" ". - . / / , , . . : ; - . , : 1 ;2 ;3 ( ); ( ) . , . W . -: :1, 1, 2, ...., ;2 ( ) x1,x2 ....., . , - , . , 1, 2, ...., , - , . x1,x2 ....., , . . - , x1,x2 ..... , . (x1, x2.....) , , , . :3, 1,2, ...., ;4 ( ) x1,x2 ....., ;5 1, Y2, .... W :W=W(1, 2, ...., 1, Y2, ....:x1,x2 .....). 1,Y2,.... , W x1, x2....., . , 1, Y2, .... . 1, 2, .... 1,Y2,....: x1,x2 ....., .1,Y2,....: , ., , 1, Y2, ....: , , , : - ; - ., , W1, W2,.., Wh , . , . , , , :k mmW WW WU........11+. , :U= 1W1 + 2W2 + + +kWk, 1, 2, ., k, . W1, , :. .....; .....; .....;1 1 2 2 k k m m m mw W w W w W w W + + . ( , , ):R1, R2, , Rm b1, b2, , bm. 1, 2, ,n . () j aij Ri (i = 1, 2, , m, j = 1, 2, , n ). j cj . (j = 1, 2, , n ). . x1, x2.....xn 1,2,, n, . :1 1 2 21 1 11,..... b x a x a x an m + + +2 2 2 22 1 21,..... b x a x a x an m + + +..............................................m n mn m mb x a x a x a + + + ,.....2 2 1 1xj 0(j= 1 ,, n). x=(x1,x2 .....xn ), W(a11, . amn ,b1, .., bm,c1,, cn, x1, x2 .....xn ).W=c1 x1+ c2 x2 + + cnxn 1 1 2 12 1 11,..... b x a x a x an n + + +2 2 2 22 1 21,..... b x a x a x an n + + + (1)..............................................m n mn m mb x a x a x a + + + ,.....2 2 1 1xj 0(j= 1 ,, n). (1) , (1):b1y1 +b2y2 +.+ bmym a11y1 + a12y2 + .+ a1nym c1a21y1 + a22y2 + .+ a2nym c2(2)...............................................am1y1 + am2y2 + ..+ amnymcnyi 0(i= 1 ,, m). y1, y2, ., ym , . , c1 x1+ c2 x2 + + cnxn b1y1 +b2y2 +.+ bmym (3) x1,x2 .....xn y1, y2, ., ym , . x*1,x*2 .....x*n y*1, y*2, ., y*m (1) (2), c1 x*1+ c2 x*2 + + cnx*n = b1y*1 +b2y*2 +.+ bmy*m (4) , . (1), (2), (3) (4) , (c1 x*1 + c2 x*2 + + cnx*n ), y1, y2, ., ym, (3) (4) . , (1) (2) . aij, bi cj (i= 1, 2, , m, j = 1, 2, , n ) xj yi (i= 1, 2, , m, j = 1, 2, , n) (1) (2) , " " 1949 . (1) (2). . m :C1, C2, ,Cm n 1, 2, ,n C1,C2,, Cm 1,2,, n. C1,C2,, Cm a1, a2, , am1,2,, n b1, b2, , bn : miinjja b1 1 C1, C2, ,Cm 1, 2, , n . Ci j cij (i= 1, 2, , m, j = 1, 2, , n ). ij , i j (i=1,2, ,m, j =1, 2,, n), {ij}, .. W = minjij ijx c1 1.. ( ) : minjij ijx c1 1 , jmiijb x 1 j =1, 2, , ninjija x 1 i =1, 2, ., mxij 0 i= 1, 2, , m, j = 1, 2, , n .xij , . cij, bj ai , cij j i. ai bj , xij 1. () , () (). , . , . , , t., , .. () , . . - . 