А.Г. Дьячков - Вероятность, основные определения и примеры, страница 6

óÕÍÍÁ + ÎÅÚÁ×ÉÓÉÍÙÈ ÎÏÒÍÁÌØÎÙÈ ÓÌÕÞÁÊÎÙÈ ×ÅÌÉÞÉÎ ∼ N (a1 ; 1 ) É ∼ N (a2 ; 2 ) Ñ×ÌÑÅÔÓÑ ÎÏÒÍÁÌØÎÏÊ ÓÌÕÞÁÊÎÏÊ ×ÅÌÉÞÉÎÏÊ:µ + ∼ N (a1 ; 1 ) + N (a2 ; 2 ) ∼ N a1 + a2 ;q12 + 22¶;ÍÁÔÅÍÁÔÉÞÅÓËÏÅ ÏÖÉÄÁÎÉÅ É ÄÉÓÐÅÒÓÉÑ ËÏÔÏÒÏÊ ×ÙÞÉÓÌÑÀÔÓÑ ÐÏ ÆÏÒÍÕÌÁÍM( + ) = a1 + a2 = M + M; D( + ) = 12 + 22 = D + D:§ 11.3.óÏ×ÍÅÓÔÎÏÅ ÒÁÓÐÒÅÄÅÌÅÎÉÅ n ÓÌÕÞÁÊÎÙÈ ×ÅÌÉÞÉÎ (1 ; 2 ; : : : ; n )óÏ×ÍÅÓÔÎÏÅ ÒÁÓÐÒÅÄÅÌÅÎÉÅ n ÓÌÕÞÁÊÎÙÈ ×ÅÌÉÞÉÎ (1 ; 2 ; : : : ; n ) ÚÁÄÁÅÔÓÑ ×ÅÒÏÑÔÎÏÓÔÎÏÊÆÕÎËÃÉÅÊ ÏÔ n ÐÅÒÅÍÅÎÎÙÈ.•÷ ÓÌÕÞÁÅ ÄÉÓËÒÅÔÎÏÊ ÍÏÄÅÌÉ ÜÔÁ ÆÕÎËÃÉÑ q = q(x1 ; x2 ; : : : ; xn ), ÎÁÚÙ×ÁÅÍÁÑ ÓÏ×ÍÅÓÔÎÙÍ ÒÁÓÐÒÅÄÅÌÅÎÉÅÍ ×ÅÒÏÑÔÎÏÓÔÅÊ, ÕÄÏ×ÌÅÔ×ÏÒÑÅÔ ÕÓÌÏ×ÉÑÍn XXq(x1 ; x2 ; : : : ; xn ) ≥ 0;i=1 xiq(x1 ; x2 ; : : : ; xn ) = 1É ÏÐÒÅÄÅÌÑÅÔ ×ÅÒÏÑÔÎÏÓÔÉ ÓÏÂÙÔÉÊPr{1 = x1 ; 2 = x2 ; : : : ; n = xn } def= q(x1 ; x2 ; : : : ; xn ):27á.