turchin valeri...
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519.2 (075.8) 22.177389 :.. , - i.-. , ., . - (I ),.. , - i.-. , ., .-. ( i i ii - ). ii i i , -i i i i i ( 1/11-7270 i 04.08.2011) ..89 i i i -. i , , i: i i i. i:IMA-, 2014. - 556 .ISBN 978-966-331-535-5i i i i- i ( ii ii ). i i i i i - . i i - i ii ii (i-, ii, ii, , , i, i -, , i , , -i, i, i, i, ii, ii,i, ). i i i. 519.2 (075.8) 22.1773 i i iISBN 978-966-331-535-5 .. , 2014 .. ,, 2014 ii
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4i i , ii - , i i . i i i ' i iii. ii i i. ( i) , , [14.i -i- i, i - i ii, i - i, i, i - i. i-i, i i i i i i- i. i , , , i, i, i i i., (3.1.2) 2 1 3, 5.12 12 5. i -i i , - i i . i-i i i . - i . i, i ii i - i - i. , i - : . . , i i, i-- , i i i- ii , i, 72, i-, 49010, , : vnturhyngmail.om
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1i1.1 ii ii . i A,B,C, . . . , i . i i- A n(A). i A, - : i A i B iii,
n(A) = n(B)( i B i,i i A). . A i B i-i . i a A i b B - (a, b). i - (a, b), a A, b B () A i B i AB. 1.1.1 . i A B iB A, A = {1, 2} i B = {3, 4, 5}. ' . AB = {(1; 3), (1; 4), (1; 5), (2; 3), (2; 4),(2; 5)}, B A = {(3; 1), (3; 2), (4; 1), (4; 2), (5; 1), (5; 2)}.5
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6 1. i k A1, A2, . . . , Ak. - i (a1, a2, . . . , ak), a1 A1,a2 A2, . . . , ak Ak, () A1, A2, . . . , Ak i
A1 A2 Ak. 1.1.2 . A1 = R1, A2 = R1, A3 == R1, A1 A2 = R1 R1 = R2 , A1 A2 A3 = R1 R1 R1 = R3 -i . ( i-). n(A B) i - AB i A i B i n(A)n(B) n(A) i A i n(B)i B:
n(AB) = n(A)n(B).i, a A i n(B) -i (a, b), b B, AB. I i A i n(A) i, n(AB) -i AB i n(B)+n(B)+. . . + n(B) = n(B)n(A) ( ii i n(A) i i i A). k -: n(A1 A2 Ak) i A1A2 Ak i A1, A2, . . .. . . , Ak i n(A1)n(A2) . . . n(Ak) -i n(A1), n(A2), . . . , n(Ak) :
n(A1 A2 Ak) = n(A1)n(A2) . . . n(Ak). 1.1.3. A = {1, 2, 3}, B = {4, 5, 6, 7}. n(AB). ' . n(AB) = n(A)n(B) = 3 4 = 12. i i. - i i. i k i. i n1 i,
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1.1. i 7n2 i i k- i, - nk i, i k i i n1n2 . . . nk i.i, A1 i - i, A2 , . . . , Ak k- i. i - (a1, a2, . . . , ak) A1A2 Ak i i k i . ii k i i i - A1 A2 Ak. ,n(A1A2 Ak) = n(A1)n(A2) . . . n(Ak) = n1n2 . . . nk. 1.1.4 . i - 0, 1, 2, 3, 4, 5, i ? ' . , i: (i ) ,, , . i - ' ( ii ), ' ( - i i , , i, ), i , . i i i 5543 = 300 . , i 0 5 300 i , .i ., n i, n-.. n- , iii ( ) i 1 n, , i i iii ii (i , iii i i i 1, 2, . . . , n ).i i, ii- , i . i : i a, b, c, . . . , f , i i, i
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8 1. i ; i , i - i i - i, i . i .. i , i ii- i i, -, .. n- n- i. 1.1.5. i - = {a, b, c}. ' . (a, b, c), (a, c, b), (b, a, c), (b, c, a), (c, a, b),(c, b, a). . Pn i n- ( i n- ) i n!,
Pn = n! 1.1.6 . i - 1, 2, . . . , 2n , i i ? ' . 1, 2, . . .. . . , 2n, i 2n 2n i, , i i - ( , i i ). i i.i i n n i ( n- ) n! , i i n - n i n! .i i (i i i-, i ) i (n!)2 .i. i n i k - k- i n-- .i n i k i, ii , i -.
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1.1. i 9 1.1.7 . = {a, b, c}. ii 3 i 2. ' . (a, b), (b, a), (a, c), (c, a), (b, c), (c, b). i. Akn i k- i n- (- i n i k) i
n(n 1) . . . (n (k 1)),Akn = n(n 1) . . . (n (k 1)). 1.1.8 . i i i 0 9 , i ii? ' . i- 3- i 0, 1, . . . , 9. ii A310 3- - i, i i10- , i 10 9 8,
A310 = 10 9 8 = 720. (ii). (ii) n i k k- in- . n i k i, i-i (i ). - i i i , - i i, ii. 1.1.9 . = {a, b, c}. i 3 i 1 i 3 2. ' . {a}, {b}, {c} i 3 -i 1, {a, b}, {a, c}, {b, c} 3 2., {a, b} i {b, a}; {b, c} i {c, b} {a, c} i {c, a} i. . Ckn i k- i- n- ( n -i k) i n!/(k!(n k)!),
Ckn =n!
k!(n k)! .
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10 1. i 1.1.10 ( i). - i i i, - m n i, in 1 i m 1 - (. 1.1.1). i i ii i - i, i i ( (0, 0)) ( (m,n))?
(0, )n
( , )m 0
0
(m,n)
. 1.1.1: i ' . ii , . - (0, 0) (m,n) n i-ii i m . i i - ii n + m, m i n i . i i i n+m, m i n . i-i m i n+m (i, , - ), i Cmn+m. . n- - m i i-, i k1, k2, . . . , km i (k1 + k2 + . . .. . .+ km = n), i
(A,B,C, . . . , S)
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1.1. i 11 i , i ii k1, k2,. . . , km i. m ii, i ii k1, k2, . . . , km -i, ii, i i ii kj-- i (j = 1, 2, . . . ,m) ii . 1.1.11 . i i = {a, b, c, d} ii A, B, C, i ii k1 = 1,k2 = 2, k3 = 1 i. ' .
({a}, {b, c}, {d}); ({a}, {c, d}, {b}); ({a}, {b, d}, {c});({b}, {a, c}, {d}); ({b}, {c, d}, {a}); ({b}, {a, d}, {c});({c}, {a, b}, {d}); ({c}, {a, d}, {b}); ({c}, {b, d}, {a});({d}, {a, b}, {c}); ({d}, {a, c}, {b}); ({d}, {b, c}, {a})., , , ({a}, {b, c}, {d})i ({d}, {b, c}, {a}) = {a, b, c, d} i. i i. - Cn(k1, k2, . . . , km) i i n- - m i, i ii k1, k2, . . . , km i (k1+k2+ . . .
. . .+ km = n), i n!/(k1!k2! . . . km!), Cn(k1, k2, . . . , km) =
n!
k1!k2! . . . km!. . () n, k1 -i () a1, k2 i () a2, . . . , km i() am (k1 + k2 + . . . + km = n) - ii n, k1 -i () a1, k2 i () a2, . . . , km i() am. n, i k1 a1, k2
a2, . . . , km am ii, ii - . . - (i) n, i k1 i () a1, k2 -
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12 1. ii () a2, . . . , km i () am (k1 + k2 + . . .. . .+ km = n), i Cn(k1, k2, . . . , km). 1.1.12. i i i in i m i , i k1 , k2, . . . , m- km? ' . i' n- -i m i:k1- i , i i, k2- , . . . , km-- i , i m-i (k1 + k2 + . . . + km = n). i - n- , , iCn(k1, k2, . . . , km). (ii) . (ii) m i n - i () n i, m i. m i n - x1 i , - x2 i i . ., xm -i m- , , -ii (x1, x2, . . . , xm) i' i , -, x1 + x2 + . . . + xm = n, i - m i n i-i (x1, x2, . . . , xm) i i' , x1 + x2 + . . . + xm = n (x1 i , x2 i . ., xm i m-).i m i n i-i, ii ii i . i i i. 1.1.13. i - 4 i a, b, c, d 2. ' . aa, bb, cc, dd, ab, ac, ad, bc, bd, dc. . fnm m i n i Cm1n+m1,
fnm = Cm1n+m1.
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1.2. i 13 n > m, m i n , i- , i Cm1n1 . 1.1.14 . i i' 'i i i x1 + x2 + . . .+ xm = n? ' . ' ix1 + x2 + . . .+ xm = n i i' ii (x1, x2, . . . , xm)i i' , x1+x2+ . . .+xm = n. - ii m i n (i ). '-i i fnm m i n -. i . - n, m (i):
x1 a1, x2 a2, i . ., xm am (x1 + x2 + . . .. . . + xm = n). , i i n i, i i x1, x2, i . ., m- xm, m i n - () i i , .. m i n , i i - i x1, x2 i . ., m- xm( i i n i, : x1 -i a1, x2 i a2 i . ., xm i am). -i n i (- ), . i i -i Cn(x1, x2, . . . , xm).1.2 i: 1.3, 1.10, 1.14, 1.16 , 1.18, 1.19 , 1.22, 1.23, 1.25.: 1.4, 1.5, 1.11, 1.15, 1.17 , 1.20, 1.24, 1.27, 1.30, 1.32. , , i i i i i , i '-: i (,
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14 1. ii). i ii i-? , ii ? ( -).1.1. i A i B n i, B C m i. i i AB C?1.2. i . i i ? ii , i i .1.3. i i 17 . i -ii i , i i?1.4. i - 0, 1, 2, 3, 4?1.5. i - 0, 1, 2, 3, 4, i ?1.6. i i i ?1.7. i 10 i. i - 6 i, i ii. i - i?1.8. i ' , i 5?1.9. i i i . i i , 26 i i?1.10. i i 16 . , i , i, ii , ii , , i i i, i. i i -i i ?1.11. i 9 i ii i 4 i?1.12. i '?1.13. i i -i 4 ii ?1.14. p1, p2, . . . , pn ii i . i-
-
1.2. i 15 ii m = p11 p
22 . . . p
nn , 1, 2, . . . , n i i ?1.15. i n i, i ?1.16. i 4 i 25 i?1.17. i 8 i 4 ( i - ). i ?1.18. i - {1, 2, 3, . . . , n} , 1, 2, 3 i ?1.19. i , i ?1.20. i , ?1.21. m i i n i +1 i 1 , i i 1. i- ?1.22. p i i q (p > q). i- i i i , i 2 i i ?1.23. i n , iii i ii i i.1) i ii .2) i i i?3) i i i?4) i , i i , i i ?1.24. i i n-?1.25. i ii - n-, i i i?1.26. n- i ii.i, i ii. i i n--?
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16 1. i1.27. , n(A1 A2 Ak) = n(A1)n(A2) . . . n(Ak).1.28. , i n-- i n!1.29. , Akn i n i k i n(n 1) . . . (n (k 1)).1.30. , Ckn k- i- n- i n!/(k!(n k)!).1.31. ,
(a+ b)n =
n
k=0
Cknakbnk.1.32. , Cn(k1, k2, . . . , km) i n- m i- ii k1, k2, . . . , km i
n!/(k1!k2! . . . km!).1.33. , i Cn(k1, k2, . . . , km) i - n k1 a1, k2 a2, . . . , km am.1.34. i i m+n+ si , i i mi, i n, i s?1.35. i i 3n -i i , n i?1.36. , fnm m i n i Cm1n+m1.1.37. , m i n (n > m), i- , i Cm1n1 .
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22.1 i i, ii i i -, i i i i. i - i , - i ( - ), - i ( i i i) i -i. i1. i i i i , . - , , , , , , , i .2. i i , . 1, 2, . . .. . . , 6 i .3. i i i . i: 0, 1, 2, . . . i -. 17
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18 2. 4. i -. i i i .5. i i - i i i i ( - i). i i ., i -i i i.i i. - ii i () ii. i ii , -i i, ( '), - ( i). i - -i. i 1 i ii - = {, , , }, 2, 3, 4ii = {1, 2, . . . , 6}, = {0, 1, . . .}, = [0,), i 5 - i .ii i- i .i i -, i i, 1, 2, . . . i. -i i i i, i A,B,C, . . . , i i i, i i - A. i 1 i, i A , B i -, . . . i A i ii , :
A = { : , i i A}.
-
2.1. i i 19 i - i ii, i i i i i . i - A, i A, , i A i, - i i. i 1 i i i {, } == {, , , }, i 2 i i {2, 4, 6} = {1, 2, . . . , 6}, i 3 i i- i i- {0, 1, 2, 3} = {0, 1, . . .}, i 4 ii i 100 i (100,) == [0,). i ii i, i ii - ( i ), i i, i ii ( - ). A i B i i -, i ii i A i B i . , i i A, i- i i B, , i A i B i : A B ( i i, i, A i B).i A i B, i A B i B A ( A iB i ), i i i-. i : A = B( i i, i, iA i B ii).i, , i i i A B, (')1i A i B i : A B ( i A i B ' A B A i B).1 i i . . 7.1. 7.
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20 2. i, , i -i A, i i B, ()i A i B i : A B ( i A iB A B A i B). i A i B i, A B = , - i ().i, , A i, B i, i i A i B i - : A\B (i i A i B iA \B A i B ).i, , A i, - i A i A (-i A A A ). A i, i A ii A, i A i B ii AB, i. ,i i, i i, ( i i . 7). i 1 i B i- , A i i . i B i A i - i = {, , , } :B = {, }, A = {, }. i, i i i, : AB = {} i i , -i , AB = {, , } -i ' , A\B = {} ' , B = {, } i -. ii i i-. 2.1.1. - (. . 2.1.1). - , , i i A, i B. i A, B, A,AB, AB, B \A i, ii i, . 2.1.1.
-
2.2. i 21A B A
A A AB B B. 2.1.1: i 2.2 i: 2.2, 2.5, 2.8, 2.11, 2.12, 2.15, 2.16, 2.17.: 2.3, 2.4, 2.6 , 2.7, 2.10, 2.13, 2.14, 2.18, 2.19. . i i - , i -, - .2.1. i, i : ) A - i i ; ) B i ii ii;) C i i -ii ii.2.2. i ii. i Ai , i i- i ,i = 1, 2, 3. Ai i i: ) A i- ; ) B ;) C i ; ) D i .2.3. A,B,C i i. -i, i , i -
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22 2. i A,B,C; i A,B,C i: ) i i A;) i A B i i i C; ) i i;) i i; ) i i;) i i.2.4. ii i i', n i . - ' i- i i, Ai, i = 1, 2, . . . , n. Ai, i = 1, 2, . . . , n, i i: A '- ( i); B ' (i i); C ' i i.2.5. m ii i, i ', n i . Aji i ' j- -i i- i, j = 1, 2, . . . ,m, i = 1, 2, . . . , n. Aji i i: A ' - (i i i); B ' i.2.6. ii i ; i i . i i: A i ' , B ii '. i i; - i, A; B.2.7. i ii. i i . i- i: A , ', i 8;B i ' 6; C - ii ' ; D i ' . i , A, B, C, D.2.8. i , i -. i i . - i i: A ' ; B ' 5. i , A, B.2.9. ii , i ii [0; 1]. i i.