1,2,, k , m . K0, . . . :. , 1, 2,, k ? . . i1,2,, k :,) 1 (iX ,) 2 (iX ,,) (kiX , i- :U = (,) 1 (iX ,) 2 (iX ,) (kiX). :U = (U1, U2, Um). , .U W, . ( ) m :W=W(U)=W(U1, U2, Um) : U1, U2, Um , W(U) ?, W(U) , u = (u1, u2, , um). , miiw W1, wi i- . :S0- ,Sw .Wi(S) , - , i- ; . , , S.ui(S) . : S , - , max .: Wi(S) ui(S) i = 1,2, ., m . i-. (i-1) S ui i- . Wi=wi(S, ui) . , , . , (i+1) . Ui i- , S S S=i (S, Ui) (i )., , i- . Ui, . ) , (~i iU S W ) ' ( ) , ( ) , (1~S W U S w U S Wi i i i i ++ (1))) , ( ( ) , ( ) , (1~i i i i i iU S W U S w U S W ++ . Ui=ui(S), (1) , .. (2) ) , ( max )) ( , ( ) (~ ~i iUi i iU S W S u S W S Wi( ) ( ) ( ) { }i i i iUU S W U S wi, , max1 ++ Ui=ui(S), max, . (2) wi(S,Ui) i(S,Ui) . Wi(S) Wi+1(S); . (2) . Wi(S), Wi+1(S). Wm(S) ( ), . ( S). , .. (3) ) , ( max ) (m mUmU S w S Wm( wm(S,U) ). Um, Sw. Um = um(S), (3) e . . . , Wm(S) Wm-1(S) um-1(S) (2). Wm-2(S) um-2(S) .. W1(S) u1(S). W1(S) ( Wi(S) i =1,2,,m), S. . .u = (u1, u2, , um) . S0 . S S0.Wmax = W1(S0)u1 = u1(S0). S0 u1 *1S ) , (1 0 1*1u S S . *1S . u2 = u2(*1S) ) , (2*1 2*2u S S .. S0 u1(S0) *1S u2(*1S) *1 mS um(*1 mS) *mSw mS S * S0, .. ~0S, ) ( max ) (1*0 1 max~0S W S W Ws s *0S .- . :1.() .. , .. 2. i- S Ui Wi = wi(S, Ui).3. i- S S) , ('i iU S S .4. (4) ( ) ( ) ( ) { }i i i iUiU S W U S w S Wi, , max ) (1 ++ 5. Wm(S) ( )( ) { }m mUmU S w S Wm, max ) ( . um(S).6. Wm(S) (4) wi(S,Ui) ) , (i iU S Wm-1(S), Wm-2(S), ... , W1(S), um-1(S), um-2(S), ... , u1(S).7. S0 . ) (0 1 maxS W W . , :S0 u1(S0) *1S u2(*1S) *1 mS um(*1 mS) *mS K0 ( ), m I II. , . . I , f(X), . : (X) < X. Y II g(Y) : (Y) < Y. , K0 I II. ; , . , m- . .1. , I II ... Ui = (Xi, Yi)U = (U1, U2,, Um)2. i- K , (i-1) . Ui i- Xi I .; Yi Yi = K - Xi. i- wi(K, Xi) = f(Xi) + g(K Xi).3. Ui K = (Xi) + (K - Xi). 4. ( ) ( ) { }i i i i iK XiX K X W X K g X f K Wi + + + + ) ( ) ( ) ( max ) (10 . Xi(K), .. .5. . . : { } ) ( ) ( max ) (0m mK XmmX K g X f K W + Xm(K) . .6. ) (K Wm ( ) ( ) { }1 1 1 101) ( ) ( ) ( max ) (1 + + + m m m m mK XmX K X W X K g X f K Wm .. Xm-1(K),Xm-2(K), ..., X1(K).7. K0 ) (0 1 maxK W W . 1- ( )1 1*1) ( X K X K + .. 