ç. äØÑÞËÏ×: ÷ÅÒÏÑÔÎÏÓÔØ, ÏÓÎÏ×ÎÙÅ ÏÐÒÅÄÅÌÅÎÉÑ É ÐÒÉÍÅÒÙ•÷ ÓÌÕÞÁÅ ÎÅÐÒÅÒÙ×ÎÏÊ ÍÏÄÅÌÉ ÜÔÁ ÆÕÎËÃÉÑ p = p(x1 ; x2 ; : : : ; xn ), ÎÁÚÙ×ÁÅÍÁÑ ÐÌÏÔÎÏÓÔØÀ ÓÏ×ÍÅÓÔÎÏÇÏ ÒÁÓÐÒÅÄÅÌÅÎÉÑ ×ÅÒÏÑÔÎÏÓÔÅÊ, ÕÄÏ×ÌÅÔ×ÏÒÑÅÔ ÕÓÌÏ×ÉÑÍZ∞ Z∞p(x1 ; x2 ; : : : ; xn ) ≥ 0;:::−∞ −∞Z∞p(x1 ; x2 ; : : : ; xn ) dx1 dx2 · · · dxn = 1−∞É ÏÐÒÅÄÅÌÑÅÔ ×ÅÒÏÑÔÎÏÓÔÉ ÓÏÂÙÔÉÊPr{a1 ≤ 1 ≤ b1 ; a2 ≤ 2 ≤ b2 ; : : : ; an ≤ n ≤ bn } def=def=Zb1 Zb2a1 a2:::Zbnanp(x1 ; x2 ; : : : ; xn ) dx1 dx2 · · · dxn :ðÒÉ ÏÐÒÅÄÅÌÅÎÉÉ ÎÅÚÁ×ÉÓÉÍÏÓÔÉ n ÓÌÕÞÁÊÎÙÈ ×ÅÌÉÞÉÎ (1 ; 2 ; : : : ; n ) ÍÙ ÏÇÒÁÎÉÞÉÍÓÑ ÎÁÉÂÏÌÅÅ ×ÁÖÎÙÍ ÄÌÑ ÐÒÉÌÏÖÅÎÉÊ ÓÌÕÞÁÅÍ n ÎÅÚÁ×ÉÓÉÍÙÈ ÏÄÉÎÁËÏ×Ï ÒÁÓÐÒÅÄÅÌ£ÎÎÙÈ(ÏÄÎÏÒÏÄÎÙÈ) ÓÌÕÞÁÊÎÙÈ ×ÅÌÉÞÉÎ.

÷ ÍÁÔÅÍÁÔÉÞÅÓËÏÊ ÓÔÁÔÉÓÔÉËÅ ÜÔÁ ÍÏÄÅÌØ ÎÁÂÌÀÄÅÎÉÊÎÁÚÙ×ÁÅÔÓÑ ×ÙÂÏÒËÏÊ ÏÂߣÍÁ n. òÁÓÓÍÏÔÒÉÍ ÐÒÏÉÚ×ÏÌØÎÕÀ ×ÅÒÏÑÔÎÏÓÔÎÕÀ ÆÕÎËÃÉÀ ÁÒÇÕÍÅÎÔÁ x, −∞ < x < ∞, Ô.Å. ÒÁÓÐÒÅÄÅÌÅÎÉÅ ×ÅÒÏÑÔÎÏÓÔÅÊ q(x) (ÄÉÓËÒÅÔÎÁÑ ÍÏÄÅÌØ) ÉÌÉÐÌÏÔÎÏÓÔØ ÒÁÓÐÒÅÄÅÌÅÎÉÑ ×ÅÒÏÑÔÎÏÓÔÅÊ p(x) (ÎÅÐÒÅÒÙ×ÎÁÑ ÍÏÄÅÌØ).ïÐÒÅÄÅÌÅÎÉÅ 3. âÕÄÅÍ ÇÏ×ÏÒÉÔØ, ÞÔÏ ÓÌÕÞÁÊÎÙÅ ×ÅÌÉÞÉÎÙ (1 ; 2 ; : : : ; n ) Ñ×ÌÑÀÔÓÑ ÎÅÚÁ×ÉÓÉÍÙÍÉ ÏÄÉÎÁËÏ×Ï ÒÁÓÐÒÅÄÅÌ£ÎÎÙÍÉ ×ÅÌÉÞÉÎÁÍÉ i ∼ q(x) (ÄÉÓËÒÅÔÎÁÑ ÍÏÄÅÌØ) ÉÌÉi ∼ p(x) (ÎÅÐÒÅÒÙ×ÎÁÑ ÍÏÄÅÌØ), ÅÓÌÉ ÓÏ×ÍÅÓÔÎÁÑ ×ÅÒÏÑÔÎÏÓÔØ q = q(x1 ; x2 ; : : : ; xn ) (ÄÉÓËÒÅÔÎÁÑ ÍÏÄÅÌØ) ÉÌÉ ÓÏ×ÍÅÓÔÎÁÑ ÐÌÏÔÎÏÓÔØ p = p(x1 ; x2 ; : : : ; xn ) (ÎÅÐÒÅÒÙ×ÎÁÑ ÍÏÄÅÌØ) ÚÁÄÁÀÔÓÑ × ×ÉÄÅ ÐÒÏÉÚ×ÅÄÅÎÉÑ n ×ÅÒÏÑÔÎÏÓÔÎÙÈ ÆÕÎËÃÉÊ:defq = q(x1 ; x2 ; : : : ; xn ) = q (x1 ) · q (x2 ) · · · · q (xn );defp = p(x1 ; x2 ; : : : ; xn ) = p(x1 ) · p(x2 ) · · · · p(xn ):éÓÐÏÌØÚÕÑ ÜÔÏ ÏÐÒÅÄÅÌÅÎÉÅ É ÆÏÒÍÕÌÕ (60 ), ÍÏÖÎÏ ÄÏËÁÚÁÔØ ÓÌÅÄÕÀÝÅÅ ÏÂÏÂÝÅÎÉÅ Ó×ÏÊÓÔ×Á 4 Ï ÒÁÓÐÒÅÄÅÌÅÎÉÉ ÓÕÍÍÙ n ÎÅÚÁ×ÉÓÉÍÙÈ ÎÏÒÍÁÌØÎÙÈ ÓÌÕÞÁÊÎÙÈ ×ÅÌÉÞÉÎ.ó×ÏÊÓÔ×Ï 5.

ðÕÓÔØ (1 ; 2 ; : : : ; n ) { ÎÅÚÁ×ÉÓÉÍÙÅ ÏÄÉÎÁËÏ×Ï ÒÁÓÐÒÅÄÅÌ£ÎÎÙÅ ×ÅÌÉÞÉÎÙ,nPÇÄÅ i ∼ N (a; ) ÄÌÑ ÌÀÂÏÇÏ i = 1; 2; : : : ; n. ôÏÇÄÁ ÉÈ ÓÕÍÍÁ Sn = i ÔÁËÖÅ Ñ×ÌÑÅÔÓÑi=1ÎÏÒÍÁÌØÎÏÊ ÓÌÕÞÁÊÎÏÊ ×ÅÌÉÞÉÎÏÊSn =nXi=1√i ∼ N (na; n);ÍÁÔÅÍÁÔÉÞÅÓËÏÅ ÏÖÉÄÁÎÉÅ É ÄÉÓÐÅÒÓÉÑ ËÏÔÏÒÏÊ ×ÙÞÉÓÌÑÀÔÓÑ ÐÏ ÆÏÒÍÕÌÁÍM Sn = M( nXi=1)i =nXi=1Mi = na;D Sn = D28( nXi=1)i =nXi=1Di = n2 :á.ç. äØÑÞËÏ×: ÷ÅÒÏÑÔÎÏÓÔØ, ÏÓÎÏ×ÎÙÅ ÏÐÒÅÄÅÌÅÎÉÑ É ÐÒÉÍÅÒÙ§ 12.þÉÓÌÏ×ÙÅ ÈÁÒÁËÔÅÒÉÓÔÉËÉ ÒÁÓÐÒÅÄÅÌÅÎÉÑ ×ÅÒÏÑÔÎÏÓÔÅʧ 12.1.íÁÔÅÍÁÔÉÞÅÓËÏÅ ÏÖÉÄÁÎÉÅ, ÄÉÓÐÅÒÓÉÑ É ÉÈ Ó×ÏÊÓÔ×Á1.

