 036 0. 006 0. 00 4. 4. 4. 4, k 4. 4 P 4 ( p ) ( n= 10)
104 k= 4), P k( p ) = P p { = k} p,, p P k( p ). 4. 2. 2,,,.,,. ( 1) : 0 1, 1 0 p q, ( 4. 30) 0< p < 1, q= 1- p., p = 0. 5,. ( 2) : 0, 1, 2,, n, 0 1 k n p 0 p 1 p k p n, ( 4. 31) p k = P { = k} = C k np k q n- k, k = 0, 1,, n, n1, 0< p < 1, q= 1- p.,,, k= 0, 1, 2,, n,. ( 3) : 0, 1, 2,, n,, 0 1 n p 0 p 1 p n,
4. 2 105 P n, ) = P {= n} = e - n, n = 0, 1, 2,, ( 4. 32) n!, > 0. P ( n, ) 0, e - > 0;, n= 0 n n! = e, 1 = n n= 0 n! e- = n= 0 P ( n, ). 4. 5. 4. 5, [ 19],. Rutherford Ruth erford( 18711937) T. N = 2608, T = 7. 5 s, 4. 6. 4. 6 k 0 1 2 3 4 5 6 7 8 9 10 N k 57 203 383 525 532 408 273 139 45 27 16
106 N = 2608 M = kn k= 10094, T k = M N = 10094 = 3. 87. ( 4. 33) 2608 = 3. 87 P ( k, ) ( 4. 32) 4. 7 P k. N P k, N k ( ). 4. 7 k 0 1 2 3 4 5 6 7 8 9 10 P k 0. 0208 0. 0807 0. 1562 0. 2015 0. 1949 0. 1509 0. 0973 0. 0538 0. 0260 0. 0112 0. 0065 N P k 54. 246 210. 47 407. 37 525. 51 508. 30 393. 55 253. 76 140. 31 67. 81 29. 21 16. 95 4. 6 N k 4. 7 N P k, 4. 6. 4. 6 ( T = 7. 5 s) V-2, V -2,
4. 2 107. 576, 537, 537/ 576= 0. 9323. = 0. 9323, P ( k, 0. 9323) ( ( 4. 32) ) 576 N P k, 4. 8, 4. 7. 4. 8 k 0 1 2 3 4 5 N P k 226. 7 211. 4 98. 5 30. 6 7. 1 1. 3 N k 229 221 93 35 7 1 4. 7 V -2 4. 8 Petri..., 4. 9 k.,.
108 4. 9 k k 0 1 2 3 4 5 6 7 N k 5 19 26 26 21 13 8 6. 1 18. 0 26. 7 26. 4 19. 6 11. 7 9. 5 N k 26 40 38 17 7 27. 5 42. 2 32. 5 16. 7 9. 1 N k 8 16 18 15 9 7 6. 8 16. 2 19. 2 15. 1 9. 0 6. 7 N k 3 7 14 21 20 19 7 9 2. 1 8. 2 15. 8 20. 2 19. 5 15 9. 6 9. 6 N k 60 80 45 16 9 62. 6 75. 8 45. 8 18. 5 7. 3 Bort kewitsch 1898 18751894. 4. 10. N = 20( ), 24, = 1. 20. P ( k, 1. 20), P k= P ( k, 1. 20), 4. 11. Bortkewitsch L V on. Das G esetz der Kleinen Zahlen. Leipzig: T eubner, 1898.