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2.2. i 232.10. I 1, 2, 3, 4, 5 , i , , . - i i . i- i: A i , B i . i , A, B.2.11. i i N , M -, n i. -i i . i -i A n i m (n N,m M,m n). -i , A.2.12. ii 7 i, i 10 -. i -, i . i - i . i i A i i . - i , A.2.13. i i i . - , . i i . i i:A i k- ii, B i- k- i, C , , D , .2.14. i i , i, i, . i , i i i i i. i i . i i: A i, B i .2.15. i i n i. i i . i i A n1 , n2 i-, . . . , n6 i (n1 + n2 + . . . + n6 = n). ii i ,A.2.16. ii 2n i, i - ii i n i i. i i . i i: A i i i , B i
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24 2. i i (i iii ). i ,A,B.2.17. i i, i r(r 12) i, i i -. i i . - i i: A i ii, B i i, C i - i, D i. i ,A,B,C,D.2.18. 1, 2, . . . , n i . - i, i- . i i . i i: A 1 i 1, B 1 i 1, n - n. i ,A,B.2.19. i i . - i i . i- i: A i , B i i ,C i ii i, D i . - i ,A,B,C,D.2.20. i i i i. i - i . i i: A i i, B 'i i- .2.21. n i i m i. i -i . i i: A i k1 , k2, . . . , m- km. i , A.2.22. n i i m (n m) i. i i . i i:A i i i . i ,A.
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3iii i3.1 Iii, i. Iii. i i, i i ii - (i i i, ii i). , ii -i B i i i, ii C i . ii - i B ii i i,i i C ii.ii i i A n(A) i A ii n i, - . - i n i; kn(A) -i, i i A. i
n(A) = kn(A)/n. n(A) i i.1. i An(A) 0. (3.1.1)25
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26 3. iii i2. i i A i Bn(A B) = n(A) + n(B). (3.1.2)3. i i
n() = 1. (3.1.3) n(A) i A ii i i , i n i i n(A) i i i n(A) ii. ii.Iii iP : A P (A), i i, , 1. - i AP (A) 0. (3.1.4)2. i () i
Ai, i = 1, 2, . . . (Ai Aj = , i 6= j)
P
(
i=1
Ai
)
=
i=1
P (Ai). (3.1.5)3. i i P () = 1. (3.1.6) P (A) i P A iiii A.Iii i.
{, P}, i i , P ii ii (i ), ii-i .
-
3.1. Iii, 27 i {, P} i A - i ' ( i i i) ii i:A =
iA{i},
P (A) =
iAP (i), (3.1.7)
1 = P () =
iP (i).ii (3.1.7) , ii- i ii i A i i i- P (i) ii i, i i A ( - ). I -, ii -i , ii ii P (),
, i ii (3.1.7). iii i {, P} - i i i ii. i . Iii p = P (A) i A i- ii i : - i A i i i ii p. 3.1.1 . i , i , ii ( i). ii i -. i i: A , 3, B . - ii i.i (i -i ). iii ? - ii i A i B.
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28 3. iii i ' . i i - i = {1, 2, . . . , 6}. Iii i p, i ii i j jp, i i
P () = 1, p+ 2p + . . .+ 6p = 1.i p = 1/21, P (j) = j/21, j = 1, 2, . . . , 6. , i-ii i {, P} -.i A i B ii i A == {3; 6} i B = {2; 4; 6} .Iii i A i B ( iii i - ii i, . (3.1.7)) - :
P (A) = P ({3; 6}) = P (3) + P (6) = 321
+6
21=
3
7;
P (B) = P ({2; 4; 6}) = P (2)+P (4)+P (6) = 221
+4
21+
6
21=
4
7. i i -i i :
= {1, 2, . . . 6}, ii i :P (i) = 1/6, i = 1, 2, . . . , 6.i
P (A) = P ({3; 6}) = P (3) + P (6) = 16+
1
6=
1
3;
P (B) = P ({2; 4; 6}) = P (2)+P (4)+P (6) = 16+
1
6+
1
6=
1
2. . , '- i i (i i). 1. - (iii i), - i i i i-i i:
P (i) = i/21, i = 1, 2, . . . , 6
-
3.1. Iii, 29( ) iP (i) = 1/6, i = 1, 2, . . . , 6( ). i -, i i i i i -i i , ii - i, i , ,i, P (i) = 1/6, i = 1, 2, . . . , 6.I , ( i i) iiii , i - i (ii- , , -)., (i) iii - i i. 2. i ii
{, P} iii i i (- i i). . iii -i {, P}, i i i i iii, P (i) = P (j), i, j = 1, 2, . . . , - . i i {, P} i -i i, i ii P (i) ii i 1/n(),
P (i) =1
n(), i = 1, 2, . . . , n(), i i A
P (A) =n(A)
n(). (3.1.8) -ii.
-
30 3. iii i 3.1.2. i- i i. . i A i i i i-i. ' . i i - - i, i (i 1 6). , ii (6, 1, 6, 3, 2, 4) - i , , ii - 6, 1, . . . , 4.i A i i i- , i, i 1, 2, . . . , 6.i, i i ii - i i i i (), , i ii iii ( ). I , - -. iii i ( - ) i P (): P () = 1/n(). ii i A ii . i , i ii (3.1.8), ii-i i A i i n(A) - i, i A, n() i i . i i, i i- 6, i - (i 1 6). i 66. - i, i A, i, i- i, i i - i 1 6 (i 1, 2, . . .. . . , 6) 6!, ii
P (A) =6!
66.
-
3.2. i 313.2 i: 3.1, 3.5, 3.6, 3.7, 3.14, 3.16, 3.21, 3.27, 3.28 .: 3.2, 3.9, 3.10, 3.11, 3.18, 3.19, 3.20, 3.21, 3.22. , i, i - ii i i i A, i- -, i i i i ii P (), . i i A i i ii - ii i {, P} ( ). i . i i i, ' i, i ( ii -).3.1. i i, i r -i, i i . ii i A, , i i i i.3.2. i i, i r -i, i i . - ii i A, , i i i i-i.3.3. ii i 2n - (i ) ii i i- n .1. ii , i ii : ) i i; ) i ii?2. ii , ii - : ) i i ( i i);) i ii; ) i i, iii 3 , ii 1?3.4. i i ' i, i 1 5. i . ii , i ' ?3.5. 1, 2, . . . , n . ii , 1 i 2 ii ?
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32 3. iii i3.6. , 20 25 - . -. ii , : 1) iii i ; 2) iii -; 3) i ( ii i ); 4) i.3.7. i 6 i. ii , i i i.3.8. I i -. i - i. ii , .3.9. i i - , , , , , , , , , . ii , ii i ?3.10. n i, i A i B, - i . ii, i A i B i r i?3.11. ii 20 i(i ), i -ii i 10 i . ii, : ) i i i ; ) i i - i ?3.12. ii , , ii 1, 2, . . . , n, - , i i i k(1 < k < n).3.13. ' i i . ii , :) ;) , , -i ?3.14. ii 7 i. i - . ii , i - i?3.15. ii , i - 12 i ii ii .3.16. N i M . - n i (n < M, n < N M). ii
-
3.2. i 33, m (m < M)? i-i , i i m ?3.17. n ii, m , -. ii , r i-i?3.18. i N . iii n . k , , i-i i r (r < k) . ii, ?3.19. i 49 i ( 1, 2, . . . , 49) 6. , ii . i i, - . ii . ii , 5, 4, 3 . ii - ?3.20. i 12 i. i-i , 1, 2, 3, 4, 5, 6 ii?3.21. i n i. i-i , n1 , n2 i, . . . , n6i (n1 + n2 + . . .+ n6 = n)?3.22. i i i -i . ii , - i i.3.23. ii i . - ii i -. i: A i , B i . - ii P (A), P (B), P (A B).3.24. i i . - ii i -. i: A ii , B i . i-i P (A), P (B), P (A B), P (B/A).3.25.i n i. -ii , i ' - .3.26. i i i. - ii , i 7.
-
34 3. iii i3.27. n i i . -ii , i i ? - ii, i - .3.28. , ii i ii -i, i i 24 i i -i i i. i . i . - , , , ii -i i, i i -.3.29. , i i {, P} i i i i - i i iiP (i) = 1/n(), i = 1, 2, . . . , n(), n() i i.3.30., i i {, P} - i A (i A )
P (A) =n(A)
n(), n() i i, n(A) i , A.3.31 ( -). n i i m i-. ii , i, i, . . . , m-ii ii k1, k2, . . . , km ?3.32. i n i. ii , i .3.33. i . iii: A i i , B - i , C i i.3.34. I i i n, i 0, 1, 2 . ii i: A ii - 0, B ii i m+ 2 i,
-
3.2. i 35 i ii, C iii m , D ii m0i, m1 , m2 i.3.35. i i . -ii i: A , , 6, B , , 2, C , , .3.36. ii , ii i:) i ; ) i i i; ) i ; ) i i ; ) .3.37. , i M1 1, M2 2, . . . , MN N , - n . ii i '- m1 1, m2 2, . . . ,' mN N .3.38. I {1, 2, . . . , N} i 1 i 2. ii i {1 < 2}.I {1, 2, . . . , N} i - 1, 2, 3. -ii , i - i i, i ii i{1 < 2 < 3}.I {1, 2, . . . , N} i - n 1, 2, . . . , n (n N). - ii , 1, 2, . . . , n '- .3.39. , i i , i i. i-. ii , :) i ; ) i -i i; ) i i.3.40 ( ). ( i). . i. i. . i i i i i i , i . i - i i: i i, i
-
36 3. iii i i, i i, . i i . i i i, i i, i i, ?i, , ii i i. ' i i i-i i, : 1) i ii i; 2) i i, . i ' i, -i n ( i n = 1000). i, i , n 2 n 1, i - , i . i i i- i i i i, .
-
4 ii4.1 ii. ii. {, P} iiii. ii P (A/B) i A ii B (P (B) > 0) P (A B)P (B)
,P (A/B) =
P (A B)P (B)
. (4.1.1) (4.1.1) P (A B) = P (A/B)P (B) (P (B) > 0). ii . i iP (A/B) =
P (A B)P (B)
=n(A B)n(B)
, ii P (A/B) i A i -i B i i n(AB) i, AB, n(B) i, B. 37
-
38 4. ii i . ii p = P (A/B) i Ai i B i ii i A, i, ii B. 4.1.1. i i -i . ii , - , i, ii i. ' . i. i i -i i, i 1, 2, . . . , 6. i i, i ii ii. ii i . B i ii -i, A . -i i ii P (A/B).P (A/B) =
P (A B)P (B)
,
P (B) =6 5 4
63; P (A B) = C
13 5 463
;
P (A/B) =P (A B)P (B)
=C13 5 4
63
/6 5 463
=1
2.Iii P (A/B) , , i i {, P} ii
P (A/B) i i i, AB, i, - B, P (A/B) =
C13 5 46 5 4 =
1
2. 4.1.2 ( ). ii , i, i
( iii 1/2), . i -, i i i , i. ii , i i i ?
-
4.1. ii. 39 ' 1. i- i . i W , B. i ii - = {WW,WB,BW}., WB i: i , . A1 i i - , A2 i .i P (A2/A1). i A1, A2, A1 A2 i :
A1 = {WW,WB}, A2 = {WW,BW}, A1 A2 = {WW}.i i i, A1 i, A1 A2 , P (A2/A1) =
P (A2 A1)P (A1)
=1/3
2/3=
1
2. ' 2. ii i, -, i, , i W . i i ii -
= {WW , W W, BW ,W B}.i A1, A2, A1 A2 i :A1 = {WW ,W W,W B}, A2 = {WW ,W W,BW },
A1 A2 = {WW ,W W}.P (A2/A1) =
P (A2 A1)P (A1)
=2/4
3/4=
2
3. ' i ii iii. 1. ' ? 2. ?
-
40 4. ii ii 1 i 2, i. i i 4.1.2.' 1 . , ii i i -i. Iii i i :P (WW ) = 1/2, P (WB) = 1/4, P (BW ) = 1/4.i i i ( iii- 1/2) ( iii 1/2), i , i , i i-ii 1/2 i iii 1/2 i. i i :
P (A1 A2) = P (WW ) = 1/2,P (A1) = P ({WW,WB}) =
= P (WW ) + P (WB) = 1/2 + 1/4 = 3/4,
P (A2/A1) =P (A2 A1)P (A1)
=1/2
3/4=
2
3. 'i 2 i -ii 1/4. , i i , i i , i -, : i ii i ( iii 1/2), i, ( iii 1/2). . , , i- , - ii i, i ii iii, i -. i - i:
= {WW,WB,BW} i = {WW ,W W, BW ,W B}.i ii i i :P (WW ) = 1/2, P (WB) = 1/4, P (BW ) = 1/4,
-
4.1. ii. 41 i :P (WW ) = 1/4, P (W W ) = 1/4,
P (W B) = 1/4, P (BW ) = 1/4.i i -. , , i i. ii. i Bi ,i = 1, 2, . . . , n, i, (Bi Bj = , i 6= j) i 'i i i ( n
i=1Bi =
)
.. B1, B2, . . . , Bn i iP (Bi) > 0, i = 1, 2, . . . , n. i - i A i ii
P (A) =n
i=1
P (A/Bi)P (Bi). ii i i- i. 4.1.3 . , i i i i. N i ii n (i ). i- i. i ii i: , ii , , ii ? ' . Ali i i- - i, i = 1, 2, Au1 1- i. i P (Al1) iP (Al2).i ii
P (Al1) =n
N.
-
42 4. iiIii i P (Al2) ii, , Al1 i Au1 i:P (Al2) = P (A
l2/A
l1)P (A
l1) + P (A
l2/A
u1)P (A
u1 ) =
=n 1N 1
n
N+
n
N 1N nN
=n
N., ii i i . i . ii , - , i i. i ii , i , i (i ), i (n 1)/(N 1),
P (Al2/Al1) = (n 1)/(N 1), i (i ),
P (Al2/Au1 ) = n/(N 1). . B1, B2, . . . , Bn - i i P (Bi) > 0, i = 1, 2, . . . , n. i ii A (P (A) > 0)
P (Bi/A) =P (A/Bi)P (Bi)
n
k=1
P (A/Bk)P (Bk)
, i = 1, 2, . . . , n. 4.1.4 ( i ). i i ii iiip (0 < p < 1) . (i ), i i iii p1, i , i - i iii p2. i, i . ii , i ii i ?
-
4.2. i i 43 ' . : Ac i -i (i ), A i i i- , Bc i i ,B i . i P (Ac/Bc). (i Ac i A i),
P (Ac/Bc) =P (Bc/Ac)P (Ac)
P (Bc/Ac)P (Ac) + P (Bc/A)P (A). i
P (Ac) = p, P (A) = 1 p, P (Bc/Ac) = p1, P (Bc/A) = p2., iiP (Ac/Bc) =
p1p
p1p+ p2(1 p).4.2 i iI, i i i, i-i i (i, i) i i. (, ii - i ii , i i i.) i i A i B i, ii
P (A/B) = P (A), ,P (A B)P (B)
= P (A). ii i :P (A B) = P (A)P (B).
-
44 4. ii. {, P} iii i. -i A i B (A , B ) ,P (A B) = P (A)P (B).i A1, A2, . . . , An -i, k = 2, 3, . . . , n i i1, i2, . . .
. . . , ik , 1 i1 < i2 < . . . < ik n iiiP (Ai1 Ai2 . . . Aik) = P (Ai1)P (Ai2) . . . P (Aik).i A1, A2, . . . , An -, - i ii s i k
P (As Ak) = P (As)P (Ak). i A i B i, i A i B; A i B i. 4.2.1 (. . ). i , i i-i , i i , -i i i . i: R -, B i, G , -i i, , iii. i, i R, B, G -i, i. ' . i . i - , . i i , P (R) = 2/4 = 1/2, P (B) = 1/2, P (G) = 1/2. i i i i i i,
P (R B G) = 1/4, P (R B) = 1/4,P (R G) = 1/4, P (B G) = 1/4.i
P (R B) = P (R)P (B), P (R G) = P (R)P (G),
-
4.3. i 45P (B G) = P (B)P (G)., i R,B,G i.
P (R B G) = 1/4 6= 1/2 1/2 1/2 = P (R)P (B)P (G). , i R,B,G i.4.3 i: 4.11, 4.12, 4.13, 4.22, 4.23, 4.24.: 4.3, 4.5, 4.7 , 4.9, 4.10 , 4.13, 4.16, 4.19, 4.24, 4.26.4.1. i A i B i. , i Ai B, A i B i.4.2. i A1, A2, . . . , An i i P (Ak) = pk, k = 1, 2, . . . , n. ii , :) i i A1, A2, . . . , An;) i i i A1, A2, . . . , An;) i i i A1, A2, . . .. . . , An?4.3. ii i, i ', i i-ii p. ' i - i i. n i . ii , ' ?4.4. , n i, , i . i . ii1 p. ii -.4.5. i ii i - ; ii - p. i i -. ii:1i i ii i i- ii i iii .
-
46 4. ii1) i i;2) i, i - ii p1.4.6. i ii i - n1 ; ii - p. - i . ii:1) i i;2) i, , , ii p1.i i, ii i P?4.7. i i : - i. ii , i i 10, i, :) 5 ;) i 5 (i 5 )?4.8. i i n . i i - ii i i iii. - . ii , i?4.9. N i n1, n2, . . . , nN , i ii m1,m2, . . . ,mN . - , . ii , i?4.10. i n1 i n2 , i i- iim1 im2. , i i. i . ii , i?4.11. i i . i-i , ', - i ?4.12. ii i i ', -ii . ' , i iii p0, , iii p1 (p1 < p0). Iii, i -, p i i , i
-
4.3. i 47 i . -ii , ' i n i .4.13. n i, i i , i . i-i i p1, - p2. Iii , i p, (1p). i . ii .4.14. , i n , i . ii , i, i ii i i iii?4.15. i i n . i ii i i iii. - i. i-i i ii i i. iii?4.16. i k1 ( ) i m1i i n1 , i k2 ( )i m2 i i n2 . I - , i. i-i , ?4.17. i A i iii p1 == 0,6, i B iii p2 =0,5, i C iii p3 =0,4. ii i ii. i, . ii: i C i i?4.18. i i 12 i, 8 i 10 . i i. ii , i i i , i, ?4.19. , i 3 ii i 2 i i, - i i , i 4 ii i 4 i i. ii i ?4.20.i . -i n i . i p1,
-
48 4. ii p2. -. ii , i?4.21. , i 3.40, i , i, , i , - i i. i, , i , - i i, i i, i, . i - i i . ii i. i : i - i n . i (n 1) , i , (n 2) , i, ( - ), i , i i , . i ? ii , - i?4.22. ii ii . Iii ii i 1/3 i i , i i i ii. i-i i: i ii i i,i ii i .4.23. i i ii - 1 . Iii i . i . ii , , i .4.24. Iii , i i - , i 0,96. , i i iii 0,98, i iii 0,05. ii , i, , ?
-
4.3. i 494.25. i i . i-i , i 'i, i-, i ?4.26. ii i i. i 5 10, 8 10. i i - i. ii i, , i. i , i i. ii , i i?4.27. : i N1 i iM1 , i N2 i i M2 , i N3 i iM3 (Ni 2, Mi 2, i = 1, 2, 3). i - i i. i, i . ii , i ) -; ) ; ) .4.28 ( i ). i - i.(, i , , i -, .) i , , 6 i, . , i (, 5 i, 3). i ? i i i. i i 1494 . i i. i , - . ' - i. i - ' i i i i. - ' i (14991557), - i i, i - i i i. ' i i 1654 . i i , 1654 . i -i.
-
50 4. ii4.29. i . , 7 i, . -, 6 i, 3. i ?4.30 ( i ). i , i, i ( iii 1/2), . - n i , i i ii n , i i. ii , , , i?4.31. , i 5 i i 2 i, . , -, 2 i, i i. ii , i .4.32.m ii i, ' iii p (- i i i i i). T i i n i. -ii i: A T ' i i i i, B T ' i i.
-
5 i5.1 i ii i-i i. I, , i i i ( i- , i, i i i ii, -i i, . . .). i ii ii- {, P}, i = () i .. ii i {, P} i = () = (1(), 2(), . . . , n()) i Rn, i i . n > 1, = () = (1(),2(), . . . , n()) i - i Rn ( -), , n = 2 i, n = 1 51
-
52 5. ...i () -.. , i i i , - . , i-i i, .i . - - i i, (, i). x Rn - = () i Rn, i {, P}, P{ = x} > 0; X
(X Rn).. i - = () i Rn iP : x P(x), x X, i X i - , iii x X iiiP(x) = P{ = x}, .i
P(xi, yj, . . . , zk) = P{1 = xi, 2 = yj, . . . , n = zk} = (1, 2, . . . , n) i Rn (n > 1) i i - 1, 2, . . . , n.i = ()i R1 i i, ii i - , ii, i :
-
5.1. i i ... 53x1 x2 . . . xn . . .
P(x1) P(x2) . . . P(xn) . . .iP : (xi, yj) P(xi, yj), (xi, yj) X R2, = (, ) i R2 i . 5.1.1. 5.1.1. i = (, )- y1 y2 ym x1 P(x1, y1) P(x1, y2) P(x1, ym)
x2 P(x2, y1) P(x2, y2) P(x2, ym) ... ... ... ... ... ...xn P(xn, y1) P(xn, y2) P(xn, ym) ... ... ... ... ... ... 5.1.1 . i, i - i i . i . ' . i
= () ii i {, P}, = {, , , }, ii - i i 1/4 ( i); 0, 1, 2, P{ = 0} = P{ : () = 0} == P () = 1/4, P{ = 1} = P (, ) = P () ++P () = 1/4 + 1/4 = 1/2, P{ = 2} = P () = 1/4., i
xi 0 1 2
P(xi) 1/4 1/2 1/4
-
54 5. ... i i i . i P i - i g()i .. i - Rn ii i {, P},P : x P(x) i, g = g(x) i Rni Rl. i B Rl
P{g() B} =
x:g(x)BP(x). (5.1.1), = (, ), g = g(x, y) i P(xi, yj) -i = (, ),
P{g() B} = P{g(, ) B} =
(xi,yj):g(xi,yj)BP(xi, yj).(5.1.2) 5.1.2 . ii i - , i ii , . i min{, }. ' . i - = (, ). = () = ((), ()) -i ii i {, P}. -i i ii - i ; , i = () , i-i i i , i , -. i i iii ( i),iii i 1/16. P(i, j) (i, j), i = (, ) (. . 5.1.2).,
P(1, 1) = P{ = (1, 1)} = P{(, ) = (1, 1)} =
-
5.1. i i ... 55= P{ : ((), ()) = (1, 1)} =
= P{, , , } == P () + P () + P () + P () =
= 1/16 + 1/16 + 1/16 + 1/16 = 1/4. 5.1.2. i = (, )- 0 1 20 1/16 1/8 1/161 1/8 1/4 1/82 1/16 1/8 1/16 i P(i, j), i, j = 0, 1, 2, - = (, ) i -i i (. (5.1.2)), , ig() = g(, ) = min{, }. B = {k}, k = 0, 1, 2, :
P{min{, } = k} =
(i,j):min(i,j)=k
P(i, j)., k = 0P{min{, } = 0} =
(i,j):min(i,j)=0
P(i, j) =
= P(0, 0) + P(0, 1) + P(0, 2) + P(1, 0) + P(2, 0) =
= 1/16 + 1/8 + 1/16 + 1/8 + 1/16 = 7/16.iP{min{, } = 1} = 1/2, P{min{, } = 2} = 1/16., i min{, }
-
56 5. ...xi 0 1 2
Pg()(xi) 7/16 8/16 1/16i i . i - i , i xi, yj i P(xi, yj) = P(xi)P(yj). (5.1.3)(i P, ii (5.1.3), ii P i P .) I -, i i i, i i- i i i ii i .i 1, 2, . . . , n -, i i i i -i ii 1, 2, . . . , n.i i - . i . I, i -i, ii , i - , i , i. 5.1.3. 1, 2, . . . , n i -i , i
P{l = k} =k
k!e, > 0, k = 0, 1, . . . (l = 1, 2, . . . , n). i i 1,
2, . . . , n. ' . i P1 , P2 , . . . , Pn - 1, 2 . . . , n i iiP(k1, k2, . . . , kn) = P{1 = k1, 2 = k2, . . . , n = kn} ii -:
P(k1, k2, . . . , kn) =n
i=1
Pi(ki) =n
i=1
ki
ki!e =
-
5.2. i i R1 57=
n
i=1ki
k1!k2! . . . kn!en; k1 = 0, 1, . . . ; k2 = 0, 1, . . . ;
. . . ; kn = 0, 1, . . .5.2 i i R1 i. i i,i i. ii - (i) i (i, -) ii i, i i - , - i.i ii n i - ii = (1, 0, 1, . . . , 1) n, -i 1 i 0 (1 i i, 0 ).Iii i iiP () = p()(1 p)n() (0 < p < 1), () ii , p ii-i i i (i - q = 1 p). {, P} ii n i.. i -i (n; p),
P(k) = Cknp
k(1 p)nk, k = 0, 1, . . . , n. (5.2.1)i Cknpk(1 p)nk Bn;p(k) (k n+ 1 , Bn;p(k) = 0). ii n i i-ii i p i ii (n; p).. i -i (i ) ( > 0),
P(k) =k
k!e, k = 0, 1, . . . (5.2.2)
-
58 5. ... ii Bn;p(k), n i k i (i i). , , ( n, p ) i . . np ( > 0), n , i k, k = 0, 1, 2, . . .
limn
Bn;p(k) =k
k!e. (5.2.3)I , n i p i i i i i. i ii
k=0
Bn;p(k)
k
k!e
2min{2, }
n( . . ). . npq , n , m,
m npnpq
C
(C i, i ),limn
Bn;p(m)/ 1
2npqexp
{
12
(m npnpq
)2}
= 1.(5.2.4)I -. m ii n i iii i p i. -i x1 < x2limn
P
{
x1 0), P(k) = (1 p)kp, k = 0, 1, . . . (5.2.6) i ii iii i p - i p (p > 0).i' i i. - i' i i (r; p),
P(k) = Cr1k+r1p
r(1 p)k, k = 0, 1, . . . (5.2.7) r- i ii - iii i p -i i' i i - (r; p)., r = 1, i' i -i i .i i. - i i (N,M,n), n M,n N M ,
P(m) = PN,M,n(m) =CmMC
nmNM
CnN, (5.2.8)
m = 0, 1, . . . , n. i i N , M i-i, (N M) i. i n(n M, n N M) i i (N,M,n). N,M , M/N p (0 < p < 1),
PN,M,n(m) Cmn pm(1 p)nm, m = 0, 1, . . . , n.
-
60 5. ...5.3 i: 5.3, 5.8(1), 5.8(2), 5.8(3), 5.13, 5.29, 5.30.: 5.1, 5.4, 5.8(1), 5.8(2), 5.14, 5.18, 5.31, 5.40.5.1. i . - , . -i = sin 3 .5.2. , i-i i i i. i = sign(cos 3 ), , .5.3. , i i 3 ii 2 i i, i i , i 1 i 2 i i.i 2 i. i . i .5.4. 1, 2, . . . , n i i -, i - p :P{l = k} = Pl(k) = (1 p)kp, k = 0, 1, 2, . . . ;
l = 1, 2, . . . , n. i i 1, 2,. . . , n.5.5. 1, 2, . . . , n i i -, i i (m, p) :
P{l = k} = Ckmpk(1 p)mk, k = 0, 1, . . . ,m;
l = 1, 2, . . . , n. i i 1, 2,. . . , n.5.6. j = j(), j = 1, 2, . . . , n, ii , P{j > 0} = p, P{j < 0} = q, P{j = 0} = f, p+q+f = 1,
j = 1, 2, . . . , n;
-
5.3. i 61s ii j , j = 1, 2, . . . , n,ii i , j , j == 1, 2, . . . , n. i i s.5.7. ii i. i-i i {, P} . = (, ) {, P} i - R2, , . i i - i (i = (, )). , i i i.5.8. i ii = (, ), - i, .1. i:) max{, };) min{, };) + .2. :) P{ 2,max{, } 4};) P{max{, } 4};) P{| | 3};) P{4 + 6};) P{ 1,max{, } 3};) P{max{, } 4};) P{min{, } 1,max{, } 5};) P{max{, } 3};) P{ 2,max{, } 3}.3. i i :) i +; i i + -?) i max{, }; i i max{, }?) i min{, }; i i min{, }?5.9. i i - .
