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ABL @O4 25I59, > :>B>@KE 8< =C6=> 8 <>6=> 1K;> 1K 3>2>@8BL, => MB>3> =5 ?@>8AE>48B. 0 >A=>20=88 ?>;O :>@@5;OF88 <>6=> 2K428=CBL 38?>B57C (4;O 35=5@0;L=>9 A>2>:C?=>AB8) > B><, GB> A2O7L <564C 2A5<8 2>7<>6=K<8 7=0G5=8O<8 X 8 Y =>A8B ;8=59=K9 E0@0:B5@. 8=59=>5 C@02=5=85 @53@5AA88 8<55B 284 y = bx + a +  !8AB5<0 =>@<0;L=KE C@02=5=89. a" n + b"x = "y a"x + b"x2 = "y" x ;O =0H8E 40==KE A8AB5<0 C@02=5=89 8<55B 284 30a + 243 b = 273 243 a + 2491 b = 2679 7 ?5@2>3> C@02=5=8O 2K@0605< 0 8 ?>4AB028< 2> 2B>@>5 C@02=5=85: >;CG05< M<?8@8G5A:85 :>MDD8F85=BK @53@5AA88: b = 0.8948, a = 1.8523 #@02=5=85 @53@5AA88 (M<?8@8G5A:>5 C@02=5=85 @53@5AA88): y = 0.8948 x + 1.8523 ;O @0AG5B0 ?0@0<5B@>2 @53@5AA88 ?>AB@>8< @0AG5B=CN B01;8FC (B01;. 2.1) "01;8F0 2.1 0AG5B=0O B01;8F0 xyx2y2x " y109100819012101441001201218144324216911811219956253630161725628927281264144964416161610131001691305825644056253630121214414414464361624911811219913516925655525252518183243243246836644810610036606123614472131516922519530900693681541111121121121000005325915813641691046536253006036010161002561602432732491320126790@0<5B@K C@02=5=8O @53@5AA88. K1>@>G=K5 A@54=85. EQ \x\to(x) = \f("xi;n) = \f(243;30) = 8.1 EQ \x\to(y) = \f("yi;n) = \f(273;30) = 9.1 EQ \x\to(xy) = \f("xiyi;n) = \f(2679;30) = 89.3 K1>@>G=K5 48A?5@A88: EQ S2(x) = \f("x2i;n) - \x\to(x)2 = \f(2491;30) - 8.12 = 17.42 EQ S2(y) = \f("y2i;n) - \x\to(y)2 = \f(3201;30) - 9.12 = 23.89 !@54=5:204@0B8G5A:>5 >B:;>=5=85 EQ S(x) = \r(S2(x)) = \r(17.42) = 4.17 EQ S(y) = \r(S2(y)) = \r(23.89) = 4.89 >MDD8F85=B :>@@5;OF88 >20@80F8O. EQ cov(x,y) = \x\to(x " y) - \x\to(x) " \x\to(y) = 89.3 - 8.1 " 9.1 = 15.59 0AAG8BK205< ?>:070B5;L B5A=>BK A2O78. "0:8< ?>:070B5;5< O2;O5BAO 2K1>@>G=K9 ;8=59=K9 :>MDD8F85=B :>@@5;OF88, :>B>@K9 @0AAG8BK205BAO ?> D>@<C;5: EQ rxy = \f(\x\to(x " y) -\x\to(x) " \x\to(y) ;S(x) " S(y)) = \f(89.3 - 8.1 " 9.1;4.17 " 4.89) = 0.76 8=59=K9 :>MDD8F85=B :>@@5;OF88 ?@8=8<05B 7=0G5=8O >B  1 4> +1. !2O78 <564C ?@87=0:0<8 <>3CB 1KBL A;01K<8 8 A8;L=K<8 (B5A=K<8). E :@8B5@88 >F5=820NBAO ?> H:0;5 '544>:0: 0.1 < rxy < 0.3: A;010O; 0.3 < rxy < 0.5: C<5@5==0O; 0.5 < rxy < 0.7: 70<5B=0O; 0.7 < rxy < 0.9: 2KA>:0O; 0.9 < rxy < 1: 25AL<0 2KA>:0O;  =0H5< ?@8<5@5 A2O7L <564C ?@87=0:>< Y D0:B>@>< X 2KA>:0O 8 ?@O<0O. @><5 B>3>, :>MDD8F85=B ;8=59=>9 ?0@=>9 :>@@5;OF88 <>65B 1KBL >?@545;5= G5@57 :>MDD8F85=B @53@5AA88 b: EQ rx,y = b\f(S(x);S(y)) #@02=5=85 @53@5AA88 (>F5=:0 C@02=5=8O @53@5AA88). 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;";J;L;N;P;hIh"3CJH*aJjhIh"3CJUaJhIh"35CJaJhIh"3CJaJhIh"3CJaJmH sH K 9999999{jjjjjD$dh$Ifa$gd+kd$$$IfTr] m4ayt+T99!9#9&9)9,9{jjjjjD$dh$Ifa$gd+kd%$$IfTr] m4ayt+T,9-9/919396989{jjjjjD$dh$Ifa$gd+kd%$$IfTr] m4ayt+T8999<9?9C9G9K9{jjjjjD$dh$Ifa$gd+kd &$$IfTr] m4ayt+TK9L9P9T9Y9^9c9{jjjjjD$dh$Ifa$gd+kd&$$IfTr] m4ayt+Tc9d9>:h::@1%1 dh`gd"3$dh`a$gd"3kd'$$IfTr] m 24ap2yt+T: ;;;8<<<R====>?