ðÕÓÔØ { ÐÒÏÉÚ×ÏÌØÎÁÑ ÓÌÕÞÁÊÎÁÑ ×ÅÌÉÞÉÎÁ, ËÏÔÏÒÁÑ ÉÍÅÅÔ ÒÁÓÐÒÅÄÅÌÅÎÉÅ ×ÅÒÏÑÔÎÏÓÔÅÊ,ÚÁÄÁ×ÁÅÍÏÅ ×ÅÒÏÑÔÎÏÓÔÎÏÊ ÆÕÎËÃÉÅÊ q(x) (ÄÉÓËÒÅÔÎÁÑ ÍÏÄÅÌØ) ÉÌÉ ÐÌÏÔÎÏÓÔØÀ ÒÁÓÐÒÅÄÅÌÅÎÉÑ ×ÅÒÏÑÔÎÏÓÔÅÊ p(x) (ÎÅÐÒÅÒÙ×ÎÁÑ ÍÏÄÅÌØ). óÉÍ×ÏÌÏÍ f (x) ÏÂÏÚÎÁÞÉÍ ÐÒÏÉÚ×ÏÌØÎÕÀÆÕÎËÃÉÀ ÄÅÊÓÔ×ÉÔÅÌØÎÏÇÏ ÁÒÇÕÍÅÎÔÁ x, −∞ ≤ x ≤ ∞, Á ÓÉÍ×ÏÌÏÍ c, −∞ ≤ c ≤ ∞, {ÐÒÏÉÚ×ÏÌØÎÕÀ ÐÏÓÔÏÑÎÎÕÀ.òÁÓÓÍÏÔÒÉÍ ÓÌÕÞÁÊÎÕÀ ×ÅÌÉÞÉÎÕ = f ( ), ËÏÔÏÒÕÀ ÍÏÖÎÏ ÎÁÚÙ×ÁÔØ ÆÕÎËÃÉÅÊ ÏÔÓÌÕÞÁÊÎÏÊ ×ÅÌÉÞÉÎÙ . íÁÔÅÍÁÔÉÞÅÓËÏÅ ÏÖÉÄÁÎÉÅ ÆÕÎËÃÉÉ ÏÔ ÓÌÕÞÁÊÎÏÊ ×ÅÌÉÞÉÎÙ ÏÂÏÚÎÁÞÁÅÔÓÑ ÓÉÍ×ÏÌÏÍ Mf ( ) É ÏÐÒÅÄÅÌÑÅÔÓÑ ÆÏÒÍÕÌÏÊPÄÌÑ ÄÉÓËÒÅÔÎÏÊ ÍÏÄÅÌÉ, x f (x)q (x)def∞Mf ( ) =  Rf (x)p(x) dx ÄÌÑ ÎÅÐÒÅÒÙ×ÎÏÊ ÍÏÄÅÌÉ.−∞åÓÌÉ ÄÌÑ ÌÀÂÏÇÏ x, −∞ ≤ x ≤ ∞, ÐÏÌÏÖÉÍ ÆÕÎËÃÉÀ f (x) = Ó, ÔÏ ÏÞÅ×ÉÄÎÏPPÄÌÑ ÄÉÓËÒÅÔÎÏÊ ÍÏÄÅÌÉ, x Ó q (x) = c · x q (x) = cdefMÓ =  R∞R∞c p(x) dx = c ·p(x) dx = c ÄÌÑ ÎÅÐÒÅÒÙ×ÎÏÊ ÍÏÄÅÌÉ.−∞−∞åÓÌÉ ÐÏÌÏÖÉÔØ ÆÕÎËÃÉÀ f (x) = (x − M )2 , ÔÏ ÄÉÓÐÅÒÓÉÑ D ÐÒÅÄÓÔÁ×ÌÑÅÔÓÑ ËÁË ÍÁÔÅÍÁÔÉÞÅÓËÏÅ ÏÖÉÄÁÎÉÅ ÜÔÏÊ ÆÕÎËÃÉÉ ÏÔ ÓÌÕÞÁÊÎÏÊ ×ÅÌÉÞÉÎÙ :D =P22 x (x − M ) q (x) = M( − M )R∞(x − M )2 p(x) dx = M( − M )2−∞ÄÌÑ ÄÉÓËÒÅÔÎÏÊ ÍÏÄÅÌÉ,ÄÌÑ ÎÅÐÒÅÒÙ×ÎÏÊ ÍÏÄÅÌÉ.÷ ÞÁÓÔÎÏÓÔÉ, Dc = M(c − Mc)2 = M(c − c)2 = M 0 = 0.