-
62 5. ... (iiii {, P}) . = (, ) {, P} i R2, i, i i, i . i , i i 5.8, ii, ii .5.10. i i i iP{ = xk} = ak, P{ = xk} = bk, k = 1, 2, . . . , n. P{ = }.5.11. i i
i , 5.3.3, i - , i i 5.8, -ii ii i. 5.3.3. i i i - 1 2 3 4 5 6 7 81 1/16 1/32 0 1/32 1/32 1/32 1/32 1/322 1/32 1/16 1/32 0 1/32 1/32 1/32 1/323 1/32 1/32 1/16 1/32 0 1/32 1/32 1/324 1/32 1/32 1/32 1/16 1/32 0 1/32 1/325.12. i i i iP{ = i} = 1/(n+1), P{ = i} = 1/(n+1), i = 0, 1, . . . , n. i = + .5.13. i i i , i i ii i . i = + .
-
5.3. i 635.14. i i i , i i ii i ., i - + = n i i (n, /(+ )), P{ = k/ + = n} = Ckn
(
+
)k (
1 +
)nk,
k = 0, 1, . . . , n.5.15. i i ii p i :P{ = k} = p(1 p)k, P{ = k} = p(1 p)k, k = 0, 1, . . . i i imax{, }; i max{, }.5.16. i i . - , i i, , i 3, i i - i; , i i. i = + .5.17. i i : i25 . i i i 50 .; 1 i, - i i 25 ., 2 i 50 . i = (1, 2).5.18. , - 1 i 2 i, i = min{1, 2} i. i, - ii 1 i 2 iii p1 i p2.5.19. i 5 -i. ii , : 1) ii ' - , ; 2) i i ' , .5.20. i i- 6 i. ii , - i .
-
64 5. ...5.21. i 6 -i. ii , i .5.22. i i i , i i, iii p - .1. ii , 10 i ii ?2. ii , - k- i i.3. ii , i 10 i - , , i l i . ii i l?5.23 ( ). i i ii. , i i,i . , . - ii , i () ir ii, , i N (N r) ii.5.24. ii i:) i i ii 6 i;) i ii 12 i;) i ii 18 i.5.25. i 10 i. - i i, .5.26. i 10i. i i, .5.27. i i, i, '- . i .5.28. ii, , ( i):1) 4 i 8 3 5?2) 3 i 6 2 4?3) 3 i 4 5 8?4) i 3 i 4 i 5 8?5) i i n i 2n i i n i 2n?
-
5.3. i 656) i i n i 2n + 1 i i ni 2n?5.29. i = 1 + 2 + . . . + r - , - i p.5.30. Iii , i k i, ik
k!e, k = 0, 1, . . . ,( > 0). ii i i- p. ii , si iii .5.31. , iii
k
k!e i k (k = 0, 1, . . .) . Iii i p. -i , ii , i .5.32. i -- iii , i j i, i - ,
pj =j
j!e, j = 0, 1, . . . ,
. i i -i i ii - , iii i i i i i i-. -i i p ii i-i. i i-i -i.5.33. i A 500 -i ' i B. i-, ii i ' i i i i . i: ) = 5;) = 8; ) = 10, ii -
-
66 5. ... ii ' i A i i B, iiii 0,01; 0,02; 0,05?5.34. , i 1000 i, ii. i . ii i . ii ii i , 9 i 10 i i - i , i? : ) i ;) i i. i, i i.5.35. ixi x1 x2 . . . xn
P(xi) p1 p2 . . . pn x (,) P{ < x}. - i i F (x) = P{ < x}, x (,). F (x)? i i F (x) i i i.5.36. , - i i i i, n,n = 1, 2, . . . , ii
P{ = n+m/ n} = P{ = m}, m = 0, 1, . . .(i ii ii i-).5.37. i i i . - , . iipk , i k- ii. i- i i ii , ?5.38. i ii i i- i. i i-i. , i, .1. ii p = 1/5, i-i: ii i?
-
5.3. i 672. i i ii i ii , ii p = 1/3i ii ii ?5.39. i iii i i i -i.1 i 4040 i, 2043 ().2 i 12 000 i, 0,5016; i -i ii 24 000 i 0,5005 (. i). i:) ii , i - i i 1/2 - i i 1/2 -, i i;) , i, ii - 0,9999, i, - , i i- i 1/2 i ii.5.40. i i - , i i i. iix2 + x+ = 0
, , i i- . ii , i:1) ii i; 2) ii i; 3) i i.5.41. iP : (xk, yl) P(xk, yl) = (, ) i R2 i .5.42. i i ii i i
-
68 5. ...P{ = k} = pk, k = 0, 1, . . . ; P{ = l} = ql, l = 0, 1, . . . i = + .5.43. - . x P{ < x}. - i i
F (x) = P{ < x}.5.44. i 10 i. i-, ii , 4 , iB10;1/2(4) = C
410(1/2)
10. i ii , iii 4- i, ii 10 i?5.45. ii - {, P} i = (), = () , i P i P i i, 6= i i .5.46. i i , iii - 1/3; i, . i .5.47. , iii i i i p, i i n i. Ii-i , i r i, iBn;p(r) = C
rnp
r(1 p)nr. ii , i ii r i ii n i?
-
6i 6.1 , i, iP : xi P(xi) = P{ = xi}.i , - i ii, i . I-i i - ( - ). ii i -i . ii -i , i, -, i i i, i i i . ,, i i i - . (i i i- i i i.). iM - = (), i-69
-
70 6. ii i {, P}, M =
()P (), i i.i i.1. i i ii i:
Mc = c (c ).2. i i i i :M( + ) =M +M.3. -i:
Ma = aM.4. i - i i i:M =MM. i - i. i - - i i i ( i i). 6.1.1. = () - i R1, P : xi P(xi) i,
g i R1 i R1. xi
g(xi)P(xi) i , Mg() =
xi
g(xi)P(xi), (6.1.1), xi
xiP(xi) i , M =
xi
xiP(xi). (6.1.2)
-
6.1. , i, 71 ii i i i- i Rn.
x1,x2,...,xn
g(x1, x2, . . . , xn)P(x1, x2, . . . , xn)i , Mg(1, 2, . . . , n) =
=
x1,x2,...,xn
g(x1, x2, . . . , xn)P(x1, x2, . . . , xn), (6.1.3) P : (x1, x2, . . . , xn) P(x1, x2, . . . , xn) i - = (1, 2, . . . , n). i . i () ii (6.1.2). 6.1.1 . , - ii . , i xi 1 2 3 4 5 6
P(xi) 1/10 1/10 1/10 1/10 1/10 1/2 i . ' . i (6.1.2)M = 1 1
10+ 2 1
10+ . . .+ 5 1
10+ 6 1
2= 4,5., , -i
xi 1 2 3 4 5 6P(xi) 1/6 1/6 1/6 1/6 1/6 1/6
-
72 6. iM = 1 1
6+ 2 1
6+ . . .+ 5 1
6+ 6 1
6= 3,5. 6.1.2. i .
M1
+ 1. ' . i i P i-Mg() i i , i (. (6.1.1)). i g(t) = 1/(1 + t), i i :
P(k) =k
k!e, k = 0, 1, . . .
M1
1 + =
k=0
1
1 + kP(k) =
k=0
1
1 + k
k
k!e =
=e
k=0
k+1
(1 + k)!=e
(e 1) = 1 e
.i . i D M( M)2 (
M( M)2
-
6.1. , i, 731. i i i:Dc = 0 (c ).2. i :
Da = a2D.3. i i i i i:D( + ) = D +D. 6.1.3. i- i i i (n; p) . ' . i
P{ = k} = P(k) = Cknpk(1 p)nk, k = 0, 1, . . . , n.i (6.1.2)M =
n
k=0
kP(k) =n
k=0
kCknpk(1 p)nk =
=
n
k=0
kn!
k!(n k)!pk(1 p)nk =
= np
n
k=1
(n 1)!(k 1)!(n k)!p
k1(1 p)nk. k 1 = s, :np
n1
s=0
(n 1)!s!((n 1) s)!p
s(1 p)(n1)s =
= np(p+ (1 p))n1 = np.
-
74 6. i,M = np.i
M2 =
n
k=0
k2Cknpk(1 p)nk = np(np p+ 1).i
D =M2 (M)2 = np(np p+ 1) (np)2 = np(1 p). 6.1.4. i i i. i i -i , . ' . , , , i == + , . - i, i i
xi 1 2 3 4 5 6P (xi) 1/6 1/6 1/6 1/6 1/6 1/6i
M =M( + ) =M +M = 7/2 + 7/2 = 7,
D =M2(M)2 = 1216+221
6+. . .+621
6(7/2)2 = 35/12. i i i i ,
D = D( + ) = D +D = 35/12 + 35/12 = 35/6., - i i i i.. ii i cov(, ) =M( M)( M).
-
6.2. i 75. ii i - i r(, ) =
M( M)( M)D
D
. i i i, r(, ) = 0, . 6.1.2 (ii ). D 0
P{| M| } D2. ii , , , i D , i i (ii i ) i ii. 6.1.3 ( ).
1, 2, . . . i ii i - i Mi = a i i iDi 0
P
{
1
n
n
i=1
i a
}
0, n .6.2 i: 6.1(1),6.2,6.4,6.5,6.16,6.18,6.21,6.27 , 6.31,6.32;: 6.3,6.6,6.13,6.15,6.17,6.20, 6.24,6.30,6.33.6.1. i i ; i, , .:1)M
1
+ 1cos
6; 2)Me sin
6.
-
76 6. i6.2. i i 2 ii 8 . i. i i ; M D.6.3. i i i i-i ixi 0 1 2 3 yj 0 1 2
P(xi) 1/2 1/6 1/6 1/6 P(yj) 1/2 1/4 1/4M
2 + 1.6.4. ii i , i, .
M(1) sin 3.6.5. i i'. ,
M =
m=1
P{ m}.6.6. i i i i-i ixi 1 2 3 4 yj 1 0 1
P(xi) 1/4 1/4 1/4 1/4 P(yj) 1/3 1/3 1/3M2 sin
(
2)
.
-
6.2. i 776.7. ixi 2 1 1 2
P(xi) 1/4 1/4 1/4 1/4 i - :1) = sin
; 2) = 2||; 3) = sin
3.6.8. i
xi 2 1 1P(xi) 1/2 1/4 1/4
M22.6.9. ixi 1 1 2
P(xi) 1/4 1/4 1/2:1)M2 sin
3; 2)M2|| cos2
12.6.10. i
xi 2 1 0 1P(xi) 1/4 1/4 1/6 1/3:1) i = ||;2) i i .
-
78 6. i6.11. ixi 2 1 1 2
P(xi) 1/6 1/6 1/2 1/6M sin2
12.6.12. i
xi 2 1 2P(xi) 1/2 1/4 1/4:
1)M2|| cos2
12; 2)M2||.6.13. i i i ii i
xi 0 1 2 3 yj 1 0 1P(xi) 1/2 1/6 1/6 1/6 P(yj) 1/4 1/2 1/4
M + 1
4 + 2.6.14. i i i ii i
xi 0 1 2 yj 1 0 1P(xi) 2/3 1/6 1/6 P(yj) 1/3 1/3 1/3
-
6.2. i 79M4 + 1
.6.15. n,(n 1), . . . ,1, 0, 1, . . . , n 1, n ii1/(2n + 1). : 1)M; 2)M ||.6.16. i . : M, M2, D.6.17. i i i , i . -:
1)M
1 + ; 2)M; 3)D; 4)D( + ).6.18. -i p. : M, M2, D.6.19. i i.1. i i i i -?2. ii , i i?6.20. -i p. :
1)Mx , |x| < 1; 2)Meit .6.21. i . : 1)Mx, |x| < 1; 2)Meit .6.22. i i -i i 100 - , iiii i 0,7.6.23. i i , ii 1/3. i, . i . M D.
-
80 6. i6.24. i 10 i. - i i, . M D.6.25. i i -i i i 5.26.6.26. i i -i i i 5.27.6.27. ii - iii i p (0 < p < 1) -i. , i r- i. i' i i (r, p):P{ = k} = Ckr+k1pr(1 p)k, k = 0, 1, . . .,
M = r1 pp
, D = r1 pp2
.6.28.1 = (, ) i i5.8 :1)M max{, }; 2)M sin(max{, }/6); 3)M min{, }.6.29. i i i i i 5.10 :
1)M min{, }; 2)M cos(max{, }/3);
3)M max{, }; 4)M sin(min{, }/4).6.30. = (1, 2, . . . , r) -iP{(1, 2, . . . , k, . . . , r) = (0, 0, . . . , 0, 1, 0, . . . , 0)} = pk, ii (0, 0, . . . , 0, 1, 0, . . . , 0) k- ii - , i i , k = 1, 2, . . . , r.1 ' 6.28 6.31 i (6.1.3).
-
6.2. i 81M exp{i(t, )} =M exp{i(t11 + t22 + . . . + trr)}.6.31. 1, 2, . . . , n i -ii i i R1;
R1 =
r
j=1
Xj , Xs Xl = , s 6= l,
P{k Xj} = pj, j = 1, 2, . . . , r; = (1, 2, . . . , r) , j- - j i ii 1, 2,. . . , n, Xj , j = 1, 2, . . . , r.
M exp{i(t, )} =M exp{i(t11 + t22 + . . . + trr)},
tu R1, u = 1, 2, . . . , r.6.32. ii k . i . i 1. . i i k ii. i 2. k i i i - i. i (i i), i k . , - i i k i i- k + 1 i. , ii p i i i ii i.1. ii , i i i k .2. i i-i , i i .6.33. i i 2 ii, 3 i i 5 . i. i- : i , , . -i i .
-
82 6. i6.34. i' i - iP{ = k} = pk, k = 0, 1, . . .i P (t), ii
P (t) =Mt =
k=0
tkpk, |t| 1, i i (i {pk}). i i , : 1 i i; 2 i;3 i i.6.35. , ii i iii -i ii ii i ( i).6.36. , i i i i ii (i i i i).6.37. , - i 1 i 2 ii 1 + 2.6.38. , i i- i ii (n; ) i (m; ) i i - (n+m; ).6.39.
= {0, 1, 2, . . .}, P () = ap, = 0, 1, . . . ,p i i (0; 1). a i P () = ap, , -ii ? = () {, P}, - ii = () = 2. M, M2.6.40.
= {0, 1, 2, . . .}, P () = a 1!, . a {, P} ii ?
-
6.2. i 83 = () {, P}, - ii = () = 2. M, D.6.41. = {0, 1, . . .}, P () = a
!, ,
i i . a {, P} ii i? = () {, P}, - ii = () = 2. M, D.6.42. ii ( 6.1.2).6.43. ( 6.1.3).6.44. P(xi, yj) i = (, ). M.6.45. i . i i , i i, i i i, i i, ii 'i, i i. i, i i i , .6.46. i .i i i i . i i i -, i i, i i. i i , i. - i i , .6.47. 1, 2, . . . , n i , i ii i, Sn = 1+2+ . . .+n. ,
DSn =
n
k=1
Dk + 2
i
-
84 6. i6.48. 1, 2 i i i i = 1+2, = 12. , r(, ) = 0.6.49. i ii i i ,i ' ii i. -, r(, ) = 0. , i i.6.50. 1,+1,2,+2 iii 1/4, = 2. i i i. , : 1) r(, ) = 0; 2) i .6.51. i i i 5.3.6.52. , i-i , i, ii . -
M, D, M, D, M( + ), D( + ).