~@AA BDB|BBBCPDDDEv dh`gd"3$dh`a$gd"3P;;;;;;;;;;;;"<$<4<6<8<:<B<D<Z<\<^<z<|<<<<<<===N=P=R=T=p=r=======>>????????z@|@AABBRBVBBBBB*D,DPDRDZDhIh"3CJNHaJhIh"35CJaJhIh"3CJH*aJhIh"3CJH*aJjhIh"3CJUaJhIh"3CJaJGZD`DDDDDDDDDDEEEE.0^`Z\ 0TVX`bjl@BLN $&24 hIh"3CJH*aJhIh"3CJNHaJUhIh"35CJaJjhIh"3CJUaJhIh"3CJaJhIh"3CJH*aJB8=59=>5 C@02=5=85 @53@5AA88 8<55B 284 y = 0.89 x + 1.85 >MDD8F85=B0< C@02=5=8O ;8=59=>9 @53@5AA88 <>6=> ?@840BL M:>=><8G5A:89 A<KA;. >MDD8F85=B @53@5AA88 b = 0.89 ?>:07K205B A@54=55 87<5=5=85 @57C;LB0B82=>3> ?>:070B5;O (2 548=8F0E 87<5@5=8O C) A ?>2KH5=85< 8;8 ?>=865=85< 25;8G8=K D0:B>@0 E =0 548=8FC 53> 87<5@5=8O.  40==>< ?@8<5@5 A C25;8G5=85< =0 1 548=8FC y ?>2KH05BAO 2 A@54=5< =0 0.89. >MDD8F85=B a = 1.85 D>@<0;L=> ?>:07K205B ?@>3=>78@C5<K9 C@>25=L C, => B>;L:> 2 B>< A;CG05, 5A;8 E=0 =0E>48BAO 1;87:> A 2K1>@>G=K<8 7=0G5=8O<8. > 5A;8 E=0 =0E>48BAO 40;5:> >B 2K1>@>G=KE 7=0G5=89 E, B> 1C:20;L=0O 8=B5@?@5B0F8O <>65B ?@825AB8 : =525@=K< @57C;LB0B0<, 8 4065 5A;8 ;8=8O @53@5AA88 4>2>;L=> B>G=> >?8AK205B 7=0G5=8O =01;N405<>9 2K1>@:8, =5B 30@0=B89, GB> B0:65 1C45B ?@8 M:AB@0?>;OF88 2;52> 8;8 2?@02>. >4AB0282 2 C@02=5=85 @53@5AA88 A>>B25BAB2CNI85 7=0G5=8O E, <>6=> >?@545;8BL 2K@>2=5==K5 (?@54A:070==K5) 7=0G5=8O @57C;LB0B82=>3> ?>:070B5;O y(x) 4;O :064>3> =01;N45=8O. !2O7L <564C C 8 E >?@545;O5B 7=0: :>MDD8F85=B0 @53@5AA88 b (5A;8 > 0  ?@O<0O A2O7L, 8=0G5 - >1@0B=0O).  =0H5< ?@8<5@5 A2O7L ?@O<0O. >MDD8F85=B M;0AB8G=>AB8. >MDD8F85=B M;0AB8G=>AB8 =0E>48BAO ?> D>@<C;5: EQ E = \f("y;"x) \f(x;y) = b\f(\x\to(x);\x\to(y)) EQ E = 0.89\f(8.1;9.1) = 0.8 >MDD8F85=B M;0AB8G=>AB8 <5=LH5 1. !;54>20B5;L=>, ?@8 87<5=5=88 % =0 1%, Y 87<5=8BAO <5=55 G5< =0 1%. @C38<8 A;>20<8 - 2;8O=85 % =0 Y =5 ACI5AB25==>. 5B0  :>MDD8F85=B EQ j = bj\f(S(x);S(y)) = 0.89\f(4.17;4.89) = 0.76 ".5. C25;8G5=85 x =0 25;8G8=C A@54=5:204@0B8G5A:>3> >B:;>=5=8O Sx ?@82545B : C25;8G5=8N A@54=53> 7=0G5=8O Y =0 0.76 A@54=5:204@0B8G=>3> >B:;>=5=8O Sy. >MDD8F85=B 45B5@<8=0F88 R2= 0.762 = 0.5839 B.5. 2 58.39 % A;CG052 87<5=5=8O E ?@82>4OB : 87<5=5=8N y. @C38<8 A;>20<8 - B>G=>ABL ?>41>@0 C@02=5=8O @53@5AA88 - A@54=OO. AB0;L=K5 41.61 % 87<5=5=8O Y >1JOA=ONBAO D0:B>@0<8, =5 CGB5==K<8 2 <>45;8. ;O >F5=:8 :0G5AB20 ?0@0<5B@>2 @53@5AA88 ?>AB@>8< @0AG5B=CN B01;8FC (B01;. 2.2) "01;8F0 2.2 F5=:0 :0G5AB20 ?0@0<5B@>2 @53@5AA88 xyy(x)(yi-ycp)2(y-y(x))2(xi-xcp)2|y - yx|:y10910.80.013.243.610.2121012.590.816.7115.210.26121812.5979.2129.2715.210.39119.913.611.20.810.0995566.339.610.119.610.0544161716.1762.410.6962.410.04898129.018.418.940.010.25445.4326.012.0516.810.36101310.815.214.843.610.17586.331.212.89.610.21566.339.610.119.610.0544121212.598.410.3515.210.0491647.2226.0110.374.410.819119.913.611.20.810.099513513.4816.8171.9924.011.7556.3316.811.769.610.27181817.9679.210.0017498.010.00232687.221.210.614.410.097410610.89.6123.043.610.86127.228.4122.844.410.4131513.4834.812.324.010.1304.5482.8120.5826.010697.220.013.164.410.2111111.693.610.488.410.06320000000536.3337.2111.069.611.118139.0115.2115.920.010.31657.2216.814.934.410.44061.859.6117.265.610.69101610.847.6127.043.610.32243273271.15633.89294.78457.099.45F5=:0 ?0@0<5B@>2 C@02=5=8O @53@5AA88. =0G8<>ABL :>MDD8F85=B0 :>@@5;OF88. ;O B>3> GB>1K ?