2-( )*1 2 2K x X .. i-( )i i iX K X K + ) (* . ( )*1 i i iK x X .. K0 x1(K0) *1K x2(*1K) *1 mK xm(*1 mK) *mK*mK . I X= {X1, X2, ,Xm}, Y ={ K0 - X1, *1K - X2, ,*1 mK - Xm}. H0V0, Hw, Vw( w ). , H H (H >H) V. , V V (V > V) . , . , () () . S VOH , , S () S0 Sw. , . . Vw V0 n1 10n V VVw Hw H0 n2 20nH HHw . n1=5, n2=4 S0 Sw. . , m = n1+n2. . S S0 Sw. m ()., ( ). , :11108791017 1413 10 11 13 1712 13 13 14 15 14 14141320 18 1615 14 1215 1714813912101011912813111215121261381271081110131115102091078889891010121113815711891081171091113121410189129111010129131314141417122010V0H0 S0HwVwSw n1=8,n2=6 m = n1+n2 = 14. S0 Sw 14. , . , W :W = 12 + 11 + 10 + 8 + 11 + 8 + 10 + 10 + 13 + 15 + 20 + 9 + 12 + 14 = 163 . ! 14 14 Sw. Sw. B1 B2 . SwB1B2171714 13 B1, (.. ) 17 , B2 (.. )14. B1 B2.SwB1B2171413171417 B1 : B1, Sw17. 14B2. , , . , 14 (B1B2)13. 13. -12 . C1,C2, C3. .SwB1B2171714 13C1C2C31215 1, Sw 15 + 17 = 30 . C1. 2 : Sw B1 B2. 13 + 17 = 30 , 17 + 14 = 31 . , 2 Sw ( ), min min(30, 31) = 30. C2.C3Sw 12+14=26. C3. .SwB1B2171714 13C1C2C312151714 303226c ( ) , ... , Sw min . - . min , , . , -( C2 ). , , minSw(min ). S0. () , . - . S0 Sw. . . 139 S0 min . - . . , . :1 ;2 6 ;3 . , .: 6 . - , .: 4 .. , , , A. . , A . A , .. () (B). , , , ...., L. , . , . : 1, 2, 3, 4, 56, . eA1,A2, ,An :1. .. Ai,Ak , . 2.A1,A2, ,An ( ).3. A1, A2, , An . , -, . , Ai , . , nA1,A2, ,An, m . , P(A), ..nmA P ) (. - . A1, A2, , A6 61) ( ) (6 1 A P A P.: , 3 ( 3) 6 (. 6). 3162) ( A P. . . .. . P(A B) = P(A) + P(B). , , , , ...., LP( .... L) = P(A) + P(B)+ ..+ P(C). B , B, , PA(B).. , , , . P(A B) = P(A).P(B). .. P(A B) = P(A).P(B) = P().P().. , ( , ), P(A B) = P(A).P(B).( . B). :P( .... L) = P(A).P(B)..P(L). . .H1,H2, ,Hn , Hi, ) ( A PiH. : ?, ) ( ) ( ) (11 1A P H P A H PH) ( ) ( ) (22 2A P H P A H PH...........................) ( ) ( ) ( A P H P A H PnH n n, ..H1,H2, ,Hn :) ( ) ( ...... ) ( ) ( ) ( ) ( ) (2 12 1A P H P A P H P A P H P A PnH n H H+ + + . . . : , . 1, 2, 3, 4, 5 6. , . , . . (- ). : . - .. Xx1,x2, ., X=xi. pi=P(X=xi) , X xi.xipi . . X M(X) + + + nii i n np x p x p x p x X M12 2 1 1.... ) (. X :( ) ( )2) ( ) ( X M X M X D . .1. .2. S , .3. ; ; () .. S , : t0, ( t>t0) (t= t0) . SS1,S2, , Sn. () ( S) ( ).:S , : S1, ; S2 , ; S3- ; S4 ; S5 .S1S2S3S4S5 S. . . : , t1, t2, . . : t. ) ( KiS , , K () S Si. K ) ( ) (2) (1....,, ,KnK KS S S ., , . : ,....... , , , ,) 4 (3) 3 (2) 2 (1) 1 (2) 0 (1S S S S S . . k S S1, ....,Sn . ) ( ) (2) (1.......,, ,KnK KS S S. : ( ) ( ) ( )) 1 (n) 1 (2 2) 1 (1 1(1) p .......,, (1) p , ) 1 (nS P S P S P p .. ; ( ) ( ) ( )) (n) (2 2) (1 1(k) p .......,, (k) p , ) (KnK KS P S P S P k p - .nik p11 ) ( k... , .