4. 2 109 4. 10 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1 1 2 1 1 3 0 4 0 1 1885 1886 1887 1888 1889 1890 1891 1892 1893 1894 0 3 2 1 0 2 1 1 0 0 4. 11 k 0 1 2 3 4 N k 6 8 3 2 1 N P k 6. 024 7. 228 4. 338 1. 734 0. 676 ( ),. ( 4) : 4. 1. : n1, n 2, m, k. k= 0, 1, 2,, mn 1;, P {= k} = Ck n 1 C m - k n 2, k = 0, 1, 2,, m, ( 4. 34) C m n + n 1 2 ( 4. 34). : N = n1+ n 2, p = n 1 n1 + n2, = n 1, ( 4. 35) N n 1 = N p ; n2 = N - n 1 = N - N p = N ( 1 - p ), ( 4. 36) ( 4. 34) P {= k} = Ck N pc m- k N ( 1- p ) C m N, k = 0, 1, 2,, m. ( 4. 37) ( 4. 34) ( 4. 37). ( 4. 34), ( 4. 37),,. 4. 12 6 14, 6
110 4. 9. 4. 12 N = 20, p = 0. 3, m= 6 k 0 1 2 3 4 5 6 P ( k) 0. 0774 0. 3099 0. 3874 0. 1878 0. 0352 0. 0022 0. 0001 4. 9 N = 20, p = 0. 3, m= 6, ( 4. 35). 36, 7, ( 4. 13 )., 7, : 12,,, 13 36,. 12 24 7,. 4. 13 001054 12 14 17 22 27 28 35-18 01 09 11 17 22 25 35-20 02 10 17 23 32 33 34-05 01 06 09 14 20 33 36-13 02 10 14 18 23 27 34-30 02 09 12 16 23 25 26-21 20 24 27 29 30 31 34-22 05 11 12 18 22 28 30-16 10 12 18 22 27 31 35-25 16 17 20 21 22 30 36-03 03 04 06 10 12 18 36-19 01 02 12 22 24 30 32-29 07 08 10 14 16 22 30-31 01 09 11 18 20 27 29-02 03 18 21 22 24 33 35-27 10 16 18 28 29 31 33-30 07 09 11 14 21 27 35-18 06 12 14 17 21 30 33-20 02 06 11 12 16 17 24-34 04 11 14 18 20 22 24-06 04 08 11 15 17 20 21-14 05 11 18 22 24 27 35-26 02 06 10 12 17 26 35-19 01 02 07 15 24 26 33-27 05 06 12 16 33 34 35-21 03 05 11 14 24 34 36-21 03 13 14 20 28 35 36-29 02 03 13 14 15 20 26-33 02 06 11 16 17 28 35-22 02 03 11 20 21 23 30-16 02 07 14 19 20 24 33-26 05 08 14 18 23 27 32-02 06 11 15 22 27 33 35-29 01 05 10 20 24 28 35-34 08 12 14 15 17 19 35-25 03 08 09 17 22 33 34-20
4. 2 111 03 10 13 24 25 27 36-08 01 14 15 17 20 29 34-21 08 12 13 15 16 22 35-01 07 15 20 25 27 29 33-21 03 08 14 21 22 25 28-13 04 08 16 26 27 30 36-25 04 12 18 21 25 33 35-24 05 10 16 18 24 26 28-32 12 19 20 23 24 26 28-04 10 19 20 25 32 35 36-12 01 02 22 28 31 32 33-29 11 24 25 26 27 33 34-36 01 11 15 16 21 23 25-24 02 17 21 23 24 28 34-13 03 04 16 25 31 33 35-24 02 11 19 22 27 30 36-26 05 06 07 09 17 31 34-11 04 14 17 26 30 32 33-28 001054 : {1, 3, 0, 0, 3, 1, 4, 2, 3, 2, 2, 3, 2, 1, 2, 1, 2, 2, 3, 2, 3, 5, 3, 3, 2, 4, 3, 3, 2, 2, 1, 2, 2, 2, 1, 4, 2, 3, 2, 3, 1, 2, 3, 3, 1, 3, 2, 3, 2, 2, 1, 1, 2, 1}. ( 4. 34),, n 1 = 12, n2 = 24, ( 4. 38) m = 7, k = 0, 1, 2, 3, 4, 5, 6, 7. ( 4. 38) ( 4. 34), ( 4. 14). N = 54, N P k.. 4. 14 k 0 1 2 3 4 5 6 7 P k 0. 0415 0. 1935 0. 3361 0. 2800 0. 1200 0. 0262 0. 0027 0. 0001 4. 15, 4. 10 4. 15 k 0 1 2 3 4 5 6 7 N k 2 11 21 16 3 1 0 0 N P k 2. 24 10. 50 18. 15 15. 12 6. 48 1. 41 0. 14 0. 00 ) 4. 10
112,,,., - 12. 5 C, 37. 5 C..,,,., - 20 C,, + 40 C., [ - 20, 40] ( ).,, - 20 + 40 0 ( C)., [ - 20, 40],., ( 4. 1). { = k}{a < < b}. 4. 3. 1 4. 3, p ( x ) ( - < x < + ), ( a, b) P {a < < b} = b a p ( x ) dx ( 4. 39),, p ( x ). ( 4. 39) P {a< < b}: 1 P {a < < b}0. ( 4. 39) P {a < < b} ( 4. 39), ( a, b) p ( x ).