@8 C@>2=5 7=0G8<>AB8  ?@>25@8BL =C;52CN 38?>B57C > @025=AB25 =C;N 35=5@0;L=>3> :>MDD8F85=B0 :>@@5;OF88 =>@<0;L=>9 42C<5@=>9 A;CG09=>9 25;8G8=K ?@8 :>=:C@8@CNI59 38?>B575 H1 `" 0, =04> 2KG8A;8BL =01;N405<>5 7=0G5=85 :@8B5@8O EQ t=01; = rxy \f(\r(n-2);\r(1 - r2xy)) 8 ?> B01;8F5 :@8B8G5A:8E B>G5: @0A?@545;5=8O !BLN45=B0, ?> 7040==><C C@>2=N 7=0G8<>AB8  8 G8A;C AB5?5=59 A2>1>4K k = n - 2 =09B8 :@8B8G5A:CN B>G:C t:@8B 42CAB>@>==59 :@8B8G5A:>9 >1;0AB8. A;8 t=01; < t:@8B >A=>20=89 >B25@3=CBL =C;52CN 38?>B57C. A;8 |t=01;| > t:@8B  =C;52CN 38?>B57C >B25@30NB. EQ t=01; = 0.76 \f(\r(28);\r(1 - 0.76\s\up6(2))) = 8.33 > B01;8F5 !BLN45=B0 A C@>2=5< 7=0G8<>AB8 =0.05 8 AB5?5=O<8 A2>1>4K k=28 =0E>48< t:@8B: t:@8B (n-m-1;/2) = (28;0.025) = 2.048 345 m = 1 - :>;8G5AB2> >1JOA=ONI8E ?5@5<5==KE. A;8 t=01; > t:@8B8G, B> ?>;CG5==>5 7=0G5=85 :>MDD8F85=B0 :>@@5;OF88 ?@87=05BAO 7=0G8<K< (=C;520O 38?>B570, CB25@640NI0O @025=AB2> =C;N :>MDD8F85=B0 :>@@5;OF88, >B25@305BAO). >A:>;L:C t=01; > t:@8B, B> >B:;>=O5< 38?>B57C > @025=AB25 0 :>MDD8F85=B0 :>@@5;OF88. @C38<8 A;>20<8, :>MDD8F85=B :>@@5;OF88 AB0B8AB8G5A:8 - 7=0G8<  ?0@=>9 ;8=59=>9 @53@5AA88 t2r = t2b 8 B>340 ?@>25@:0 38?>B57 > 7=0G8<>AB8 :>MDD8F85=B>2 @53@5AA88 8 :>@@5;OF88 @02=>A8;L=0 ?@>25@:5 38?>B57K > ACI5AB25==>AB8 ;8=59=>3> C@02=5=8O @53@5AA88. =B5@20;L=0O >F5=:0 4;O :>MDD8F85=B0 :>@@5;OF88 (4>25@8B5;L=K9 8=B5@20;). EQ (r - t:@8B \f(1-r2;\r(n)); r + t:@8B \f(1-r2;\r(n))) >25@8B5;L=K9 8=B5@20; 4;O :>MDD8F85=B0 :>@@5;OF88 EQ (0.76 - 2.048\f(1-0.762;\r(30)); 0.76 + 2.048\f(1-0.762;\r(30))) r(0.61;0.92) =0;87 B>G=>AB8 >?@545;5=8O >F5=>: :>MDD8F85=B>2 @53@5AA88. 5A<5I5==>9 >F5=:>9 48A?5@A88 2>7<CI5=89 O2;O5BAO 25;8G8=0: EQ S\s\up4(2)y = \f("(y\s\do4(i) - y\s\do4(x))2;n - m - 1) EQ S\s\up4(2)y = \f(294.78;28) = 10.53 S2y = 10.53 - =5>1JOA=5==0O 48A?5@A8O (<5@0 @071@>A0 7028A8<>9 ?5@5<5==>9 2>:@C3 ;8=88 @53@5AA88). EQ Sy = \r(S2 y ) = \r(10.53) = 3.24 Sy = 3.24 - AB0=40@B=0O >H81:0 >F5=:8 (AB0=40@B=0O >H81:0 @53@5AA88). Sa - AB0=40@B=>5 >B:;>=5=85 A;CG09=>9 25;8G8=K a. EQ Sa = Sy \f( \r( "x2);n S(x)) EQ Sa = 3.24 \f( \r(2491);30 " 4.17) = 1.29 Sb - AB0=40@B=>5 >B:;>=5=85 A;CG09=>9 25;8G8=K b. EQ Sb = \f( Sy; \r(n) S(x)) EQ Sb = \f( 3.24; \r(30) " 4.17) = 0.14 @>25@:0 38?>B57 >B=>A8B5;L=> :>MDD8F85=B>2 ;8=59=>3> C@02=5=8O @53@5AA88. 1) t-AB0B8AB8:0. @8B5@89 !BLN45=B0. t:@8B (n-m-1;/2) = (28;0.025) = 2.048 EQ tb = \f(b;Sb) EQ tb = \f(0.89;0.14) = 6.3 >A:>;L:C 6.3 > 2.048, B> AB0B8AB8G5A:0O 7=0G8<>ABL :>MDD8F85=B0 @53@5AA88 b ?>4B25@6405BAO (>B25@305< 38?>B57C > @025=AB25 =C;N MB>3> :>MDD8F85=B0). EQ ta = \f(a;Sa) EQ ta = \f(1.85;1.29) = 1.43 >A:>;L:C 1.43 < 2.048, B> AB0B8AB8G5A:0O 7=0G8<>ABL :>MDD8F85=B0 @53@5AA88 a =5 ?