(k) p .......,, (k) p ), (n 2 1k p . , . . , . . Pij SiSj . Pii Si(.. ). nn n nnnijP P PP P PP P PP....... ... .... ..........2 12 22 211 12 11 : ijP, (k) p .......,, (k) p ), (n 2 1k p k. ,( )) 1 ( ) (/kikj ijS S P P.. ) (kjS, ) 1 ( kiS. (|0|) S Sm .. 0 (0) p .......,, 0 (0) p , 0 ) 0 (n 2 1 p. S1, S2, , Sn Pm1, Pm2, .,Pmn. p1(1), ., pn(1). , p1(1)= Pm1, p2(1)= Pm2, ., pn(1)= Pmn. p1(2), .,pn(2) :p1(2) = p1(1)P11 + p2(1)P21 + + pn(1)Pn1p2(2) = p1(1)P12 + p2(1)P22 + + pn(1)Pn2pn(2) = p1(1)P1n + p2(1)P2n + + pn(1)Pnn njij j iP k p k p1) 1 ( ) (.: t1, t2, t3 t4. . . pi(t) t Si (i=1, 2, , n).:nit p11 ) (.Pij ij. ( )t t Pj itij 0lim ( ) t Pij - S t Si t Sj. : ( ) t t Pij ij . ij - , . :S1S2S3S412 31234224 34 4 : pi(t) ( ). S .S1S2S3S4S5123213345345 :( ) ) () (1 13 121t pdtt dp + ) ( ) () (3 32 1 122t p t pdt t dp + ( ) ( ) t p t p t pdtt dp5 53 1 13 3 34 323) ( ) () ( + + + ) ( ) () (3 34 4 454t p t pdt t dp + ) ( ) () (4 45 5 535t p t pdtt dp + . t=0 p1=1p2=p3=p4=p5=0 ( ).. , . , . , . , , . SS1,S, .,Sn, . , . constij , p1(t)p2(t),,pn(t), t 1 ) (1niit p S t. p1 (t) p2(t),,pn(t) ? , . : S , . ? , . (p1, p2, , pn), i itp t p ) ( lim. :( )1 34 120 p + 3 32 1 120 p p + ( )5 54 1 13 1 34 320 p p p + + + 3 34 4 450 p p + 4 45 5 530 p p + () S, . () , :S1S2SkSn-1Sn............: , ; . : S0;S1 , 2 ; S2 2 , ; S3 . S0S1S2S3 ()ij .S1S2SkSn-1Sn............12212332k-1kkk-1kk+1k+1kn-1nnn-1n-2n-1n-1n-2 :2 21 1 12p p 3 32 2 23p p ..k kk k k kp p1 1 1 ..n nn n n np p1 1 1 1 ) (1niit p. :121122p p121 3212 23232233p p p 121 2 1 112 1 2 1111......p p pk k kkk k k kkkkk kk . , 1.21 2 1 112 1 2 121 3212 2321121..........11 + + + +n n nnn n n npS1S2S3231232S4 p1, p2, p3 p4. , :S1S2SkSn...23 k-1k k+1kn-1n12n11 12 2 23p p 2 23 3 34p p ..k kk k k kp p1 1 1 ..1 12 1p pn n 11niip. p1 :123122p p13412134 2323 12234233p p p p .1112p pkkk+1112p pnn ,_+ + + + + 1 1 34 231211 1...1 111n n np . : S1 ,; S2 , ; S3 , ; S4 , . , ; 6 , . .S1S2S3S4448 224 S1S2S4S5S3 ( , )1. 4 t1, t2, t3 t4.(S)S1 ; S2; S3 ; S4 .S1S2S3S4 :1 0 0 07 . 0 3 . 0 0 02 . 0 4 . 0 4 . 0 01 . 0 2 . 0 4 . 0 3 . 