4. 3 113. 2 P {- < < + }= 1., {- < < + }, 1. 3 P {a b}= P {a < b} = P {a < b}= P {a < < b}.,,.,,, 3., p ( x ). U[ a, b] [ a, b], p ( x ) = 1, a x b, b - a 0,, ( 4. 40) [ a, b], 4. 11( a). [ a, b]. ( b- a ) 1 b- a = 1. 4. 11( a) U[ a, b]
114 U 4. 16, U [ 0, 1].,,,,.,,. 4. 16 U[ 0, 1] ( n= 200) 0. 001251 0. 539354 0. 425153 0. 517014 0. 834620 0. 563585 0. 142338 0. 802881 0. 662984 0. 707999 0. 193304 0. 462081 0. 517106 0. 426222 0. 600238 0. 808741 0. 235328 0. 989990 0. 104678 0. 747215 0. 585009 0. 862239 0. 751549 0. 949339 0. 252724 0. 479873 0. 209601 0. 345561 0. 921384 0. 144475 0. 350291 0. 779656 0. 168981 0. 549547 0. 001617 0. 895962 0. 843654 0. 657308 0. 345988 0. 061007 0. 822840 0. 996796 0. 491897 0. 471725 0. 806238 0. 746605 0. 999695 0. 063540 0. 374981 0. 852626 0. 174108 0. 611499 0. 699759 0. 846980 0. 210578 0. 858943 0. 392438 0. 504807 0. 316874 0. 115604 0. 710501 0. 266213 0. 147496 0. 456099 0. 553209 0. 513535 0. 297281 0. 949583 0. 271889 0. 014252 0. 303995 0. 840144 0. 141575 0. 982971 0. 113773 0. 014985 0. 023743 0. 905118 0. 297800 0. 454512 0. 091403 0. 375866 0. 692892 0. 739189 0. 752220 0. 364452 0. 092624 0. 303049 0. 567278 0. 686148 0. 147313 0. 677206 0. 426557 0. 195990 0. 543443 0. 165899 0. 056215 0. 070376 0. 761315 0. 073885 0. 988525 0. 008789 0. 966613 0. 839442 0. 436720 0. 445692 0. 918790 0. 683187 0. 397656 0. 201941 0. 119083 0. 275887 0. 153233 0. 500900 0. 696219 0. 004669 0. 272897 0. 877255 0. 890164 0. 290353 0. 008911 0. 587909 0. 821680 0. 027467 0. 377880 0. 691183 0. 582049 0. 994629 0. 531663 0. 837611 0. 191351 0. 572588 0. 571184 0. 726493 0. 177892 0. 050508 0. 601764 0. 484939 0. 817194 0. 531327 0. 607166 0. 205359 0. 475265 0. 194067 0. 166234
4. 3 115 ) 0. 743736 0. 155553 0. 843043 0. 663045 0. 468459 0. 503922 0. 626759 0. 450789 0. 457961 0. 732017 0. 657613 0. 352123 0. 949156 0. 405591 0. 197851 0. 057039 0. 744438 0. 279580 0. 842158 0. 607685 0. 108280 0. 568743 0. 123325 0. 783319 0. 599048 0. 682241 0. 109928 0. 802606 0. 385235 0. 755852 0. 743126 0. 519883 0. 735008 0. 721915 0. 314066 0. 301950 0. 608966 0. 475295 0. 941069 0. 875973 0. 572405 0. 123020 0. 286081 0. 726676 0. 361339 0. 367809 0. 336314 0. 955901 0. 151555 0. 834681 0. 140263 0. 925718 0. 225105 0. 035096 0. 733085 4. 17 4. 11( b ) [ 0, 1] 10 ( 20). 4. 17 [0, 0. 1) [ 0. 1, 0. 2) [ 0. 2, 0. 3) [0. 3, 0. 4) [0. 4, 0. 5) [0. 5, 0. 6) [0. 6, 0. 7) [0. 7, 0. 8) [ 0. 8, 0. 9) [ 0. 9, 1. 0] 19 27 16 19 18 23 19 20 23 16 4. 11( b) U[ 0, 1] ( n= 200) 200 U [ 0, 1],. N ( a, 2 ),,,.