>4B25@6405BAO (?@8=8<05< 38?>B57C > @025=AB25 =C;N MB>3> :>MDD8F85=B0). -B> >7=0G05B, GB> 2 40==>< A;CG05 :>MDD8F85=B>< a <>6=> ?@5=51@5GL. >25@8B5;L=K9 8=B5@20; 4;O :>MDD8F85=B>2 C@02=5=8O @53@5AA88. ?@545;8< 4>25@8B5;L=K5 8=B5@20;K :>MDD8F85=B>2 @53@5AA88, :>B>@K5 A =0456=>ABL 95% 1C4CB A;54CNI8<8: (b - t:@8B Sb; b + t:@8B Sb) (0.89 - 2.048 " 0.14; 0.89 + 2.048 " 0.14) (0.6;1.19) ! 25@>OB=>ABLN 95% <>6=> CB25@640BL, GB> 7=0G5=85 40==>3> ?0@0<5B@0 1C4CB ;560BL 2 =0945==>< 8=B5@20;5. (a - t:@8B Sa; a + t:@8B Sa) (1.85 - 2.048 " 1.29; 1.85 + 2.048 " 1.29) (-0.8;4.5) ! 25@>OB=>ABLN 95% <>6=> CB25@640BL, GB> 7=0G5=85 40==>3> ?0@0<5B@0 1C4CB ;560BL 2 =0945==>< 8=B5@20;5. "0: :0: B>G:0 0 (=>;L) ;568B 2=CB@8 4>25@8B5;L=>3> 8=B5@20;0, B> 8=B5@20;L=0O >F5=:0 :>MDD8F85=B0 a AB0B8AB8G5A:8 =57=0G8<0. 2) F-AB0B8AB8:0. @8B5@89 $8H5@0. EQ R2 = 1 - \f("(yi - yx)2; "(yi - \x\to(y))2) = 1 - \f(294.78;633.89) = 0.535 EQ F = \f(R2;1 - R2)\f((n - m -1);m) EQ F = \f(0.535;1 - 0.535)\f((30-1-1);1) = 32.21 "01;8G=>5 7=0G5=85 :@8B5@8O A> AB5?5=O<8 A2>1>4K k1=1 8 k2=28, FB01; = 4.2 >A:>;L:C D0:B8G5A:>5 7=0G5=85 F > FB01;, B> :>MDD8F85=B 45B5@<8=0F88 AB0B8AB8G5A:8 7=0G8< (=0945==0O >F5=:0 C@02=5=8O @53@5AA88 AB0B8AB8G5A:8 =0456=0). !2O7L <564C F-:@8B5@85< $8H5@0 8 t-AB0B8AB8:>9 !BLN45=B0 2K@0605BAO @025=AB2><: EQ t2r = t2b = \r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xi>;-2>, fixi * fi0:>?;5==0O G0AB>B0, S|x - xA@|*f(x - xA@)2*f'0AB>B0, fi/n274108420.27102.680.13284112816.2766.150.132912993.079.40.033330390126.212.810.1314124164.274.550.1332396190.20.01330.1344136237.7314.950.1335135242.938.60.033337274269.8748.680.0667382762811.8770.410.066740140297.9362.940.033342142309.9398.670.0333 30962 100.53499.871;O >F5=:8 @O40 @0A?@545;5=8O =0945< A;54CNI85 ?>:070B5;8: >:070B5;8 F5=B@0 @0A?@545;5=8O. !@54=OO 2725H5==0O EQ \x\to(x) = \f( "x " f;"f) EQ \x\to(x) = \f(962;30) = 32.07 >40 >40 - =081>;55 G0AB> 2AB@5G0NI55AO 7=0G5=85 ?@87=0:0 C 548=8F 40==>9 A>2>:C?=>AB8. 0:A8<0;L=>5 7=0G5=85 ?>2B>@5=89 ?@8 x = 2 (f = 4). !;54>20B5;L=>, <>40 @02=0 2 5480=0 5480=>9 (Me) =07K205BAO 7=0G5=85 ?@87=0:0, ?@8E>4OI55AO =0 A5@548=C @0=68@>20==>9 (C?>@O4>G5==>9) A>2>:C?=>AB8. 0E>48< xi, ?@8 :>B>@>< =0:>?;5==0O G0AB>B0 S 1C45B 1>;LH5 "f/2 = 16. -B> 7=0G5=85 xi = 3. "0:8< >1@07><, <5480=0 @02=0 3 >:070B5;8 20@80F88. 1A>;NB=K5 ?>:070B5;8 20@80F88. 07<0E 20@80F88 - @07=>ABL <564C <0:A8<0;L=K< 8 <8=8<0;L=K< 7=0G5=8O<8 ?@87=0:0 ?5@28G=>3> @O40. R = Xmax - Xmin R = 42 - 27 = 15 !@54=55 ;8=59=>5 >B:;>=5=85 - 2KG8A;ONB 4;O B>3>, GB>1K CG5ABL @07;8G8O 2A5E 548=8F 8AA;54C5<>9 A>2>:C?=>AB8. EQ d = \f("|xi - \x\to(x)| " f;"f) EQ d = \f(100.53;30) = 3.35 064>5 7=0G5=85 @O40 >B;8G05BAO >B 4@C3>3> =5 1>;55, G5< =0 3.35 8A?5@A8O - E0@0:B5@87C5B <5@C @071@>A0 >:>;> 55 A@54=53> 7=0G5=8O (<5@0 @0AA5820=8O, B.5. >B:;>=5=8O >B A@54=53>). EQ D = \f("(xi - \x\to(x))2 f;"f) EQ D = \f(499.87;30) = 16.66 !