0ijp: S1.. p1(0)=1, p2(0)=0, p3(0)=0, p4(0)=0. : S 4 S1, S2, S3 , S4., .S1S2S3S4221231 p1, p2, p3 p4.: . ) ( ) ( 5) (3 11t p t pdtt dp+ ) ( 2 ) ( 2 ) () (3 1 22t p t p t pdt t dp+ + ) ( 2 ) ( 3 ) ( 3) (4 1 33t p t p t pdtt dp+ + ) ( ) ( 2) (2 44t p t pdtt dp+ :3 15 0 p p + 3 1 22 2 ) ( 0 p p t p + + 4 1 32 3 3 0 p p p + + 2 42 0 p p + :(*) 14 3 2 1 + + + p p p p. :1 4 1 2 1 36 , 12 , 5 p p p p p p . (*) + + + 1 6 5 121 1 1 1p p p p41,245,21,2414 3 2 1 p p p p. . :S1 ; S2, , ; S3 , ; S4 , . . 6 . (0,5 .). 1 . . . S1S2S3S41t, 2t, 3t 4t .211 t( . )4812 t ( )413 t ( )2414 t ( ) 12, 13, 14, 41 1121t , 2231t , 3341t , 4411t 2 23 1 12p p 3 34 2 23p p 4 41 3 34p p 1 12 4 41p p (*) 14 3 2 1 + + + p p p p. :123122p p;134123p p; 141124p p *) p1 11 1 1141 34 2312 1,_+ + + p p p1 = 0.615, p2 =0.026, p3 = 0,308, p4 = 0,0051 (). ; ; ; .. , - . . (), . ( ), . , (.. ) . . , , , , - . : , ,, (, ). 1 , 1- ;2 , ;3 ;4 , . . :1 . .2 . , . , .2 , . , . :n ; - ;- ( , ). , . n=1 ().:3 . ;4 . q. S, :S0- ;S1 -.S1S0 :1 00p pdtdp + 0 11p pdtdp + p0(t) p1(t) S0 , S1 , p0(t) + p1(t) = 1 t. , 0p q + 0p q 1 00 p p + 0 10 p p + p0(t) + p1(t) = 1 q A .. + A. Pq P 1 (.. P= p1) , . n- . S , :S0 ;S1 , .;..... ... ............ ...Sk k , . ;..... ... ............ ...Sn . S S1SkSn............ S0S22 3 k (k+1) n , . 1 00p pdtdp + 2 0 112 ) ( p p pdtdp + + + .1 1) 1 ( ) (+ + + + + k k kkp k p p kdtdp .1 + n nnp p ndtdp . p0, ,pn, 0 :p0 +p1++ pn =10!) / (pkpkk k= 1,,n ! ! 2 ! 11120npn,_+ + +,_+ +P= pn ;, q=1-pn; ) 1 (np q A n=1, , Mm. S S0 ;S1 , ;S2 , ;..... ... ............ ...Sk , k-1 ;..... ... ............ ...Sm+1 , m .S1SkSm+1............S0S2 , ( ) 0 1p p022p p,_.0p pkk,_.011p pmm++,_1 20... 11+,_+ + ,_+ +mpP= pm+1 ;, q=1-pm+1; q A n, , m . S :S0 ;S1 , . ;..... ... ............ ...Sk k ,. ;..... ... ............ ...Sn ;Sn+1 , ;..... ... ............ ...Sn+r , r ;..... ... ............ ...Sn+m , m ; S S1SkSn............S02 k(k+1) nSn+1n......nSn+rn......nSn+mn 0!) / (pkpkk k= 1,,n 0!) / (11pn npnn++ , 0!) / (222pn npnn++ ,.., 0!) / (pn npmm nm n++ ! ! ! ! ! 2 ! 11122 1 20n n n n n n npmm n n n n + + +,_+ +,_+,_+,_+ + +,_+ + P= pm+n= ! n nmm n+,_, q=1-pm+n; q A ()1. : .- , . 8 , 0 ( ) 5 , 1 t 1) . q;2) A;3) P.: S1S0S0 ; S1 .8 , 0 ,667 , 01 t ( ) 445 , 0 + q364 , 0 445 , 0 . 