-
7ii i7.1 , -i .i - () i i i-. i - i A,B,C, . . . (-) . - A, : A (: A); A, : / A (: - A)., i , - . A - B, : A B (:A i B). A - B (A B) i B A (B A) (i , i -i A i B ), A i B : A = B. ii A = B, i-, A B i B A, 85
-
86 7. i i iA B, i B A. i A i B i -., i, i i i A B, - ' A i B i A B., i, i i A, i B, A i B i A B., i, i i A, B, i A i B i A \B., i , i - i A, A i A., A i B (i), A B = . 7.1.1. {An} ii - (Ai Aj = , i 6= j) i
Bn =
i=n
Ai, n = 1, 2, . . ., n=1
Bn = . ' . , n=1
Bn 6= , i - n=1
Bn. i B1 = i=1
Ai i, , - i Ai, i = 1, 2, . . . , iAi Aj = , i 6= j, i i i, An0 .
/
i=n0+1
Ai = Bn0+1. n=1
Bn.
-
7.1. , - 87 . i A i , :1 i - A A
A A;2 i - A,B A ' A B A.. K i . , i K (, - K), A(K), :1 i K, K A(K);2 i -i ii i A, i K, A(K) A.- . i F i - , :1 i A F A F;2 i - i ii - Ai F, i = 1, 2, . . . , '
i=1Ai F. i - F i.. K i . -, i K (--, K), -
(K), :1 i K, K (K);2 i -i -i F, i
K, (K) F. 7.1.2. K1 i K2 i- , K1 K2. , (K1) (K2). ' . K2 (K2), - K1 K2, K1 (K2). I i (K1) - -, K1, (K2) i -, i K1, (K1) (K2).i R1. - R1 - B1,
-
88 7. i i i K ii [a, b). -B1 R1. 7.1.3. , R1 :
1) {a} a R1;2) i ;3) i (a, b);4) i i i;5) i i. ' . , - - i i i i, i-
i=1
Ai =
i=1
Ai.1) , {a} = n=1
[a, a + 1/n). i i ii [a, a + 1/n) B1, B1 i i i i, {a} =
n=1
[a, a+ 1/n) B1.2) i A i i=1
{ai},i i {ai} B1 i i = 1, 2, . . . (. 1), A B1.3) i (a, b) = [a, b) {a}, [a, b) B1, {a} B1i B1 i i , (a, b) B1.4) i O R1 i ' i i i- ii: O = i(ai, bi). iii (ai, bi) B1, O =
i(ai, bi) B1.5) B1 i.
-
7.2. i 89 i , - . i i i- Rn, , .i R2. - R2 - B2, K i [a1, b1) [a2, b2).i Rn. - Rn - Bn, K ii [a1, b1) [a2, b2) . . .. . . [an, bn).i i.- i(X, ) - B(X), i (X, ) i.7.2 i i i i F - i .Iii. Iii, -i Fi , i' i- i.I , iii P i , -i F i , , :
1 A FP (A) 0;
2 i Ai F, i = 1, 2, . . . , , Ai Aj = , i 6= j,
P
(
i=1
Ai
)
=
i=1
P (Ai);
3 P () = 1., = Rn, F = Bn, ii iii i Rn.i 1, 2 - i1, 2, 3 ii
-
90 7. i i ii i i. i . . .i 1, 2, 3 ii - i'i, i i ,. (3.1.1), (3.1.2), (3.1.3).. Iii i {,F, P}, i i,F - i , P iii -i F.Iii i {,F, P} - . -i i - ii , F -- i i, i -i, P iii F i ii i.i ii. i ii i -i ii i i 1, 2, 3.1. Iii i i i:
P () = 0.2. Iii - i A F 1:P (A) 1.3. A B (A,B F), ii iii B i A i ii i i:
P (B \ A) = P (B) P (A).4. P (A) = 1 P (A).5. i A,B FP (A B) = P (A) + P (B) P (A B) (7.2.1)( ). 7.2.1. :
P (A1 A2 . . . An) =
-
7.2. i 91=
1i1nP (Ai1)
1i1
-
92 7. i i i ' . i -, i n n i, ii . , i i i ii-i, - . , ii i - i i n! Ais i, , is ii is, i Ai1 Ai2 . . .Aik , k = 1, 2, . . . , n, i i1 ii i1, i2 ii i2, . . . , ik ii ik (nk nk i i i). ii i
A1 A2 . . . An. (. (7.2.2)):P (A1 A2 . . . An) =
=
1i1nP (Ai1)
1i1
-
7.2. i 93 P (Ai1 Ai2 . . . Aik). Ai1 Ai2 . . .. . .Aik ii n, i-i i1 i1, ii i2 i2, . . . , ii ik ik, - n k n k i, , -i i. i, , (nk)! ii
P (Ai1 Ai2 . . . Aik) =(n k)!n!
.
1i1
-
94 7. i i i7.3 i: 7.1, 7.12, 7.13, 7.14, 7.15(1, 2), 7.16(1), 7.19.: 7.2, 7.9, 7.10, 7.15(3, 4, 5), 7.16(2), 7.20.7.1. , :) AB = A(B\(AB)), A(B\(AB)) = ;) A (iBi) = i(A Bi), , Bi Bj == , i 6= j, (A Bi) (A Bj) = , i 6= j.7.2. ii i {An} - :
An = {(x, y) : x2 + y2 1/n2}, n = 1, 2, . . . iA =
n=1
An, B =
n=1
An.7.3. ii:) A \B = A \ (A B) = (A B) \B;) A (B \ C) = (A B) \ (A C);) (A \ C) (B \ C) = (A B) \ C.7.4. An = [ 12n, 1n) , n = 1, 2, . . . -A =
n=1
An, B =
n=1
An.7.5. 1 {An} i ii .,
n=1
An =
n=1
Bn, B1 = A1, Bn = An \ n1i=1
Ai, Bn - i.
-
7.3. i 952 {Ai} ii ,Ai Ai+1, i = 1, 2, . . .,
i=1
Ai =
i=1
(Ai \ Ai1), A0 = , (Ai \Ai1) (Aj \ Aj1) = , i 6= j.7.6. I ii. ,
IA =
IA;
IA =
IA.7.7. A , A,B A. -, AB A ( i i).7.8. A . :) Ak A, k = 1, 2, . . . , n, n
k=1
Ak A;) Ak A, k = 1, 2, . . . , n, nk=1
Ak A;) A; A.7.9. , :) i i ;) K i i i , i K;) - i .7.10. , K i - , i K.7.11. F - , An F, n == 1, 2, . . . ,
n=1An F.7.12. , :) i i -;) K i i i -, i K;
-
96 7. i i i) - i - -.7.13. , K i - - (K), i K.7.14. K i ii (a, b) i i. , - (K), i K, - R1.7.15. , -, i -i i i, - R1 :1) K1 = {[a, b]; a, b R1};2) K2 = {(, x];x R1};3) K3 = {(, x);x R1};4) K4 = {(x,+);x R1};5) K5 = {[x,+);x R1}.7.16. , - -i i :1) i ;2) .7.17. A R1, - ii ' ii [a, b),(, b), [a,+), a, b R1., A - R1.7.18. F - i , B i-, i F. , FB A B, A F, -. i . B.7.19. B1 - R1,B i, i B1. B1B B A, A B1. , B1B -. i . B.7.20.B1 - R1. B1[a,b) B [a, b), B B1, [a, b) i i. , B1[a,b) -. i . [a, b).7.21. K Ai ,
-
7.3. i 97i = 1, 2, . . . , n, AiAj = , i 6= j, i n
i=1Ai = . -, K.7.22. K Ai , i =
= 1, 2, . . . , Ai Aj = , i 6= j, i i=1
Ai = . , K. -, K.7.23. , Rn :) {a}, a Rn;) i ;) i ;) .7.24. , i iiIa1,a2,...,an =
= {(x1, x2, . . . , xn) : x1 < a1, x2 < a2, . . . , xn < an},
ai R1, i = 1, 2, . . . , n, - Bn.7.25. , K i Rn - - Bn.7.26. , K Rn - - Bn.7.27. (X, ) -i. , :1) ;2) i ;3) .7.28. (X, ) -i, F , V - B(x, r) = {y : (x, y) < r}, W - B(x, r) = {y : (x, y) r}., i i - i (X, ).7.29. {An} ii i ; limAn i i i , i i An; limAn i i i, i i An, i .
-
98 7. i i i, limAn limAn;
limAn =
n=1
m=n
Am; limAn =
n=1
m=n
Am.7.30. {An} iii , i n i An+1 An.,
limAn = limAn =
n=1
An.7.31. {An} i-i i , i n i An An+1.,
limAn = limAn =
n=1
An.7.32. I A -i IA, iiIA() =
{1, A;0, / A. i :
1) IAB = IAIB ;2) IA = 1 IA;3) IAB = IA + IB , A B = ;4) Ai Aj = , i 6= j, A =
k=1
Ak, IA = k=1
IAk ;5) {Ak} ii i
A =
k=1
Ak, limkIAk = IA.
-
8iii8.1 . i B Rn (, ii- [a, b], [a1, b1] [a2, b2], i[a1, b1][a2, b2][a3, b3] i . i.). i - B i i: - () i A B, i ii, A, i ii L(A) A. , - , i B, {B,BnB , P}, BnB i B, P iii i BnB , A BnB - ii
P (A) =L(A)
L(B), L i Rn. ( L i-i [a1, b1] [a2, b2] . . . [an, bn] i n
i=1(bi ai),, L([a, b]) = b a, L([a1, b1] [a2, b2]) = (b1 a1)
(b2 a2)). i iii -99
- 100 8. i ii iii (i, - B 0 < L(B)
-
8.1. . 101[0; 1] [0; 1], ii i {B,B2B , P}, B == [0; 1] [0; 1], B2B - i [0; 1] [0; 1], P ii B2B .
P (A) =L(A)
L(B)=
3/4
1=
3
4. 8.1.2. 1 . ii, , i , . ' . 1 i- ii [0; 1] i i. - ii [0; 1] : - i ii ( i, i - x, y). ii:i i ( min{x, y}), i (- |x y|), i ( 1max{x, y}). I iii , - , i i:
{min{x, y} < |x y|+ (1max{x, y});|x y| < min{x, y}+ (1max{x, y});1max{x, y} < min{x, y}+ |x y|.
(8.1.1) ii [0; 1] i - i [0; 1] [0; 1] (. 8.1.1). ii [0; 1], ii (8.1.1), i-i (x, y) [0; 1] [0; 1], - i i ii. A. -iA = A ({(x, y) : x y} {(x, y) : x > y}) =
= (A {(x, y) : x y}) (A {(x, y) : x > y}) =
-
102 8. i ii= {(x, y) : x < 1/2, x+ 1/2 > y, y > 1/2}
{(x, y) : y < 1/2, x 1/2 < y, x > 1/2} .
x
y
0
1
1
1/2
1/2. 8.1.2: 8.1.2, ii i ii ii , [0; 1] [0; 1] - A (. . 8.1.2, A). i , ii :P (A) =
L(A)
L([0; 1] [0; 1]) =1/4
1=
1
4. 8.1.3 ( ). - , i i-i 2a i . 1 2l (2l < 2a). ii , .1 i : 1 (-) ii i ii 2a, - ; 2 , - , ii i i [0, ];3 i x i i .
-
8.1. . 103 ' . x i i - , i i ( i i i (. . 8.1.3)).' l sin '
x
2a
. 8.1.3: i i i, -, (, x), [0, ],x [0, a]. (, x) -i i , x (, x), - B = [0, ] [0, a] (. . 8.1.4). : B - (, x), i i- i. , -, i i i (. .8.1.3) i - B = [0, ] [0, a] (. . 8.1.4). i i i, -ii x l sin (. . 8.1.3) , , B = [0, ] [0, a] i A, x = l sin i i O(. . 8.1.4). i , ii p , A(ii ),
-
104 8. i ii ii. p =
1
a
0
l sind =2l
a.i, ii - .
0
x
a
x = l sin '
'. 8.1.4: i i, n i (n -), m i . , n m/n i ii p, - ii 2la mn . i 2l
a
n
m.
-
8.2. i 1058.2 i: 8.1(1, 2, 3, 10), 8.2(1, 2, 3), 8.4(1, 7, 11, 14), 8.12 ,8.18, 8.20, 8.29, 8.31.: 8.3(3, 6, 8), 8.4(2, 10, 15), 8.11 , 8.16, 8.17 , 8.19, 8.25,8.28, 8.30.8.1. [0;1[0;1 . ii , (x, y) ii:1) xy 1/2; 6) x2 + y2 < 1/4;2) min{ex, y} e1/2 ; 7) x+ y 1/3;3) min{y x2, x y2} 0; 8) y + 1/2 1/x;4) max{y ex, y 3/4} 0; 9) y < x2/2;5) y > x2 + 1/9; 10) |y x| 2/3.8.2. [0;1[0;1 . (x, y). ii i:1) i x i y i i1/2;2) i i i 1/4;3) i i (1,1) ii 1;4) i i (0,1) i 1.8.3. ii [0;1 . x i , y i. ii i:1) max{x, y} < 1/2; 8) max{x2, y} < a, 0 < a < 1;2) max{x, y} > 1/3; 9) max{x, y2} > a, 0 < a < 1;3) min{x, y} < 1/4; 10) yex 1;4) min{x, y} > 1/2; 11) y +1 x2 1 0;5) x+1 y2 1; 12) y +x x2 1.6) y +2x x2 1; 13) x2 y2 x+ y 0.7) x+y y2 1; 14) (y x)(y 1/2) 0.