@54=55 :204@0B8G5A:>5 >B:;>=5=85 (A@54=OO >H81:0 2K1>@:8). EQ  = \r(D) = \r(16.662) = 4.08 064>5 7=0G5=85 @O40 >B;8G05BAO >B A@54=53> 7=0G5=8O 32.07 =5 1>;55, G5< =0 4.08 B=>A8B5;L=K5 ?>:070B5;8 20@80F88.  >B=>A8B5;L=K< ?>:070B5;O< 20@80F88 >B=>AOB: :>MDD8F85=B >AF8;;OF88, ;8=59=K9 :>MDD8F85=B 20@80F88, >B=>A8B5;L=>5 ;8=59=>5 >B:;>=5=85. >MDD8F85=B 20@80F88 - <5@0 >B=>A8B5;L=>3> @071@>A0 7=0G5=89 A>2>:C?=>AB8: ?>:07K205B, :0:CN 4>;N A@54=53> 7=0G5=8O MB>9 25;8G8=K A>AB02;O5B 55 A@54=89 @071@>A. EQ v = \f(;\x\to(x)) = \f(4.08;32.07)100% = 12.73% >A:>;L:C v d" 30%, B> A>2>:C?=>ABL >4=>@>4=0, 0 20@80F8O A;010O. >;CG5==K< @57C;LB0B0< <>6=> 4>25@OBL. 8=59=K9 :>MDD8F85=B 20@80F88 8;8 B=>A8B5;L=>5 ;8=59=>5 >B:;>=5=85 - E0@0:B5@87C5B 4>;N CA@54=5==>3> 7=0G5=8O ?@87=0:0 01A>;NB=KE >B:;>=5=89 >B A@54=59 25;8G8=K. EQ Kd = \f(d;\x\to(x)) = \f(3.35;32.07)100% = 10.45% >:070B5;8 D>@<K @0A?@545;5=8O. >MDD8F85=B >AF8;;OF88 - >B@0605B >B=>A8B5;L=CN :>;51;5<>ABL :@09=8E 7=0G5=89 ?@87=0:0 2>:@C3 A@54=59. EQ Kr = \f(R;\x\to(x)) = \f(15;32.07)100% = 46.78% !B5?5=L 0A8<<5B@88 !8<<5B@8G=K< O2;O5BAO @0A?@545;5=85, 2 :>B>@>< G0AB>BK ;N1KE 42CE 20@80=B>2, @02=>AB>OI8E 2 >15 AB>@>=K >B F5=B@0 @0A?@545;5=8O, @02=K <564C A>1>9. 081>;55 B>G=K< 8 @0A?@>AB@0=5==K< ?>:070B5;5< 0A8<<5B@88 O2;O5BAO <><5=B=K9 :>MDD8F85=B 0A8<<5B@88. As = M3/s3 345 M3 - F5=B@0;L=K9 <><5=B B@5BL53> ?>@O4:0. s - A@54=5:204@0B8G5A:>5 >B:;>=5=85. M3 = 1342.02/30 = 44.73 EQ As = \f(44.73;4.083) = 0.66 >;>68B5;L=0O 25;8G8=0 C:07K205B =0 =0;8G85 ?@02>AB>@>==59 0A8<<5B@88 F5=:0 ACI5AB25==>AB8 ?>:070B5;O 0A8<<5B@88 405BAO A ?><>ILN A@54=59 :204@0B8G5A:>9 >H81:8 :>MDD8F85=B0 0A8<<5B@88: EQ sAs = \r(\f(6(n-2);(n+1)(n+3))) A;8 2K?>;=O5BAO A>>B=>H5=85 |As|/sAs < 3, B> 0A8<<5B@8O =5ACI5AB25==0O, 55 =0;8G85 >1JOA=O5BAO 2;8O=85< @07;8G=KE A;CG09=KE >1AB>OB5;LAB2. A;8 8<55B <5AB> A>>B=>H5=85 |As|/sAs > 3, B> 0A8<<5B@8O ACI5AB25==0O 8 @0A?@545;5=85 ?@87=0:0 2 35=5@0;L=>9 A>2>:C?=>AB8 =5 O2;O5BAO A8<<5B@8G=K<. 0AG5B F5=B@0;L=KE <><5=B>2 ?@>2>48< 2 0=0;8B8G5A:>9 B01;8F5: xi(x - xA@)3*f(x - xA@)4*f27-520.272636.0228-269.011093.9929-28.8488.4430-26.4854.7331-4.855.1832-0.0008895.9E-53428.9155.883525.2474.0437240.131184.6538417.762478.7140499.313961.1642980.139735.99 1342.0221368.8EQ sAs = \r(\f(6(30-2);(30+1)(30+3))) = 0.41  0=0;878@C5<>< @O4C @0A?@545;5=8O =01;N405BAO ACI5AB25==0O ;52>AB>@>==OO 0A8<<5B@8O (0.66/1.63 = 0.41<3) @8<5=ONBAO B0:65 AB@C:BC@=K5 ?>:070B5;8 (:>MDD8F85=BK) 0A8<<5B@88, E0@0:B5@87CNI85 0A8<<5B@8N B>;L:> 2 F5=B@0;L=>9 G0AB8 @0A?@545;5=8O, B.5. >A=>2=>9 <0AAK 548=8F, 8 =57028AOI85 >B :@09=8E 7=0G5=89 ?@87=0:0. 0AAG8B05< AB@C:BC@=K9 :>MDD8F85=B 0A8<<5B@88 8@A>=0: EQ Asp = \f(\x\to(x) - Mo;) = \f(32.07-2;4.08) = 7.37 ;O A8<<5B@8G=KE @0A?@545;5=89 @0AAG8BK205BAO ?>:070B5;L M:AF5AA0 (>AB@>25@H8==>AB8). -:AF5AA ?@54AB02;O5B A>1>9 2K?04 25@H8=K M<?8@8G5A:>3> @0A?@545;5=8O 225@E 8;8 2=87 >B 25@H8=K :@82>9 =>@<0;L=>3> @0A?@545;5=8O. '0I5 2A53> M:AF5AA >F5=8205BAO A ?