óÌÅÄÏ×ÁÔÅÌØÎÏ, ÉÍÅÅÔ ÍÅÓÔÏôÅÏÒÅÍÁ 1.1) íÁÔÅÍÁÔÉÞÅÓËÏÅ ÏÖÉÄÁÎÉÅ ÐÏÓÔÏÑÎÎÏÊ ×ÅÌÉÞÉÎÙ ÒÁ×ÎÏ ÜÔÏÊ ÐÏÓÔÏÑÎÎÏÊ ×ÅÌÉÞÉÎÅ, Ô.Å. Mc = c.2) äÉÓÐÅÒÓÉÑ D ≥ 0, ÇÄÅ ÚÎÁË ÒÁ×ÅÎÓÔ×Á, ÅÓÌÉ É ÔÏÌØËÏ ÅÓÌÉ ÓÌÕÞÁÊÎÁÑ ×ÅÌÉÞÉÎÁ = c = M { ÐÏÓÔÏÑÎÎÁÑ ×ÅÌÉÞÉÎÁ.áÎÁÌÏÇÉÞÎÙÍ ÏÂÒÁÚÏÍ ÍÏÖÎÏ ÐÒÏ×ÅÒÉÔØ É ÄÒÕÇÉÅ ×ÁÖÎÙÅ ÆÏÒÍÕÌÙ ÄÌÑ ÍÁÔÅÍÁÔÉÞÅÓËÏÇÏ ÏÖÉÄÁÎÉÑ É ÄÉÓÐÅÒÓÉÉ, ËÏÔÏÒÙÅ ÏÐÉÓÙ×ÁÅÔôÅÏÒÅÍÁ 2.1) M(c · ) = c · M ,M( + c) = M + c, M( − M ) = 0;2) D( + c) = D , D(c · ) = c2 · D , D = M( − M )2 = M( 2 ) − (M )2 ;3) åÓÌÉ M = 0, ÔÏ D = M( 2 ).29á.ç. äØÑÞËÏ×: ÷ÅÒÏÑÔÎÏÓÔØ, ÏÓÎÏ×ÎÙÅ ÏÐÒÅÄÅÌÅÎÉÑ É ÐÒÉÍÅÒÙäÏËÁÚÁÔÅÌØÓÔ×Ï.1)M(c · ) =Z∞M( + c) =Xxc · x q(x) = c ·Z∞(x + c) p(x) dx =−∞Xxx q(x) = c · M;Z∞x p(x) dx +−∞c p(x) dx = M + c;−∞Á ÐÏÓÌÅÄÎÅÅ ÒÁ×ÅÎÓÔ×Ï M( − M ) = 0 Ñ×ÌÑÅÔÓÑ ÞÁÓÔÎÙÍ ÓÌÕÞÁÅÍ ÐÒÅÄÙÄÕÝÅÇÏ, ÅÓÌÉÐÏÌÏÖÉÔØ ÐÏÓÔÏÑÎÎÕÀ c = −M .2)Z∞D( + c) =[(x + c) − M( + c)]2 p(x) dx =−∞D(c · ) =Z∞(x − M )2 p(x) dx = D;−∞XX[c · x − M(c · )]2 q(x) = c2 [x − M( )]2 q(x) = c2 · D:xxäÌÑ ÎÅÐÒÅÒÙ×ÎÏÇÏ ÓÌÕÞÁÑ, ÐÒÉÍÅÎÑÑ ÉÚ×ÅÓÔÎÙÅ Ó×ÏÊÓÔ×Á ÏÐÒÅÄÅÌ£ÎÎÏÇÏ ÉÎÔÅÇÒÁÌÁ,ÐÒÏ×ÅÒÑÅÍD =Z∞−∞=Z∞−∞Z∞(x − M )2 p(x) dx =x2 p(x) dx − 2M[x2 − 2xM + (M )2 ] p(x) dx =−∞Z∞x p(x) dx + (M )2−∞Z∞p(x) dx =−∞= M( 2 ) − 2(M )2 + (M )2 = M( 2 ) − (M )2 :3) åÓÌÉ M = 0, ÔÏ D = M( 2 ) − (M )2 = M( 2 ) − 0 = M( 2 ).ôÅÏÒÅÍÁ 2 ÄÏËÁÚÁÎÁ.âÕÄÅÍ ÇÏ×ÏÒÉÔØ, ÞÔÏ ÓÌÕÞÁÊÎÁÑ ×ÅÌÉÞÉÎÁ , ÉÍÅÀÝÁÑ "ÓÔÁÎÄÁÒÔÎÙÅ" ÐÁÒÁÍÅÔÒÙM = 0 É D = M( 2 ) = 1;Ñ×ÌÑÅÔÓÑ ÎÏÒÍÉÒÏ×ÁÎÎÏÊ.