-
106 8. i ii8.4. ii [2; 2] . x i , y i. ii i:1) x+ |x| = y + |y|; 8) (y 2x)(y + 2x) 0;2) x |x| = y |y|; 9) (|x|+ |y| 1)(|x| + |y| 2) 0;3) [y] = [x]; 10) |x 1|+ |y 1| 1;4) [y] = [x]; 11) (y x)(y + x 2) 0;5) [y] = [x 1]; 12) ([y] [x])({y} {x}) 0;6) {y} {x}; 13) (x signx)2 + (y sign y)2 1;7) |x|+ |y| 1; 14) |x sign x|+ |y sign y| 1;15) (|x 1|+ |y 1| 1)(|x 1|+ |y 1| 2) 0;16) ((x signx)2 + (y sign y)2 1)(|x sign x|++ |y sign y| 1) 0;([a] i a, {a} = a [a] - a).8.5. i i r. i i a i b
(a < b, 2r < a). ii , - , i, - i i.8.6. iii l i. ii , i i kl (0 < k < 1)?8.7. ii . - i i ii, ii . ii, i i, ( 1) 1 , 2 .8.8. i - i d. i i a (a > d). ii , i ?8.9. i R . ii , i - n-, ?8.10. i i . ii
-
8.2. i 107, : ) i i Ox i ii, r (r < 1); ) i- i (1; 0) r?8.11. i R . ii , i i i : 1) r? 2) i i r?8.12. i R i O i N . ii , :1) i i ii r (r < R);2) i i - r (r < R);3) i i r(r < R) i O;4) i- r (r < R) i O;5) i i- r (r < R), i i i R;6) i r (r < R), i i i R;7) i i a, i i i R;8) - i a, i i i R.8.13. ii AB a - i ' . i-i , i ii, ii b (b < a) i A, ii, iii b.8.14. ii (i i -) i , iii, i i, i i. i i, i- i, i -i i i i i, i -i ( iii - , i). i -, , ii
-
108 8. i ii ii, ii( ii). ii ii i-i i i, , i. ( - ii ii iii i ii - ii i.) . i 60 iii i i2 i-i i . ii iii , i , i, - . i, ii ii, i 0,5 , ii, i i ii - ii 2 .8.15. ii [0; 1] ,, , i .1. ii , iii - , , .2. ii , + + 3/2.3. ii , max{, , } t,0 < t < 1.8.16. i , i i i -, , i i - i ii - .1. , i t (t < 1), i i i ii, ii , i i?2. ii , i- ii: ) i i i; ) i i i , i q (q < 1)?8.17. i R i O - i N . ii , :1) i i ii r (r < R);2, i i . i ii, i i i, i-, i i, i i.
-
8.2. i 1092) i i - r (r < R);3) i i r(r < R) i O;4) ir (r < R) i O;5) i i- r (r < R), i i i R;6) i r (r < R), i i i R;7) i i - a, i i i R;8) - i i a, b, c, i i i R.8.18. ii , - iii, 1, - ?8.19. ii [1; 1] i -. p q . -ii , i x2 + px + q = 0:1) ii i; 2) i i.8.20. ii . ii , i i ?8.21. . - . ii , .8.22. x = (x1, x2) , - [0; 1] [0; 1]. ri Ar = {|x1x2| r} i Br = {x1+x2 3r} i?8.23. x = (x1, x2) , - [0; 1] [0; 1],A1 = {(x1, x2) : x1 1/2}, A2 = {(x1, x2) : x2 1/2},
A3 = {(x1, x2) : (x1 1/2)(x2 1/2) 0}., -i i i A1, A2, A3 i, i .
-
110 8. i ii8.24. i : 1) i 1; 2) i i 1. i 1. ii , i?8.25. ii [0; 1] i - . ii , :1) , i, i;2) ?8.26. ii i ii . ii , , i i-i, ii ?8.27. , i r i h, - . ii -, .8.28. ii [0; 1], i ii, . -ii i:1) i ;2) ;3) ;4) i .8.29. i . ii , - ?8.30. i . - ii , :1) , i 30;2) i ii i 30;3) 90.8.313. i A,B,C,D. - ii , ii AC i BD -.8.32. i ii i i i-, i ii i . i i ii3 . . // . 1991. 1. . 4753.
-
8.2. i 111i , i- i , i i i-. ii , i i i , i -i R, i-.8.33. i i, i- R i i, i r i i ii d 0 R r d ., R, r, d i ii -ii ii ii [50,0; 51,0, [40,0; 41,0,[9,5; 10,0 i . ii , i i, = 0, 5 .8.34. ii [0, nd], kd,
k = 0, 1, . . . , n, , ii . ii , ii i, i 1 i-i ii [0, d]; 2 ii ii [0, l](0 < l < d).8.35. . ii , i i i, i 1/3, i, i i i i i i 1/6?8.36. i i i, i- i 2a. i r (r < a). ii , ?8.37. ii L - . ii , i, i l (l < L/2) i ii.8.38. A ii i i i 1. ii i:) i i A i - x;) i i A - x;
-
112 8. i ii) i i A - x;) i i A i - x.8.39. A ii i i 1 i 2. ii - i:) i i A - x;) ii i A i x (x < 1/5).8.40. A ii i i i a. ii , i- i A -, i i i A ii .8.41. X ii i i A = {(x, y) : |x| a, |y| a}. i-i , i X i b, , i i- i A.8.42. ii [0; 1], n i (iii), i n . ii i:1) i ;2) i ;3) i i ;4) i i ;5) i ;6) i s .
-
9i 9.1 i i ii i i i i - (. . 5) i i i i , - i. , -i , i i . i. .. {,F, P} iii i. i = () i R1 , x R1
{ : () < x} F. = () , i- B (){ : () B} F.i . i i ,
+ , , , / ( 6= 0).113
-
114 9. i i i -. i, ,g 1 i R1 i R1, g() . 1, 2, . . . , n, . . . i , -
supnn, inf
nn, limn, limn,, lim n ( i) .i i . - i-
P : xi P(xi) = P{ = xi}. i i i i, - ii , -i [a, b).. iF(x) = P{ : () < x}, x R1, i i .i i i F(x) - .1. 0 F(x) 1.2. F(x) : x1 < x2,
F(x1) F(x2).3. F(x) i.4. limx
F(x) = 0, limx+
F(x) = 1.5. i a i b (a < b)P{ [a, b)} = F(b) F(a).1i g : R1 R1 , i B g1(B) . ii , i .
-
9.1. i i ii i 1156. i x0P{ = x0} = F(x0 + 0) F(x0 0).iP(B) = P{ : () B}, B B1, i , - i .ii i . - i i F (x) - i
F (x) =
x
p(t)dt, x R1, (9.1.1) , - i ( ), ip(x) ii i - . ii i p(x) P{ [a, b)}:
P{ [a, b)} =b
a
p(x)dx. (9.1.2) ii (9.1.1) , i xd
dxF (x) = p(x).ii i p(x) i'- i
+
p(x)dx = 1.
-
116 9. i i g() ii -i . i R1 i p(x) ii -i, g(x) i R1 i R1. i
P{g() B} =
x:g(x)B
p(x)dx, (9.1.3)P{g() < t} =
x:g(x)
-
9.2. i i ii i ... 117. iP : B P(B) = P{ : () B} =
= P{ : (1(), 2(), . . . , n()) B}, B Bn, i Rn, - i = (1, 2, . . .. . . , n).. i i F (x1, x2, . . .. . . , xn) = (1, 2, . . . , n) - i
F (x1, x2, . . . , xn) =
=
x1
x2
. . .
xn
p(t1, t2, . . . , tn)dtndtn1 . . . dt1, (9.2.1) , = (1, 2, . . . , n) i ( ), i p(x1, x2, . . . , xn) ii -i, i ii i - 1, 2, . . . , n.ii i p(x1, x2, . . . , xn) - = (1, 2, . . . , n) i' i+
+
. . .
+
p(t1, t2, . . . , tn)dtndtn1 . . . dt1 = 1. i g() ii -i . = (1, 2, . . . , n) - i Rn, p(x1, x2, . . . , xn) ii i, g(x1, x2, . . . , xn) -i Rn i Rl, B Rl. iP{g() B} =
x:g(x)B
p(x)dx, x Rn, (9.2.2)
-
118 9. i P{ B} =
B
p(x)dx, x Rn. (9.2.3) i = (1, 2, . . . , n) , -i g() (5.1.1) i (5.1.2).i i . i - 1, 2, . . . , n i R1 -, i x1, x2, . . . , xn R1P{1 < x1, 2 < x2, . . . , n < xn} =
= P{1 < x1}P{2 < x2} . . . P{n < xn}. 1, 2, . . . , n i ii ii i i iip1(x1), p2(x2), . . . , pn(xn), i i ii -i p(x1, x2, . . . , xn) 1, 2, . . . , n,i i i ii p1(x1),p2(x2), . . . , pn(xn):
p(x1, x2, . . . , xn) = p1(x1)p2(x2) . . . pn(xn). 9.2.1. 1, 2, . . . , n i - ii i , i ii- i p(x):p(x) =
122
exp
{
(x a)2
22
}
. i i 1, 2, . . .. . . , n. ' . i i 1, 2, . . .. . . , n i i, i i ii i p(x1, x2, . . . , xn), i i- i i ii p(x1), p(x2), . . . , p(xn) 1, 2, . . . , n :
p(x1, x2, . . . , xn) =n
i=1
p(xi) =
-
9.2. i i ii i ... 119=
n
i=1
122
exp
{
(xi a)2
22
}
=
=
(122
)n
exp
{
122
n
i=1
(xi a)2}
. 9.2.2. i i i ii i i p(s) i p(t). - ii i = + . ' . i i F(z) = + . F(z) = P{ < z} = P{ + < z}. P{+ < z}. i i i, i i ii i p(s, t) i
p(s, t) = p(s)p(t).i (9.2.2), x (s, t) R2, g(x) i g(s, t) = s + t, B i (, x).
F(z) = P{ + < z} =
=
(s,t):s+t
-
120 9. i ,F(z) =
z
+
p(u t)p(t)dt
du. (. (9.1.1)) ip(u) =
+
p(u t)p(t)dt (9.2.4) ii i = + .ii p(u), ii (9.2.4), i p(t) i p(t). 9.2.3 . i, i = 1, 2, . . . , n, -i i , i i- F (x). i i min{1, 2, . . . , n}. ' .
= min{1, 2, . . . , n}.i i i -F(x) = P{ < x}.i
1 F(x) = 1 P{ < x} = P{ x} == P{min{1, 2, . . . , n} x} =
= P{1 x, 2 x, . . . , n x} =n
i=1
P{i x} =
=
n
i=1
(1 F (x)) = (1 F (x))n( i 1, 2, . . .. . . , n). ,
F(x) = 1 (1 F (x))n.
-
9.3. i i R1 1219.3 ii R1 i. i i (a;2) (i -i, i Na;2), ii ip(x) =
122
exp
{
(x a)2
22
}
.ii i. ii i ii [a; b], iiip(x) =
{1
b a, x [a; b];0, x 6 [a; b].-i. -i i (; ), ii -i
p(x) =
{
()x1 exp {x} , x > 0;
0, x 0, > 0, > 0. i. i i ( > 0), ii i
p(x) =
{
exp{x}, x > 0;0, x 0. i -i i - (1; ).i . -i (m; ), ii
-
122 9. i ip(x) =
{m
(m 1)!xm1 exp {x} , x > 0;0, x 0.i -i i
(m; ), m = 1, 2, . . .i 2. i 2i n ii, ii ip(x) =
(12
)n/2
(n2
) xn/21 exp{
12x}
, x > 0;0, x 0.i 2 i n ii -ii (n/2; 1/2).i i. ii a, ii i
p(x) =1
a
a2 + x2.i i. i i i - (;2), ii i
p(x) =
122x
exp
{
(lnx )2
22
}
, x > 0;0, x 0.i . -i (; ), ii -i
p(x) =
{
x+1, x > ;
0, x , > 0, > 2.
-
9.4. i 1239.4 i: 9.2, 9.4(5), 9.5, 9.6, 9.12 , 9.14, 9.19, 9.23, 9.28.: 9.1, 9.4(14, 6), 9.7, 9.10, 9.16, 9.24, 9.25, 9.29, 9.38,9.42.9.1. i ii -i
p(x) =
{
0, x 6 [1; 1];1 |x|, x [1; 1]. P{2 > 1/4}.9.2. ii i ii [1; 3]. P{|| 1/2}.9.3. i i iip(x) =
1
1
1 + x2.: 1)P{3 1}; 2)P{|| 3}.9.4. ii [0; 1 . i , i. i ii :1) = max{; }; 4) = + ;2) = min{; }; 5) = max{2; };3) = ; 6) = | |.9.5. ii [0, l] ,
. i i ii i .9.6. ii i ii [1; 3]. i i ii i = 2.9.7. ii i ii [2; 2]. ii i = ||.9.8. ii i ii [0; 1]. ii i - = 1/ ( = 0, = 0).
-
124 9. i 9.9. ii i ii [2; 1]. ii i = 1/2.9.10. ii i ii [0; 2]. ii i = | 1|.9.11. i ii ii [a, b]. ii i , = e.9.12. i - . ii i - = 1/(1 ).9.13. F (x) i i - . i i - = .9.14. F (x) i i - . i i - = sign .9.15. i 1. i i = 1 e.9.16. p(x) ii i . ii ii - 1) = ||; 2) = a, a 6= 0.9.17. F (x) i i - . i i - = 2.9.18. i . ii ii : 1) = | 1|; 2) = ( 1)3.9.19. i i F (x) . i -i = F ().9.20. p(x) ii i . ii ii -: 1) = 2 + 1; 2) = 2.9.21. F (x) i i - . i ii -: 1) = e; 2) = ||.
-
9.4. i 1259.22. ii [0; 1] . i, i. 0 x 1: 1)P{| | < x}; 2)P{ < x}.9.23. i, i = 1, 2, . . . , n, i -i , i i F (x). i i max{1, 2, . . . , n}.9.24. i, i = 1, 2, . . . , n, i i ii i i Fi(x),
i = 1, 2, . . . , n. i i -:1) max{1, 2, . . . , n}; 2) min{1, 2, . . . , n}.9.25. i, i = 1, 2, . . . , n, i -i ii i i i pi(x),
i = 1, 2, . . . , n. ii ii :1) max{1, 2, . . . , n}; 2) min{1, 2, . . . , n}.9.26. i, i = 1, 2, . . . , n, i -i , i ii i p(x). ii i :1) max{1, 2, . . . , n}; 2) min{1, 2, . . . , n}.9.27. ii -i i p(x) = ae|x|, > 0. : 1) ii- a; 2) i i .9.28. ii [0; 1] , i i . i i .9.29. ii [0; l] , i i . i i .9.30. ii [0;T ] i . i i ii i ii i .9.31. 1, 2, . . . , n i -ii i , i ii i
p(x):
-
126 9. i 1) p(x) =
{1
b a, x [a; b];0, x 6 [a; b];
2) p(x) =
{
exp{x}, x > 0;0, x 0;
3) p(x) = 12a exp
{
1a |x b|};
4) p(x) =
1a exp
{
1a(x b)}
, x > b;0, x b. i ii i - 1, 2, . . . , n.9.32. i - (0; 1). i i
= 1/.9.33. i - (0; 1). i i = 1/2.9.34. ( i), i i i., N0;2-i i. i - i: ) i i; ) iii.9.35. ii i i [0; 1]. i :1) = 1 ; 2) = ln .9.36. i Na;2 . -, = ( a)/ i- N0;1.9.37. i N0;1. i = a+ ( > 0).9.38. i N0;1. i + = max{0, }.9.39. i N0;2 . - i + = max{0, }.