><>ILN ?>:070B5;O: EQ Ex = \f(M4;s4) - 3 ;O @0A?@545;5=89 1>;55 >AB@>25@H8==KE (2KBO=CBKE), G5< =>@<0;L=>5, ?>:070B5;L M:AF5AA0 ?>;>68B5;L=K9 (Ex > 0), 4;O 1>;55 ?;>A:>25@H8==KE (A?;NA=CBKE) - >B@8F0B5;L=K9 (Ex < 0), B.:. 4;O =>@<0;L=>3> @0A?@545;5=8O M4/s4 = 3. M3 = 21368.8/30 = 712.29 EQ Ex = \f(712.29;4.084) - 3 = 2.5656 - 3 = -0.43 '8A;> 3 2KG8B05BAO 87 >B=>H5=8O 4/ 4 ?>B><C, GB> 4;O =>@<0;L=>3> 70:>=0 @0A?@545;5=8O 4/ 4 = 3. "0:8< >1@07><, 4;O =>@<0;L=>3> @0A?@545;5=8O M:AF5AA @025= =C;N. AB@>25@H8==K5 :@82K5 >1;040NB ?>;>68B5;L=K< M:AF5AA><, :@82K5 1>;55 ?;>A:>25@H8==K5 - >B@8F0B5;L=K< M:AF5AA><. Ex < 0 - ?;>A:>25@H8==>5 @0A?@545;5=85 'B>1K >F5=8BL ACI5AB25==>ABL M:AF5AA0 @0AAG8BK20NB AB0B8AB8:C Ex/sEx 345 sEx - A@54=OO :204@0B8G5A:0O >H81:0 :>MDD8F85=B0 M:AF5AA0. EQ sEx = \r(\f(24n(n-2)(n-3);(n+1)2(n+3)(n+5))) A;8 >B=>H5=85 Ex/sEx > 3, B> >B:;>=5=85 >B =>@<0;L=>3> @0A?@545;5=8O AG8B05BAO ACI5AB25==K<. EQ sEx = \r(\f(24 * 30(30-2)(30-3);(30+1)2(30+3)(30+5))) = 0.7 >A:>;L:C sEx < 3, B> >B:;>=5=85 >B =>@<0;L=>3> @0A?@545;5=8O AG8B05BAO =5 ACI5AB25==K<. @>25@:0 38?>B57 > 2845 @0A?@545;5=8O. 2. @>25@8< 38?>B57C > B><, GB> % @0A?@545;5=> ?> 70:>=C C0AA>=0. EQ pi = \f(i;i!) e- 345 pi  25@>OB=>ABL ?>?040=8O 2 i-9 8=B5@20; A;CG09=>9 25;8G8=K, @0A?@545;5==>9 ?> 38?>B5B8G5A:><C 70:>=C. 0) 0E>48< ?> 7040==><C M<?8@8G5A:><C @0A?@545;5=8N 2K1>@>G=CN A@54=NN (x = 32.067). 1) @8=8<05< 2 :0G5AB25 >F5=:8 ?0@0<5B@0  @0A?@545;5=8O C0AA>=0 2K1>@>G=CN A@54=NN xA@ = 32.067. !;54>20B5;L=>, ?@54?>;0305<K9 70:>= C0AA>=0 8<55B 284: EQ pi = \f(32.067i;i!) e-32.067 2) 0945< ?> D>@<C;5 C0AA>=0 25@>OB=>AB8 Pi, ?>O2;5=8O @>2=> i A>1KB89 2 n 8A?KB0=8OE. 0E>48< B5>@5B8G5A:85 G0AB>BK ?> D>@<C;5 npi i = 0: p0 = 0, np0 = 0 i = 1: p1 = 0, np1 = 0 i = 2: p2 = 0, np2 = 0 i = 3: p3 = 0, np3 = 0 i = 4: p4 = 0, np4 = 0 i = 5: p5 = 0, np5 = 0 i = 6: p6 = 0, np6 = 0 i = 7: p7 = 0, np7 = 2.0E-6 i = 8: p8 = 0, np8 = 1.0E-5 i = 9: p9 = 1.0E-6, np9 = 3.5E-5 i = 10: p10 = 4.0E-6, np10 = 0.000113 i = 11: p11 = 1.1E-5, np11 = 0.000328 2) KG8A;O5< A;0305<K5 AB0B8AB8:8 8@A>=0 ?> D>@<C;5 (5 AB>;15F B01;8FK): EQ K = "\f((ni - n pi)2;n pi) i01;N405<0O G0AB>B0 nipi68405<0O G0AB>B0 npi!;0305<K5 AB0B8AB8:8 8@A>=0 Ki040001400021000330004400053000640053333325.337102.0E-6416664.678201.0E-5416662.67921.0E-63.5E-5113956.111014.0E-60.0001138886.891111.1E-50.0003283044.92 30  54292540.59?@545;8< 3@0=8FC :@8B8G5A:>9 >1;0AB8. "0: :0: AB0B8AB8:0 8@A>=0 87<5@O5B @07=8FC <564C M<?8@8G5A:8< 8 B5>@5B8G5A:8< @0A?@545;5=8O<8, B> G5< 1>;LH5 55 =01;N405<>5 7=0G5=85 K=01;, B5< A8;L=55 4>2>4 ?@>B82 >A=>2=>9 38?>B57K. >MB><C :@8B8G5A:0O >1;0ABL 4;O MB>9 AB0B8AB8:8 2A5340 ?@02>AB>@>==OO: [Kkp;+"). Q 3@0=8FC Kkp = 2(k-r-1;) =0E>48< ?> B01;8F0< @0A?@545;5=8O 2 8 7040==K< 7=0G5=8O< s, k (G8A;> 8=B5@20;>2), r=1 (?0@0<5B@ ). Kkp(0.05;10) = 18.30704; K=01; = 54292540.59 01;N405<>5 7=0G5=85 AB0B8AB8:8 8@A>=0 ?>?0405B 2 :@8B8G5A:CN >1;0ABL: =01; > Kkp, ?