8 , 0 q A P= 1-q = 0,5452. ( , n =3) S1S0S22 3S38 , 0 , 667 , 01 t,2 , 1 0!) / (pkpkk k= 1, 2, 3p0=0,312; p1=0,374; 2=0,224; p3=0.090q=1-p3=0,910 728 , 0 q A P= p3 = 0,093. . , . 1 ( ). 1,25 . :5 P;6 , ;7 ;8 ( ).. S1S0S22 3S48 , 025 , 11 , 25 , 18 , 01 2011+mp,0p pkk , P = p4p0 = 0,122; p1 = 0,152; 2 = 0,191; p3 = 0,238; p3 = 0,297q =1 - P = 0,703 703 , 0 q A 4. . 2 , . 2 (.. 2 ). t 2. ..S1S3S4S0S22S52 2 2 n=2; m=3, 2 ; 5 , 01 t; p1=0! 1p;pn=0! pnn; pn+1=0!pnnn; pn+m=0pnmm n+; 4 p0 = 0,008; P = 0,512q=1- P = 0,488976 , 0 q A ( ) , , . .. ., , . : , , . , . .. Ai,Bj , . , , . , . , (.. ). : - ( ), , ? . A1Am 1.n . SA = (p1,,pm) A, A1Am p1,,pm miip11. , SA = (q1,,qn)njjq11. , . , *AS*BS, v . mxn mxn, m A1Am n 1.n . {aij}. *AS= (p1,,pm) *BS = (q1,,qn). *AS. - v B v, *BS. *AS, B Bj , mj m j ja p a p a .....1 1+ (j=1,..,n) aj v .*AS .. v a p amiij i 1 j = 1,,n vpxii i =1,,m 1 ) 1 (1mij ijx a i =1,,m. miip11, vxmii1) 2 (1. (v), min ) 3 (1miix (1) .. 1 ) 4 (1mi ijx a, j = 1,,n, 0 ix. (3) (4) .B(v (.. ), njj ijnjjy ay111max) 5 (, i = 1,..,n, 0 jyvqyjj, j =1, ,, n (3) (4) (5) . (3) (4) .. .ija x max min1 , x2=x3=xm=0., (5) . . B1B2B3A12 -3 4A2-3 4 -5A34 -5 6 5 ( )B1B2B3A17 -2 9A22 9 0A39 0 11 *AS , ..v = v + 5 v ., 2011 x, 1012 x, 2013 x.. 51ix, .. v = 5 .. v= v -5=0 41'1 1 v x p 21'2 2 v x p 41'3 3 v x p.. *AS = { }41,21,41. B*BS = { }41,21,41. 2x2-2x2. B1B2A1a11a12A2a21a22 *AS= {p1,p2} v p a p av p a p a + +2 22 1 122 21 1 1112 1 + p p , 21 12 22 1121 221a a a aa ap +21 12 22 1112 112a a a aa ap +21 12 22 1121 12 11 22a a a aa a a av + *BS = {q1, q2} 21 12 22 1112 221a a a aa aq +q2 =1 - q1.B1B2A1-1 1A21 -1 ( ). 211 p, 212 p,v = 0, 211 q, 212 q, 2x2. . 2x2 .B1B2A1a11a12A2a21a22 .A1 SAA2NB1La21a22a12a11xyB2K0B2B1p1p2 . (x=0) A1, (x=1) A2. 0 1 . } , {2 1p p SA p1 p2 . 1.A1A22, a11 a21 A2( , e B2). a12a22 A1 A2( ,B B2). N B1B1 B2B2 v *AS A. B { }2 1*, q q SB 1 221kB kBkBq+, 1 222LB LBLBq+ 2xn mx2 - 2xnmx2.2xn. A1 A2 A, B1B2, B3B4,..,Bn. B1B1,B2B2, ..BnBn.- v *AS ( 2x2). , , (), , . : , .. B . A A1A2..Am . , n 1....n, . {aij}, j aij , , -. ? , (Ai) (j) , .(rij)(ij) , , j , , i. ij j ija r , ijjja max 0 ijr. : ( )1,..,n, , Q1=P(1), ..., Qn=P(n)njjQ11. :njij j ia Q a1.ia() Ai0. Ai0 , in iia a,..., 10max. () , ,.. njij j ir Q r1 i =1, .. ,n iiir r min0 . , , .. W a aj i ijj i 0 0min max..Ai0 W . ,- - .. 