-
9.4. i 1279.40. F (x) i i - . i i F(x) = ( a)+ = max{0, a} (a ). i i i F (x) -i i i F(x).9.41. iii p(x). i i - = ( a)+ = max{0, a} (a ). -?9.42. F (x) i i - . i i - = min{, L} (L ).9.43. iii p(x). i i - = min{, L} (L ). -?9.44. ii i i i, :1) i (a;2);2) ii i ii [a; b]; ii[0; 1];3) -i;4) i i;5) i ;6) i .9.45. i i ii ii ii , i i 9.44.9.46. i Na;2 . :1) P{a a+ }; 2) P{a 2 a+ 2};3) P{a 3 a+ 3}; 4) P{a 4 a+ 4}. i . -i (. . 22.1.1).9.47. 1, 2, . . . , n i i -, i - .
-
128 9. i , = min{1, 2, . . . , n} i n.9.48. 1, 2, . . . , n i i -, i - . i = max{1, 2, . . . , n}.9.49. n i. i i i- . i
1, 2, . . . , n i i - i . , i i i. i i i -i .9.50. i i = i(), i = 1, 2, 3, -i ii i {,F, P} ( = [0; 1],F = B[0;1], P = L) i:
1) 1 = 1() =
{, [0; 1/3);
+ 1/3, [1/3; 2/3); 1/3, [2/3; 1];
2) 2 = 2() =
{ + 2/3, [0; 1/3);
, [1/3; 2/3); 2/3, [2/3; 1];
3) 3 = 3() =
, [0; 1/4);1/4, [1/4; 2/4);
1/4, [2/4; 3/4);1/2, [3/4; 1]. i i i.9.51. i
i = i(), i = 1, 2, j = j(), j = 1, 2,i ii -i {,F, P}. i 1 i 2 i, i ii 1 i 2. i i - 1) 11 i 22; 2) 1 + 1 i 2 + 2?
-
10i10.1 , i,i - i -. i i- i i (. . 6.1 . 6). = () ii- i {,F, P} i R1.. i M -i' M =
()P (d) =
= limn
(2nn
j=1
j 12n
P
{
:j 12n
() < j2n
}
+
+nP{ : () n})
.129
-
130 10. i- -i ii i' + = max{0, } i = max{0,},
= + . i M , i, iiM =M+ M, i M+ i M i +.i i:1. i i ii i:
M =Mc = c (c ).2. i i i i :M( + ) =M +M.3. -i:
Ma = aM.4. i - i i i:M =M M. i - . i (i i i ), i i, - i . 10.1.1. = () - i R1, g i R1 i R1.
-
10.1. , i, 131 ip(x) ii i, ii +
g(x)p(x)dx
Mg() =
+
g(x)p(x)dx, (10.1.1) i i +
xp(x)dx
M =
+
xp(x)dx. (10.1.2) i,P : xi P(xi), xi X, ii
xi
g(xi)P(xi)
Mg() =
xi
g(xi)P(xi), ii xi
xiP(xi)
M =
xi
xiP(xi).i. i D M(M)2 (M(M)2
-
132 10. ii i:1. i i i:Dc = 0 (c ).2. i :
Da = a2D.3. i i i i i:D( + ) = D +D. 10.1.1 . ii [0; 1] - . i ii i . - i i i , i i i i ii. ' . - ii [0; 1] , i = max{, 1 } i ii, = 2 , i i . i i - = max{, 1 }. , x < 1/2
P{ < x} = 0, x > 1P{ < x} = 1. 1/2 < x 1
P{ < x} = P{max{, 1 } < x} = P{ < x, 1 < x} =
= P{1 x < < x} = (x (1 x))/(1 0) = 2x 1(P{1x < < x} ii,i ii [0; 1] ). - , i i F(x) =
{0, x 1/2;
2x 1, 1/2 < x 1;1, x > 1,
-
10.1. , i, 133 i ii ii [1/2; 1]. i i i :F(x) = P{ < x} = P{2 < x} = F
( x
2
)
=
=
0, x/(2) 1/2;2(x2
)
1, 1/2 < x/(2) 1;1, x/(2) > 1.
F(x) =
{0, x ;
x 1, < x 2;1, x > 2.i, i ii i ii
[1/2; 1], M =M2 = 2M = 3/2., M i i = 2max{, 1 } - i (. (10.1.1). i ii ii [0; 1],
M =M2max{, 1 } = 2+
max{x, 1 x}p(x)dx =
= 2
1
0
max{x, 1 x}dx = 3/2. 10.1.2. -i . i = [], - M ([x] i x).
-
134 10. i ' . = [] 0, 1, 2, . . . ( ). i:P(k) = P{ = k} = P{[] = k} = P{k < k + 1} =
=
k+1
k
exdx = ek(1 e) = p(1 p)k, p = 1 e. , = [] i p = 1 e. i i - i (. (6.1.2) i 6.18):M =
k=0
kP(k) =
k=0
k(1 p)kp = 1 pp
=e
1 e . 10.1.3. i- i i , i (a;2). ' . ii i (a;2) p(x) =
122
exp
{
(x a)2
22
}
. M( a). (10.1.1)( i (x a)/ = t):M( a) =
+
(x a)p(x)dx =
=
+
(x a) 122
exp
{
(x a)2
22
}
dx =
-
10.1. , i, 135=
12
+
x a
exp
{
12
(x a
)2}
dx =
=2
+
t exp
{
t2
2
}
dt =
=2
limn
[n,n]
t exp
{
t2
2
}
dt = 0(i i i i i i - i i). ,M( a) = 0, ,
M = a.i,D =M( M)2 =M( a)2 =
=
+
(x a)2 122
exp
{
(x a)2
22
}
dx,i i i (x a)/ = t 22
+
t2 exp
{
t2
2
}
dt = 2
2
+
t d exp
{
t2
2
}
=
=22
+
exp
{
t2
2
}
dt = 2(i i i i- 2). ,D = 2.
-
136 10. i 10.1.4. i-, i i i iip(x) =
{
()x1 exp {x} , x > 0;
0, x 0(p(x) ii -i (; )). ' . i ii p(x) - i (10.1.2)
M =
+
xp(x)dx =
+
0
x
()x1 exp {x}dx =
=( + 1)
()
+
0
+1
( + 1)x exp {x}dx =
=( + 1)
() 1 = ()
()=
. , i
+
0
+1
( + 1)x exp {x}dxi i i i ii --i ( + 1, ).i
M2 =( + 1)
2.i
D =M2 (M)2 = 2.
-
10.2. i 137 10.1.5. i i -i M i ii f(x), x R1, i - i x = a. M. ' . i i i f(x) - i x = a, f(a+ t) = f(a t)(, f(a + t) i f(a t) i). ii, i f(a+ t) .i,M =
+
xf(x)dx. ii i x = t+ a. +
xf(x)dx =
+
(t+ a)f(t+ a)dt =
= a
+
f(t+ a)dt+
+
tf(t+ a)dt = a 1 + 0 = a.I +
tf(t + a)dt i i i i i i., - i i i Mi ii f(x), x R1, i i- x = a, M = a.
-
138 10. i10.2 i: 10.1(2), 10.2(1), 10.6(1), 10.7, 10.12, 10.13, 10.16,10.19(1), 10.19(3), 10.21, 10.20(5).: 10.1(1), 10.2(2), 10.6(2, 3), 10.8, 10.10(1), 10.16(1),10.14, 10.17(2), 10.19(1), 10.19(3), 10.20(1, 2), 10.22(2, 3),10.26.10.1. , ii -i i: 1) [a; a]; 2) [a; b]. Mi D.10.2. i ii i [0; 1]. i : 1) = ln(1/); 2) = sin2 ;3) = e.10.3. i ii i [a; b]. :1) M2, a = 0, b = 3;2) Me, a = 0, b = 1;3) M( 1)2, a = 1, b = 4;4) Me||, a = 1, b = 1;5) Me2, a = 0, b = 1/2.10.4. i ii -i
p(x) =1
(1 + x2). 1) M min{||, 1}; 2) M min{||,3}.10.5. i - (0;2). Me.10.6. i iii
p(x) =1
(1 + x2). i - : 1) = (2 + 1)I[0;3](); 2) = 2I[1;3]() ;3) = I[1/3;3](2); 4) = I[1;1]()
-
10.2. i 139 IA(x) i A i, A 1, A 0.10.7. i - . : 1)M; 2)D; 3)P{ > 1}; 4)Mk.10.8. i (i ) , - i = 0, 003. i i, i . i .10.9. i i i - = i ii i ii [0; 1] i [1; 3] i-i.10.10. i iiip(x) =
{0, x 6 [a, a+ 2);
x a, x [a, a+ 1);x+ a+ 2, x [a+ 1, a + 2).: 1)M; 2)M2.10.11. i iii
p(x) =1
2e|x|. MI[0;4](2).10.12. ii i
p(x) =
2e|x|, > 0(i i ). M i D.10.13. A, ii i -i i R , . i i i iii ii i A i i i Ox. i M?10.14. P ii i i i- R. i i P .
-
140 10. i i i F (x) i ii -i p(x) . ii F (x) p(x). M i D.10.15. A ii i i - i . i A i Ox.: 1) i i ||; 2) ii -i ||; 3) M ||; 4) P{|| > 1/2}.10.16. -, 1) i ii i [a; b], a > 0,a < b;2) -i (; );3) i . i i - i i-.10.17. i , 1) i ii i [a; b], a > 0,a < b;2) -i (; );3) i . i i i.10.18. ,i:1) ii i [a; b], a > 0, a < b;2) . i ' , i .10.19. ii [0; 1] . - i i . i i .1. , i i i:) ii; ) i ii.2. i , i i i:) ii; ) i ii.3. i i , i i:) ii; ) i ii.
-
10.2. i 1414. ' , i i:) ii; ) i ii.10.20. i i i -. i i -: 1 min{, }; 2 max{, }; 3 ; 4 /( + 1);5 exp{}; 6 exp{min{, }}, : ) i i-i ii i [0; 1]; ) i i-i i [0; 1], i [0; 2];) i ii i [0; 1], - i ; ) i - i ( 1, 2, 3, 6).10.21. 1, 2, . . . , n i i -, ii ip(x) =
{0, x ;
exp{ x}, x > . M min{1, 2, . . . , n}.10.22. 1, 2, . . . , n i i -, i ii ii[a; b]. i i -:1) min{1, 2, . . . , n}; 2) max{1, 2, . . . , n}; 3) 1n
n
i=1i.10.23. 1, 2, . . . , n i i -, ii i
p(x) =
{
0, x 6 [ h; + h];1/2h, x [ h; + h]. i i -:
1) min{1, 2, . . . , n}; 2) max{1, 2, . . . , n};3) (max{1, 2, . . . , n} min{1, 2, . . . , n})/2.10.24. 1, 2, . . . , n i i -, ii ip(x) =
{0, x ;
1 exp
{
1(x )}
, x > .
-
142 10. i i i -:1) =
1
n
n
i=1
i; 2) min{i};
3) 1 = min{i} min{i}
n; 4) 2 = 1.10.25. i 1, 2, . . . , n ii 1/. i - = 1n n
i=1i.10.26. ii [0;T ] i . i i . i ii ii i , M, D, Mn.10.27. P ii i i
x2 + y2 = 1. i i- OP P i Ox. ii i i .10.28. i i R i. i i ii i i M.10.29. ii i i (0; 0) i (0;R) , ii -i ii (0;R). - x2 + y2 = R2 i Oy. i i i .10.30. Ax = {(u, v) : u + v < x} R2, x i, i . MIAx(, ), i:1) i Q = (, );2) i F i G i ;3) ii f i g i .
-
10.2. i 14310.31. i N0;1. - i + = max{0, }.10.32. i N0;2 . - i + = max{0, }.10.33. i, - i i i ii
p(x) =
{
exp{x}, x > 0;0, x 0,
> 0 (p(x) ii i - ).10.34. i, ii i iip(x) =
{m
(m 1)!xm1 exp {x} , x > 0;0, x 0, > 0 (p(x) ii i (m; ), m = 1, 2, . . .).10.35. i, ii i ii
p(x) =
(12
)n/2
(n2
) xn/21 exp{
12x}
, x > 0;0, x 0(p(x) ii 2-i n ii).10.36. i, ii i ii
p(x) =
122x
exp
{
(lnx )2
22
}
, x > 0;0, x 0,
-
144 10. i > 0 (p(x)ii i - i (;2)).10.37. i, ii i iip(x) =
{
x+1, x > > 0;
0, x , > 2 (p(x) ii i (; )).10.38. i i i , i ii ii [1; 2], i . -i i i i :1) 1 = ; 2) 2 = + ; 3) 3 = /.10.39. A ii i i - i . i A i Ox.: 1) i i ; 2) ii -i ; 3) M; 4) P{ > 1/2}.10.40. = (, ) - i ii f(x, y). M, M.10.41. 1, 2, . . . , n i -ii ( i F ) i i R1,
R1 =
r
i=1
Xi, Xi Xj = , i 6= j,
i 1, 2, . . . , n, - Xi, pi = F (Xi) ii, , k Xi, i = 1, 2, . . . , r. Mi, Di, i = 1, 2, . . . , r.
-
1111.1 ii ii. Iii i R1 - i', , i iF -i B1 R1.I , iii i R1 -ii -i B1 (. . 7.2 . 7).. F iii i R1.i F (x), x R1, ii
F (x) = F ((, x)), i i F .i F ii F (x).. i i F (x) iF (x) =
x
f(y)dy, i F , -i f ii i F .Iii i F , i i i i X R1145
-
146 11. xi, , F ({xi}) > 0,
xiXF ({xi}) = 1, xi i F i -, i F i X., i
X
g(y)F (dy)i i g(y) i F i
X
g(y)f(y)dy, F i i ii- f , i
xiXg(xi)F ({xi}), F i, -i X.. i R1 i - R1 i F iii i R1.. i iii -i F i u(x), - x R1 ii
u(x) =
R1
(x y)F (dy). F :u = F .
-
11.1. ii ii 147. ii ii G i F ii i Q, i -i Q(x) i i G(x) -i F :Q(x) =
R1
G(x y)F (dy). ii i G i-ii i F F G., i F - i f ii, Q(x) =
R1
G(x y)F (dy) =
R1
G(x y)f(y)dy. (11.1.1) i ii ii i - i. I , F , G,Q iiii i,
F G = G F,
(F G) Q = F (G Q). iii ii , -. i , i .. V = FG ii i G iii i F i i ii v ii g i G i F :v(x) =
R1
g(x t)F (dt). (11.1.2), F i G i, - : iii -ii G i F ii i i g i f - i, i ii v i
-
148 11. i i g i f :v(x) =
R1
g(x y)f(y)dy =
R1
f(x y)g(y)dy. (11.1.3) i g i f v = f g = g f. 11.1.1. , -ii i i , :f; f; = f+;. ' . ii -i - (; )
f;(x) =
{
()x1 exp {x} , x > 0;
0, x 0, > 0, > 0. x 0, ii f; f; = f+; . x > 0
f; f;(x) =+
0
f;(x y)f;(y)dy =
=
x
0
f;(x y)f;(y)dy =
=
x
0
()(x y)1e(xy)
()y1eydy =
=+
()()ex
x
0
(x y)1y1dy.