äÌÑ ÐÒÏÉÚ×ÏÌØÎÏÊ ÓÌÕÞÁÊÎÏÊ ×ÅÌÉÞÉÎÙ Ó ÍÁÔÅÍÁÔÉÞÅÓËÉÍ ÏÖÉÄÁÎÉÅÍ a = M É ÄÉÓÐÅÒÓÉÅÊ 2 = D = M( − a)2 ××ÅÄ£Í ÂÅÚÒÁÚÍÅÒÎÕÀ ×ÅÌÉÞÉÎÕ − M−a ∗ def= √=;DÎÁÚÙ×ÁÅÍÕÀ ÎÏÒÍÉÒÏ×ËÏÊ . éÚ ÔÅÏÒÅÍÙ 2 ×ÙÔÅËÁÅÔóÌÅÄÓÔ×ÉÅ. íÁÔÅÍÁÔÉÞÅÓËÏÅ ÏÖÉÄÁÎÉÅ M ∗ = 0, Á ÄÉÓÐÅÒÓÉÑ D ∗ = M ( ∗ )2 = 1.äÏËÁÚÁÔÅÌØÓÔ×Ï. ðÒÉÍÅÎÑÑ ÓÏÏÔ×ÅÔÓÔ×ÕÀÝÉÅ ÕÔ×ÅÒÖÄÅÎÉÑ ÔÅÏÒÅÍÙ 2, ÐÒÏ×ÅÒÑÅÍD = 1: − M )− M )=M ∗ = M(√= 0;M ( ∗ )2 = D ∗ = D( DDDóÌÅÄÓÔ×ÉÅ ÄÏËÁÚÁÎÏ.30á.ç.

äØÑÞËÏ×: ÷ÅÒÏÑÔÎÏÓÔØ, ÏÓÎÏ×ÎÙÅ ÏÐÒÅÄÅÌÅÎÉÑ É ÐÒÉÍÅÒÙ2. ðÕÓÔØ (; ) { ÐÒÏÉÚ×ÏÌØÎÁÑ ÐÁÒÁ ÓÌÕÞÁÊÎÙÈ ×ÅÌÉÞÉÎ Ó ÓÏ×ÍÅÓÔÎÙÍ ÒÁÓÐÒÅÄÅÌÅÎÉÅÍq(x; y) (ÄÉÓËÒÅÔÎÁÑ ÍÏÄÅÌØ) ÉÌÉ ÐÌÏÔÎÏÓÔØÀ ÓÏ×ÍÅÓÔÎÏÇÏ ÒÁÓÐÒÅÄÅÌÅÎÉÑ p(x; y) (ÎÅÐÒÅÒÙ×ÎÁÑ ÍÏÄÅÌØ). óÉÍ×ÏÌÏÍ f (x; y) ÏÂÏÚÎÁÞÉÍ ÐÒÏÉÚ×ÏÌØÎÕÀ ÆÕÎËÃÉÀ Ä×ÕÈ ÄÅÊÓÔ×ÉÔÅÌØÎÙÈÁÒÇÕÍÅÎÔÏ× x, −∞ ≤ x ≤ ∞ É y, −∞ ≤ y ≤ ∞.òÁÓÓÍÏÔÒÉÍ ÓÌÕÞÁÊÎÕÀ ×ÅÌÉÞÉÎÕ = f (; ), ËÏÔÏÒÕÀ ÍÏÖÎÏ ÎÁÚÙ×ÁÔØ ÆÕÎËÃÉÅÊ ÏÔÐÁÒÙ ÓÌÕÞÁÊÎÙÈ ×ÅÌÉÞÉÎ (; ).