>MB><C 5ABL >A=>20=8O >B25@30BL >A=>2=CN 38?>B57C. 0==K5 2K1>@:8 @0A?@545;5=K =5 ?> 70:>=C C0AA>=0. ;O @57C;LB0B>2 2B>@>3> >?@>A0 ?@>2545< 0=0;>38G=K5 8AA;54>20=8O >AB@>8< 48A:@5B=K9 20@80F8>==K9 @O4. ;O MB>3> >BA>@B8@C5< @O4 ?> 2>7@0AB0=8N 8 ?>4AG8B05< :>;8G5AB2> ?>2B>@5=8O 4;O :064>3> M;5<5=B0 @O40. 182426272727282828292929293032323232333336373738394041434446"01;8F0 4;O @0AG5B0 ?>:070B5;59. xi>;-2>, fixi * fi0:>?;5==0O G0AB>B0, S|x - xA@|*f(x - xA@)2*f'0AB>B0, fi/n18118114.47209.280.03332412428.4771.680.03332612636.4741.820.033327381616.489.650.128384913.459.850.12941161313.8748.070.1330130142.476.080.0333324128181.870.870.1333266201.070.570.066736136213.5312.480.033337274239.0741.10.066738138245.5330.620.033339139256.5342.680.033340140267.5356.750.033341141278.5372.820.0333431432810.53110.950.0333441442911.53133.020.0333461463013.53183.150.0333 30974 154.81211.471;O >F5=:8 @O40 @0A?@545;5=8O =0945< A;54CNI85 ?>:070B5;8: >:070B5;8 F5=B@0 @0A?@545;5=8O. !@54=OO 2725H5==0O EQ \x\to(x) = \f( "x " f;"f) EQ \x\to(x) = \f(974;30) = 32.47 >40 >40 - =081>;55 G0AB> 2AB@5G0NI55AO 7=0G5=85 ?@87=0:0 C 548=8F 40==>9 A>2>:C?=>AB8. 0:A8<0;L=>5 7=0G5=85 ?>2B>@5=89 ?@8 x = 2 (f = 4). !;54>20B5;L=>, <>40 @02=0 2 5480=0 5480=>9 (Me) =07K205BAO 7=0G5=85 ?@87=0:0, ?@8E>4OI55AO =0 A5@548=C @0=68@>20==>9 (C?>@O4>G5==>9) A>2>:C?=>AB8. 0E>48< xi, ?@8 :>B>@>< =0:>?;5==0O G0AB>B0 S 1C45B 1>;LH5 "f/2 = 16. -B> 7=0G5=85 xi = 3. "0:8< >1@07><, <5480=0 @02=0 3 >:070B5;8 20@80F88. 1A>;NB=K5 ?>:070B5;8 20@80F88. 07<0E 20@80F88 - @07=>ABL <564C <0:A8<0;L=K< 8 <8=8<0;L=K< 7=0G5=8O<8 ?@87=0:0 ?5@28G=>3> @O40. R = Xmax - Xmin R = 46 - 18 = 28 !@54=55 ;8=59=>5 >B:;>=5=85 - 2KG8A;ONB 4;O B>3>, GB>1K CG5ABL @07;8G8O 2A5E 548=8F 8AA;54C5<>9 A>2>:C?=>AB8. EQ d = \f("|xi - \x\to(x)| " f;"f) EQ d = \f(154.8;30) = 5.16 064>5 7=0G5=85 @O40 >B;8G05BAO >B 4@C3>3> =5 1>;55, G5< =0 5.16 8A?5@A8O - E0@0:B5@87C5B <5@C @071@>A0 >:>;> 55 A@54=53> 7=0G5=8O (<5@0 @0AA5820=8O, B.5. >B:;>=5=8O >B A@54=53>). EQ D = \f("(xi - \x\to(x))2 f;"f) EQ D = \f(1211.47;30) = 40.38 !@54=55 :204@0B8G5A:>5 >B:;>=5=85 (A@54=OO >H81:0 2K1>@:8). EQ  = \r(D) = \r(40.382) = 6.35 064>5 7=0G5=85 @O40 >B;8G05BAO >B A@54=53> 7=0G5=8O 32.47 =5 1>;55, G5< =0 6.35 B=>A8B5;L=K5 ?>:070B5;8 20@80F88.  >B=>A8B5;L=K< ?>:070B5;O< 20@80F88 >B=>AOB: :>MDD8F85=B >AF8;;OF88, ;8=59=K9 :>MDD8F85=B 20@80F88, >B=>A8B5;L=>5 ;8=59=>5 >B:;>=5=85. >MDD8F85=B 20@80F88 - <5@0 >B=>A8B5;L=>3> @071@>A0 7=0G5=89 A>2>:C?=>AB8: ?>:07K205B, :0:CN 4>;N A@54=53> 7=0G5=8O MB>9 25;8G8=K A>AB02;O5B 55 A@54=89 @071@>A. EQ v = \f(;\x\to(x)) = \f(6.35;32.47)100% = 19.57% >A:>;L:C v d" 30%, B> A>2>:C?=>ABL >4=>@>4=0, 0 20@80F8O A;010O. >;CG5==K< @57C;LB0B0< <>6=> 4>25@OBL. 8=59=K9 :>MDD8F85=B 20@80F88 8;8 B=>A8B5;L=>5 ;8=59=>5 >B:;>=5=85 - E0@0:B5@87C5B 4>;N CA@54=5==>3> 7=0G5=8O ?@87=0:0 01A>;NB=KE >B:;>=5=89 >B A@54=59 25;8G8=K. EQ Kd = \f(d;\x\to(x)) = \f(5.16;32.47)100% = 15.89% >:070B5;8 D>@<K @0A?