-
11.2. i 149 ii i y = xt:+
()()ex
x
0
(x y)1y1dy =
=+
( + )x+1ex
( + )
()()
1
0
t1(1 t)1dt =
= f+;(x)( + )
()()
1
0
t1(1 t)1dt. ,f; f;(x) = f+;(x)
( + )
()()
1
0
t1(1 t)1dt.i f; f; i f+; ii, i ii i i( + )
()()
1
0
t1(1 t)1dt = 1.,f; f; = f+;.11.2 i ' i - ii ii i i - .
-
150 11. . i i - i i i iii.I , i i i - i F i G ii, Q(x) ii , Q(x) =
R1
G(x y)F (dy).i i (11.1.2), (11.1.3) i -: , , , i ii i ii ii i - i i.ii i - i i i- ii i. I , i i i i ii p(t) i p(t) ii, ii u(x)i + i i i p(t)i p(t), u(x) =
R1
p(x y)p(y)dy =
R1
p(x y)p(y)dy. (11.2.1) ii i v(x)ii = v(x) =
R1
p(x+ y)p(y)dy. (11.2.2) 11.2.1. i -i , ii i [0; 1].
-
11.2. i 151:1 ii i = + ;2 i i = + ;3 iii P{| + 1/2| < 1}. ' . 1 i i i-i i :
f(y) =
{
1, y [0; 1];0, y 6 [0; 1],
f(y) =
{
1, y [0; 1];0, y 6 [0; 1],ii i = + i i i (. (11.2.1)):
f(x) =
R1
f(x y)f(y)dy =1
0
f(x y)dy =x
x1
f(t)dt( i i x y = t). i i x R1, i-: x < 0, xx1
f(t)dt =x
x10dt = 0; 0 x < 1, x
x1f(t)dt =
0
x10dt+
x
0
1dt = x; 0 x 1 < 1, xx1
f(t)dt =1
x11dt +
x
1
0dt =
= 2 x; x 1 1, xx1
f(t)dt =x
x10dt = 0. , ii i
= + f(x) =
0, x < 0;x, 0 x < 1;
2 x, 1 x < 2;0, x 2.
-
152 11. 2 i i F(x) ii f(t) i :
F(x) =
x
f(t)dt =
=
x
0dt = 0, x < 0;
0
0dt+
x
0
tdt = x2/2, x [0, 1);1
0
tdt+x
1
(2 t)dt = (x 2)2/2 + 1, x [1, 2);0
0dt+
1
0
tdt+2
1
(2 t)dt = 1, x 2.3 i ii i f(t) iii , B, :
P{ B} =
B
f(t)dt(. (9.1.5)). ,P {| + 1/2| < 1} = P {| 1/2| < 1} =
= P {1/2 < < 3/2} =
=
3/2
1/2
f(t)dt =
0
1/2
0dt+
1
0
tdt+
3/2
1
(2 t)dt = 78. 11.2.2 . 1, 2, . . . , n ii , i N0;1. - i
=n
i=1
2i .
-
11.2. i 153 ' . , -i N0;1, = 2 -i - (1/2; 1/2). iF(x) = P{ < x} = P{2 < x}. x 0 F(x) = P{2 < x} i i, x > 0
F(x) = P{|| 0; 3) [a; a], a > 0; 4) [1/2; 1/2]. ii i = + .11.3. i i i ,ii ii ii i [0; 1] i[0; 2]. ii i p(x) = + .11.4. i i i ; i ii i [1; 1], -i [0; 1].:1) P{2 + > 1/2};2) P{ + > 1};3) P{| + | > 1/2};4) P{| | < 1/2};5) P{2 > 0};6) P{||+ > 1}.11.5. i i i -ii ii i: 1) [a; b], a < b; 2) [0; a],a > 0; 3) [a; a], a > 0; 4) [0; 1]. ii i p(x) ii = .11.6. ii [0; 1] i . i , i. i i ii i- i .11.7. , ii [0; 1], , -i i . i = + .11.8. ii [0; 1] i ,
-
11.3. i 155 i , i. 0 < x < 1 P{| | < x}.11.9. ii [0; 1] ii ii. - i ii i .11.10. 1 i 2 i i i i
pi(x) =
{
ieix, x > 0;0, x 0;
i > 0, i = 1, 2; 1 6= 2. ii ii: 1) 1+2; 2) ii2 1.11.11. i i i ,iei i - . ii ii :1) + ; 2) ; 3) | |.11.12. 1 i 2 i i ,ii ii 1 i 2. ii i 1 + 2.11.13. i i iii i i p(x) = exp{|x|}/2. ii ii :1) + ; 2) .11.14. i i i ; i ii i [a; a], - i . iii = + .11.15. 1 i 2 i i -; 1 i ii i [1; 1], 2 i = 1. i-i i = 1 + 2.11.16. ii i- i [h;h], i i- F (x), i i. i i ii i ( i) = + .11.17. i i i . i
P{
= (1)k}
= G(
{(1)k})
= 1/2, k = 0, 1,
-
156 11. i Q. i i = + .11.18. i i i ; i ii i [0; 1], i
P{ = k} = (1/4)k(3/4)1k , k = 0, 1. i = + .11.19. i n - , i . i i ii i - (. 9.3).11.20. 1, 2, . . . , n i ii -ii [0; 1] i . ii = 12 . . . n. M.11.21. i i i -i (, ) i (, ). i-i i + .11.22. i i i 2-i ii n im i-i. , + 2-i n+m ii.11.23. i i i, --i (, ), -i . i - = + .11.24. , - ii [0; 1], i, i i . i = + .11.25. F i Q iiii i R1,
u(s) i R1. ,
R2
u(x+ y)(F Q)(d(x, y)) =
R1
u(s)(F Q)(ds).
-
12ii ii12.1 . iii i R1 (.. 11.1 . 11). i F i i iF (x) = F ((, x)) i ii
F ([a; b)) = F (b) F (a)( a < b). F ({x0}) i F i -i {x0} i i F (x) :F ({x0}) = F (x0 + 0) F (x0 0).. i F R1 iii i (iii i),
F (R1) = 1, i , F (R1) < 1. ii i: F iii i, F (+) = 1 i F () = 0; F iii- i, i iF (+) < 1 F () > 0.i iii i i i iii ii.157
-
158 12. ii ii. x0 -i F , F ({x0}) > 0. i F -i, i i i.. I I [a; b) (i i) i ii F , a i b i F ..ii ii {Fn} i i F , n , Fn(I) F (I) i i i I -i F , :Fn F
limnFn = F. F iii i, , ii {Fn} i F -, F iii i, -. i i ii -ii:
1 ii i ii {Fn(x)} i- i i F (x) i i -i , ii ii {Fn}i i F ( ).2 i iii i F i - i a i i 2, 0, i- i i, i a. 12.1.1 . {Fn} ii -ii i i i
Fn(x) =
{0, x < 0;nx, 0 x 1/n;1, x > 1/n.i ii {Fn} ii.
-
12.1. . 159 ' . , n , i-i i {Fn(x)} i iF (x) =
{1, x > 0;0, x 0i i, i x = 0, -i i, i x = 0 ( x = 0 - F (x)).i, i, i - x, i i , n i ii Fn(x) = 0, lim
nFn(x) = 0 = F (x). i, i x, i , N (n N) ii 1/n < x. n N Fn(x) -i 1, i , lim
nFn(x) = 1 = F (x). , n ,Fn(x) F (x) i i x R1 , , 0.i, i 1 ii i-i, Fn F, n ., ii ii {Fn} i - i i, i 0. 12.1.2. {Qn} ii -ii i i
qn(x) =n2
exp
{
(x (1)n)2n2
2
}
, n = 1, 2, . . .i ii {Qn} ii. ' . ii 2 n ii ii {Qn} i i i, -i 1, i i, i 1. ii {Qn} i.
-
160 12. ii ii 12.1.3. Fh iiii ii ifh(x) =
12h
exp
{
(x a)2
2h2
}
.i Fh ii 1 h ; 2 h 0. ' . 1 , i F (x) = c,x R1, c i [0; 1], i ii i, i- i. i, F ([a; b)) i F i [a; b) i F (b) F (a) = 0, F (A) = 0i A A i ' - ii [a; b) (a i b ' - i), , i ,i (A) = B1. , ii {Fn(x)}i ii i i F (x) = c (c i [0; 1]), Fh h i i F , i i.i,Fh(x) =
x
fh(t)dt =12h
x
exp
{
(t a)2
2h2
}
dt =
=12
(xa)/h
exp
{
u2
2
}
du( i i (t a)/h = u). i xlimh
Fh(x) =12
0
exp
{
u2
2
}
du =1
2. Fh h i i, - i i.
-
12.1. . 1612 h 0 i' ii Fh i - i i, i a ( ii i i)..ii ii {Fn} i- i F n i -i U , i u U
R1
u(x)Fn(dx)
R1
u(x)F (dx) n .i C(; +) - i R1; C0[; +] i, lim
x+u(x) = 0, lim
xu(x) = 0. 12.1.1. i ii ( )ii ii ii {Fn} i
F ii {Fn} F i C0[; +] i .I ii ii ii {Fn} i F ii {Fn} F i- C(; +). 12.1.4. {Fn} ii i-ii ii i i
fn(x) =n2
exp
{
x2n2
2
}
, n = 1, 2, . . .limn
R1
eitxFn(dx). ' . ii ii {Fn} i- i i F , i 0 (i 2 ii -ii). 12.1.1 ii {Fn} - i F i C(; +). I i
-
162 12. ii iieitx C(; +),
limn
R1
eitxFn(dx) =
R1
eitxF (dx) = eit0F ({0}) = 1. 12.1.5. F iii i i m i i 2. , a > 0F{x : |xm| a}
2
a2. ' .
2 =
R1
(xm)2F (dx)
x:|xm|a
(xm)2F (dx)
x:|xm|a
a2F (dx) = a2
x:|xm|a
F (dx) = a2F{x : |xm| a}. 12.1.6. {Fn} ii - ii ii i an, i i a, i i 2n, i i 0., ii {Fn} i - i i, i a. ' . , ii{Fn(x)} i i i i Fa(x)i i, i a,
Fa(x) ={
0, x a;1, x > a, i x 6= a. t > 0, x = a2t. i an a, n , n
Fn(x) = Fn(a2t) = Fn((, a2t)) Fn((, ant)) =
= Fn{y : y < an t} Fn{y : |y an| t} 2nt2, Fn(a 2t) 0, n ( - 12.1.5).i i, Fn(a+ 2t) 1,
n .
-
12.2. i 16312.2 i: 12.1, 12.3, 12.5, 12.7(2, 5), 12.9, 12.11, 12.12, 12.15 .: 12.2, 12.4, 12.6, 12.7(1, 3, 4), 12.10, 12.16 .12.1. {Na;2n} ii -ii ii i a i 2n. -, 2n i , {Na;2n} i i i, -i a.12.2. Fa i i, - i a. i ii iiii {Fn} 1 n +; 2 n .12.3. F (x) ii. i ii ii -ii {Fn}, i iFn(x) =
0, x 1/n;F (x) F (1/n)F (1/n) F (1/n) , 1/n < x 1/n;
1, x > 1/n.12.4. {Fn} ii ii i -i iFn(x) =
{0, x 1/n;
n(x+ 1/n)/2, 1/n < x 1/n;1, x > 1/n.', i ii ii {Fn}.12.5. {Fn} ii ii i i-
fn(x) =
{
n/2, x [1/n; 1/n];0, x 6 [1/n; 1/n].', i ii ii {Fn}.
-
164 12. ii ii12.6. {Fn} ii ii ii- i ifn(x) =
0, x 1/n;n2(x+ 1/n), 1/n < x 0;n2(x 1/n), 0 < x 1/n;
0, x > 1/n.', i {Fn}, n .12.7. F (x) i i ii i.i ii i i- ii, i -i:1) Fn(x) = F (x+ 1/n), n = 1, 2, . . . ;2) Gn(x) = F (x+ n), n = 1, 2, . . . ;3) Sn(x) = F (x n), n = 1, 2, . . . ;4) Pn(x) = F (x/n), n = 1, 2, . . . ;5) Qn(x) = F (x+ (1)nn), n = 1, 2, . . .12.8. ', i ii i-i {Pn} ii i ipn(x) =
n2
exp
{
(x 1)2n2
2
}
, n = 1, 2, . . .12.9. ', i ii i-i:1) Fn :
(n n1/2 1/2
)
, n = 1, 2, . . . ;
2) Gn :
(
1/n 1/n1/2 1/2
)
, n = 1, 2, . . .12.10. {Fn} ii ii i-i i ifn(x) =
{
n/2, x [(1)n 1/n; (1)n + 1/n];0, x 6 [(1)n 1/n; (1)n + 1/n].', i {Fn}, n .
-
12.2. i 16512.11. Nx;2(y) =
12
y
exp
{
(t x)2
22
}
dt; x, y R1, > 0.lim0
Nx;2(y).12.12. {Fn} ii iii -ii i ifn(x) =
n2
exp
{
(x 1)2n2
2
}
, n = 1, 2, . . .limn
R1
sinxFn(dx).12.13. {Fn} ii iii -ii i i i i 12.12.limn
R1
cos xFn(dx).12.14. F iii i,Nx;2(y) =
12
y
exp
{
(t x)2
22
}
dt; x, y R1, > 0.lim0
+
Nx;2(y)F (dx), y i F .
-
166 12. ii ii12.15. F(y) =
k:0k 0;
F(y) = 0, y 0, > 0.lim0
F(y).12.16. F(y) =
k:0k 0;
F(y) = 0, y 0, > 0.lim0
+
ey2/2F(dy).12.17. {Fn} ii ii i i-
fn(x) =
{
n/2, x [(1)n/n 1/n; (1)n/n + 1/n];0, x 6 [(1)n/n 1/n; (1)n/n + 1/n]ii.', i ii {Fn}
n .12.18. ii {Fn} iii -ii i i F i i i n Fn({k}) F ({k}) k. , Fn F .
-
13i13.1 , i,. , F i. i - (i F ) i (t), i t R1 ii(t) =Meit =
R1
eitxF (dx). i F (i ) ii f , (t) =Meit =
R1
eitxf(x)dx; i F , F : xk F ({xk}) > 0, k = 1, 2, . . . ;
xk
F ({xk}) = 1,167
-
168 13. i(t) =Meit =
xk
exp{itxk}F ({xk}).i i i i i.i i - i. i - i i - i i. i - i ii i-i i i i i. ( ii - i). n- i- F i, i n- i - i(t) =
+
eitxF (dx)i F , i ii i:(n)(t) = in
+
eitxxnF (dx). i . (n)(0) = in
+
xnF (dx) = inMni (t) i (t) = 1 +
t
1!(1)(0) +
t2
2!(2)(0) + . . . +
tn
n