0 0max minj i ijj ir r R (. Ai0 , j0 - ) 2x2 2x2 BjAiB1B2A1a11a12A2a21a22 , , .. ij ija a min max max min ( ), . . . - . A :( )2 1,*p p SA ( )2 1,*q q SB *AS - v v p a p av p a p a + +2 22 1 122 21 1 1112 1 + p p , 21 12 22 1121 221a a a aa ap +21 12 22 1112 112a a a aa ap +21 12 22 1121 12 11 22a a a aa a a av + ( )2 1,*q q SBv q a q av q a q a + +2 22 1 122 21 1 1112 1 + q q.. 21 12 22 1121 221a a a aa aq +1 21 q q .: BjAiB1B2A1-1 1A21 -1 1 1 . . - : ,21,21, 0 ,21,212 1 2 1 q q v p p( )21,21*AS ( )21,21*BS 2: . III; I, II. B. . II , II, 0,3. I, . 1-(1-0,3)2=0,51 , (I IIs)0,8. ( ) , . ; . . ? ?. .1. 1, 1 I; I-. , - - , . a11 = 0,51 + (1-0,51)(1-0,8) = 0,6080,51 (1-0,51) - (1-0,8) I2. 2, 1 II, I; a21 = 1 ?3. 1, 2 - I, II;a12 = 1 2.4. 2, 2 - II, II;a22= 0,3 + (1-0,3)(1-0,8) = 0,440,3 ; (1-0,3) II; (1-0,8) II; BjAiB1B2iA10.608 1 0.608A21 0.44 0.44j1 1j i min max < ! p1 = 0.558 p2 = 0.412v =0.768 q1 = 0.588q2 = 0.412 . ) 0,608.1 + 1.2 = v ) 1.1 + 0,44.2 = v) : (1, 2), 1;) : (1, 2), 2. A1 SAA2NB1La21a22 a12a11xyB2K0B2B1p1p20.412 0.588= 0.768 , *AS(1, 2) , . 8BS- . . !!! 2xn mx2 2xn. B1B2B3 Bn1a11a12a13 a1n2a21a22a23 a2n n : (1, 2), B1, B2, ., Bn.SANB1xyB20B2B1p1p2*BnB3B3Bn. () , ( ) . 2 ; . . . : I,II III, .IIIIII . . . , . - . . :1 2 ;2 . :1 (1 + 1 + 1) - ;2(2 +1+0 ) , , ;3 (3 + 0 + 0) , .. , .. 1 .1. 11 a11= ?2. 21 a21= 1 3. 12 a12= 2/3 (2/3.1/2=1/3)4. 22 a22= 2/3 (2/3.1/2=1/3) (1-1/3=2/3)5. 13 a13= 15. 23 a23= 2/3 123iA10 2/3 1 0A21 2/3 2/3 2/3j1 2/3 1maxI =minj =2/3. 2; 2. B1xyB20B2B1p1p2B3B31112/3SA* (Portfolio)I. =(C, Q, S) (policy), C, Q S S 0 "" (liability ), (), ( , , ).Q 0 (risk premium); ( ) ( ), 0 . ( ) (claim size). . - , S Q , F(v)=Fc(v)=P{cv}, Fc(0)=0.= Q/S (relative premium)U= C/S (increase portfolio) (collection) i { } ( ) S Q CNi i, ,1 i i = 1, 2,, N NiiC C1 ;NiiQ Q1 ;NiiS S1 () . C ( ) :) ( ) ( ) ( v c P v F v F C . (uncovarage risk) R()=) ( Q c P . N S, . p(0, 1). S. 0 S 1- p K N P. . = Np, DK = Np(1-p). { }k N k kNp p C kS C P ) 1 ( = 1, , N 2) 1 ( S p Np C D S Np C E : . : i Si Ci a [0, Si]. Ci = riSi (i= 1, , N), ri [0,1]. . , . , . , . . , .. , . , .. . : 1. , , , .. ( ) y y p ' ( ) 0 y y p > ( )2. , , , , .. ( ) x xp '( ) 0 x p , ( ) ( ) y p xp >, .. . - , . . . * p p