ïÞÅ×ÉÄÎÏ, ÞÔÏ ÒÁÓÐÒÅÄÅÌÅÎÉÅ ×ÅÒÏÑÔÎÏÓÔÅÊ = f (; ) ÏÄÎÏÚÎÁÞÎÏ ×ÙÞÉÓÌÑÅÔÓÑ ÐÏ ÓÏ×ÍÅÓÔÎÏÍÕ ÒÁÓÐÒÅÄÅÌÅÎÉÀ ÐÁÒÙ (; ). íÁÔÅÍÁÔÉÞÅÓËÏÅ ÏÖÉÄÁÎÉÅÆÕÎËÃÉÉ ÏÔ ÐÁÒÙ ÓÌÕÞÁÊÎÙÈ ×ÅÌÉÞÉÎ (; ) ÏÂÏÚÎÁÞÁÅÔÓÑ ÓÉÍ×ÏÌÏÍ Mf (; ) É ÏÐÒÅÄÅÌÑÅÔÓÑÆÏÒÍÕÌÏÊMf (; ) def=P Pf (x; y) q(x; y) x yR∞ R∞f (x; y) p(x; y) dx dy−∞ −∞ÄÌÑ ÄÉÓËÒÅÔÎÏÊ ÍÏÄÅÌÉ,ÄÌÑ ÎÅÐÒÅÒÙ×ÎÏÊ ÍÏÄÅÌÉ.(1)åÓÌÉ × (1) ÐÏÌÏÖÉÔØ f (x; y) = x + y, ÔÏ ÄÌÑ ÄÉÓËÒÅÔÎÏÇÏ (ÁÎÁÌÏÇÉÞÎÏ É ÄÌÑ ÎÅÐÒÅÒÙ×ÎÏÇÏ) ÓÌÕÞÁÑ ÉÍÅÅÍM( + ) ==XxxXyXXxyq(x; y) +(x + y) q(x; y) =XyyXxq(x; y) =XXxXxyx q(x; y) +x q(x; :) +XyXXxyy q(x; y) =y q(:; y) = M + M:óÌÅÄÏ×ÁÔÅÌØÎÏ, ÍÁÔÅÍÁÔÉÞÅÓËÏÅ ÏÖÉÄÁÎÉÅ ÓÕÍÍÙ Ä×ÕÈ ÓÌÕÞÁÊÎÙÈ ×ÅÌÉÞÉÎ ÒÁ×ÎÏ ÓÕÍÍÅ ÉÈÍÁÔÅÍÁÔÉÞÅÓËÉÈ ÏÖÉÄÁÎÉÊ, Ô.Å.M( + ) = M + M:(2)üÔÏÔ ÒÅÚÕÌØÔÁÔ ÐÏ ÉÎÄÕËÃÉÉ, ÏÞÅ×ÉÄÎÙÍ ÏÂÒÁÚÏÍ, ÏÂÏÂÝÁÅÔÓÑ ÎÁ ÓÕÍÍÕ ÌÀÂÏÇÏ ÞÉÓÌÁÓÌÁÇÁÅÍÙÈ.

îÁÐÒÉÍÅÒ, ÄÌÑ ÓÌÕÞÁÑ ÔÒÅÈ ÓÌÁÇÁÅÍÙÈ, ÐÒÉÍÅÎÉ× Ä×Á ÒÁÚÁ ÆÏÒÍÕÌÕ (2), ÐÏÌÕÞÉÍM(1 + 2 + 3 ) = M[(1 + 2 ) + 3 ] = M[(1 + 2 )] + M3 = M1 + M2 + M3 :ôÁËÉÍ ÏÂÒÁÚÏÍ, ÓÐÒÁ×ÅÄÌÉ×ÁôÅÏÒÅÍÁ 3. íÁÔÅÍÁÔÉÞÅÓËÏÅ ÏÖÉÄÁÎÉÅ ÓÕÍÍÙ ÓÌÕÞÁÊÎÙÈ ×ÅÌÉÞÉÎ ÒÁ×ÎÏ ÓÕÍÍÅ ÉÈÍÁÔÅÍÁÔÉÞÅÓËÉÈ ÏÖÉÄÁÎÉÊ, ÉÎÁÞÅM(1 + 2 + · · · + n ) = M1 + M2 + · · · + Mn :äÌÑ ÎÅÚÁ×ÉÓÉÍÙÈ ÓÌÕÞÁÊÎÙÈ ×ÅÌÉÞÉÎ ÔÁËÖÅ ×ÅÒÎÁôÅÏÒÅÍÁ 4. íÁÔÅÍÁÔÉÞÅÓËÏÅ ÏÖÉÄÁÎÉÅ ÐÒÏÉÚ×ÅÄÅÎÉÑ Ä×ÕÈ ÎÅÚÁ×ÉÓÉÍÙÈ ÓÌÕÞÁÊÎÙÈ×ÅÌÉÞÉÎ É ÒÁ×ÎÏ ÐÒÏÉÚ×ÅÄÅÎÉÀ ÉÈ ÍÁÔÅÍÁÔÉÞÅÓËÉÈ ÏÖÉÄÁÎÉÊ, ÉÎÁÞÅM( · ) = M · M:31(2)á.ç.