116 p x ) = 1 2 e- 1 2 x - a 2, - < x < +, ( 4. 41) a, a,, N ( a, 2 ). 4. 4. p ( x ) a = 0 4. 12, : ( 1), ; ( 2) a, p ( x ) ; ( 3),,,,, 1. 4. 12 ( a = 0) ( 17491827) Gauss ( 17771855), ( ) Kramp 1799, Q uetelet ( 17961874). 1893 K. Pearson ( normal distribution),.,,.,,,
4. 3 117 ( ) 100, 4. 18 ( mv, 10). 4. 18 100 0. 1-1. 0 1. 9-0. 1 0. 0 0. 3-1. 2 0. 0-0. 4 0. 1 1. 5 0. 3 1. 0-1. 3 0. 5-1. 2-3. 4-3. 0-0. 5 1. 9 0. 2 0. 1 0. 7 1. 3 2. 4-0. 5 0. 5-3. 5 0. 4 0. 7 2. 0-0. 4-1. 3-1. 9-0. 5-1. 5-0. 1-1. 1 0. 0 0. 2-2. 3 0. 5 0. 7-2. 1-0. 6-0. 4 2. 4 1. 5 1. 6 0. 6-0. 1 0. 5-0. 1 1. 1 2. 5-2. 6-0. 3 1. 2-0. 8-2. 4 0. 7 1. 2 0. 5 0. 0-0. 5-0. 3-1. 8 0. 2-1. 9-0. 8-0. 4-1. 1 2. 9-1. 1 0. 4 0. 0-0. 4-0. 3 1. 7-1. 5-1. 0 1. 1 0. 0-1. 1 0. 9 1. 7-0. 3 2. 1 0. 7 0. 7-0. 6 2. 3 2. 0-1. 1 1. 2 1. 0 0. 1-0. 5-0. 3-0. 2, n = 100, 9, 7: I 1= ( -, - 2. 45), I 2= [ - 2. 45, - 1. 75), I 3= [ - 1. 75, - 1. 05), I 4= [ - 1. 05, - 0. 35), I 5= [ - 0. 35, 0. 35), I 6= [ 0. 35, 1. 05), I 7= [ 1. 05, 1. 75), I 8= [ 1. 75, 2. 45), I 9= [ 2. 45, + )., I k ( k= 1, 2,, 9) f k, 4. 19, 4. 13. 4. 19 I 1 I 2 I 3 I 4 I 5 I 6 I 7 I 8 I 9 4 6 11 16 24 17 11 8 2 4. 13,.
118., E( b), p ( x ) be - bx, x > 0, p ( x ) = ( 4. 42) 0,, b> 0, - < x < +, b> 0.,,,. b, p ( x ), ( 4. 14). 4. 14 b 4. 20 137 ( : ). 4. 20 ( N = 137) 65. 02 9. 90 29. 72 61. 10 16. 92 14. 38 24. 13 16. 99 29. 33 4. 39 9. 80 85. 96 22. 50 37. 19 32. 31 8. 40 35. 03 41. 70 6. 08 4. 90 6. 28 20. 40 1. 80 7. 90 2. 50 15. 05 29. 27 11. 10 11. 08 26. 10 17. 50 23. 05 23. 12 3. 00 12. 88 13. 18 9. 00 44. 09 4. 00 45. 45 33. 69 21. 92 17. 00 3. 40 16. 30 6. 60 11. 36 42. 30 8. 00 7. 40 14. 98 6. 05 44. 94 40. 14 60. 05 1. 50 29. 58 18. 30 6. 00 31. 10
4. 3 119 4. 80 16. 34 3. 20 24. 53 6. 67 7. 72 49. 40 10. 03 16. 30 23. 60 12. 70 5. 00 25. 35 7. 92 64. 80 1. 39 3. 00 13. 60 0. 90 20. 20 27. 20 21. 93 13. 28 0. 90 10. 09 5. 00 27. 45 35. 60 4. 22 2. 00 20. 90 2. 00 11. 07 8. 97 4. 15 8. 70 3. 50 17. 24 60. 34 3. 30 27. 48 32. 00 55. 48 15. 12 5. 61 12. 40 0. 95 11. 80 18. 60 37. 34 2. 00 34. 07 9. 10 11. 59 0. 70 28. 00 13. 20 2. 00 4. 50 3. 97 3. 66 6. 25 3. 90 19. 60 16. 88 2. 00 2. 80 25. 16 2. 86 5. 70 10. 25 4. 05 9. 00 4. 20 3. 50 1. 90 2. 76 ( ),,.,,. 4. 21 4. 15. 4. 21 P k ( 4. 42) b= 1/ x = 0. 05821 ( x ) I k. 4. 21 I k [ 0, 8) [ 8, 16) [ 16, 24) [ 24, 32) [ 32, 40) [ 40, 48) [ 48, 56) [ 56, 64) [ 64, 72) [ 72, 100) P k 0. 3722 0. 2336 0. 1467 0. 0921 0. 0578 0. 0363 0. 0228 0. 0143 0. 0090 0. 0122 N P k 50. 991 32. 014 20. 10 12. 616 7. 920 4. 972 3. 121 1. 959 1. 230 1. 667 N k 52 29 20 15 7 6 2 3 2 1 ) 4. 15 4. 21 4. 15,.,