äØÑÞËÏ×: ÷ÅÒÏÑÔÎÏÓÔØ, ÏÓÎÏ×ÎÙÅ ÏÐÒÅÄÅÌÅÎÉÑ É ÐÒÉÍÅÒÙäÏËÁÚÁÔÅÌØÓÔ×Ï. åÓÌÉ × ÏÐÒÅÄÅÌÅÎÉÉ (1) ÐÏÌÏÖÉÔØ f (x; y) = x · y, ÔÏ ÄÌÑ ÄÉÓËÒÅÔÎÏÇÏ(ÁÎÁÌÏÇÉÞÎÏ É ÄÌÑ ÎÅÐÒÅÒÙ×ÎÏÇÏ) ÓÌÕÞÁÑ ÉÍÅÅÍM( · ) =XXxy(x · y) · q(x; y):÷ ÓÉÌÕ ÏÐÒÅÄÅÌÅÎÉÑ ÎÅÚÁ×ÉÓÉÍÏÓÔÉ, ÓÏ×ÍÅÓÔÎÏÅ ÒÁÓÐÒÅÄÅÌÅÎÉÅ q(x; y) = q(x; :) · q(:; y). ðÏÜÔÏÍÕM( · ) =XXxy"[x · q(x; :)] · [y · q(:; y)] =Xx# "x · q(x; :) ·Xy#(y · q(:; y) = M · MôÅÏÒÅÍÁ 4 ÄÏËÁÚÁÎÁ.òÁÓÓÍÏÔÒÉÍ ÔÅÐÅÒØ ×ÏÐÒÏÓ Ï ÄÉÓÐÅÒÓÉÉ ÓÕÍÍÙ Ä×ÕÈ ÓÌÕÞÁÊÎÙÈ ×ÅÌÉÞÉÎ É . ÷×ÅÄ£Í,ÄÌÑ ËÒÁÔËÏÓÔÉ, ÏÂÏÚÎÁÞÅÎÉÑ ÍÁÔÅÍÁÔÉÞÅÓËÉÈ ÏÖÉÄÁÎÉÊ a = M É b = M. ðÒÉÍÅÎÑÑÏÐÒÅÄÅÌÅÎÉÅ ÄÉÓÐÅÒÓÉÉ É ÔÅÏÒÅÍÙ 1-3, ÍÏÖÅÍ ÎÁÐÉÓÁÔØD( + ) = M[( + ) − M( + )]2 = M[( + ) − (a + b)]2 = M[( − a) + ( − b)]2 == M[( − a)2 + ( − b)2 + 2( − a)( − b)] = M( − a)2 + M( − b)2 + 2M[( − a)( − b)] == M( − M )2 + M( − M)2 + 2M[( − M )( − M)] = D + D + 2cov(; );ÇÄÅ ××ÅÌÉ ÞÉÓÌÏcov(; ) def= M[( − M )( − M)] =XXxy(x − a)(y − b) q(x; y);ÎÁÚÙ×ÁÅÍÏÅ ËÏ×ÁÒÉÁÃÉÅÊ ÐÁÒÙ ÓÌÕÞÁÊÎÙÈ ×ÅÌÉÞÉÎ (; ).