@545;5=8O. >MDD8F85=B >AF8;;OF88 - >B@0605B >B=>A8B5;L=CN :>;51;5<>ABL :@09=8E 7=0G5=89 ?@87=0:0 2>:@C3 A@54=59. EQ Kr = \f(R;\x\to(x)) = \f(28;32.47)100% = 86.24% !B5?5=L 0A8<<5B@88 !8<<5B@8G=K< O2;O5BAO @0A?@545;5=85, 2 :>B>@>< G0AB>BK ;N1KE 42CE 20@80=B>2, @02=>AB>OI8E 2 >15 AB>@>=K >B F5=B@0 @0A?@545;5=8O, @02=K <564C A>1>9. 081>;55 B>G=K< 8 @0A?@>AB@0=5==K< ?>:070B5;5< 0A8<<5B@88 O2;O5BAO <><5=B=K9 :>MDD8F85=B 0A8<<5B@88. As = M3/s3 345 M3 - F5=B@0;L=K9 <><5=B B@5BL53> ?>@O4:0. s - A@54=5:204@0B8G5A:>5 >B:;>=5=85. M3 = 2064.9/30 = 68.83 EQ As = \f(68.83;6.353) = 0.27 >;>68B5;L=0O 25;8G8=0 C:07K205B =0 =0;8G85 ?@02>AB>@>==59 0A8<<5B@88 F5=:0 ACI5AB25==>AB8 ?>:070B5;O 0A8<<5B@88 405BAO A ?><>ILN A@54=59 :204@0B8G5A:>9 >H81:8 :>MDD8F85=B0 0A8<<5B@88: EQ sAs = \r(\f(6(n-2);(n+1)(n+3))) A;8 2K?>;=O5BAO A>>B=>H5=85 |As|/sAs < 3, B> 0A8<<5B@8O =5ACI5AB25==0O, 55 =0;8G85 >1JOA=O5BAO 2;8O=85< @07;8G=KE A;CG09=KE >1AB>OB5;LAB2. A;8 8<55B <5AB> A>>B=>H5=85 |As|/sAs > 3, B> 0A8<<5B@8O ACI5AB25==0O 8 @0A?@545;5=85 ?@87=0:0 2 35=5@0;L=>9 A>2>:C?=>AB8 =5 O2;O5BAO A8<<5B@8G=K<. 0AG5B F5=B@0;L=KE <><5=B>2 ?@>2>48< 2 0=0;8B8G5A:>9 B01;8F5: xi(x - xA@)3*f(x - xA@)4*f18-3027.6543799.9824-606.935138.6626-270.421748.7327-490.12679.2428-267.341194.1429-166.65577.7130-15.0137.0232-0.410.19330.30.163644.11155.8637186.33844.738169.42937.4539278.871821.9640427.533220.6941621.385302.43431168.6912310.15441534.1417693.73462478.6533544.33 2064.9131007.12 EQ sAs = \r(\f(6(30-2);(30+1)(30+3))) = 0.41  0=0;878@C5<>< @O4C @0A?@545;5=8O =01;N405BAO ACI5AB25==0O ;52>AB>@>==OO 0A8<<5B@8O (0.27/0.67 = 0.41<3) @8<5=ONBAO B0:65 AB@C:BC@=K5 ?>:070B5;8 (:>MDD8F85=BK) 0A8<<5B@88, E0@0:B5@87CNI85 0A8<<5B@8N B>;L:> 2 F5=B@0;L=>9 G0AB8 @0A?@545;5=8O, B.5. >A=>2=>9 <0AAK 548=8F, 8 =57028AOI85 >B :@09=8E 7=0G5=89 ?@87=0:0. 0AAG8B05< AB@C:BC@=K9 :>MDD8F85=B 0A8<<5B@88 8@A>=0: EQ Asp = \f(\x\to(x) - Mo;) = \f(32.47-2;6.35) = 4.79 ;O A8<<5B@8G=KE @0A?@545;5=89 @0AAG8BK205BAO ?>:070B5;L M:AF5AA0 (>AB@>25@H8==>AB8). -:AF5AA ?@54AB02;O5B A>1>9 2K?04 25@H8=K M<?8@8G5A:>3> @0A?@545;5=8O 225@E 8;8 2=87 >B 25@H8=K :@82>9 =>@<0;L=>3> @0A?@545;5=8O. '0I5 2A53> M:AF5AA >F5=8205BAO A ?><>ILN ?>:070B5;O: EQ Ex = \f(M4;s4) - 3 ;O @0A?@545;5=89 1>;55 >AB@>25@H8==KE (2KBO=CBKE), G5< =>@<0;L=>5, ?>:070B5;L M:AF5AA0 ?>;>68B5;L=K9 (Ex > 0), 4;O 1>;55 ?;>A:>25@H8==KE (A?;NA=CBKE) - >B@8F0B5;L=K9 (Ex < 0), B.:. 4;O =>@<0;L=>3> @0A?@545;5=8O M4/s4 = 3. M3 = 131007.12/30 = 4366.9 EQ Ex = \f(4366.9;6.354) - 3 = 2.6779 - 3 = -0.32 '8A;> 3 2KG8B05BAO 87 >B=>H5=8O 4/ 4 ?>B><C, GB> 4;O =>@<0;L=>3> 70:>=0 @0A?@545;5=8O 4/ 4 = 3. "0:8< >1@07><, 4;O =>@<0;L=>3> @0A?@545;5=8O M:AF5AA @025= =C;N. AB@>25@H8==K5 :@82K5 >1;040NB ?>;>68B5;L=K< M:AF5AA><, :@82K5 1>;55 ?;>A:>25@H8==K5 - >B@8F0B5;L=K< M:AF5AA><. Ex < 0 - ?;>A:>25@H8==>5 @0A?@545;5=85 'B>1K >F5=8BL ACI5AB25==>ABL M:AF5AA0 @0AAG8BK20NB AB0B8AB8:C Ex/sEx 345 sEx - A@54=OO :204@0B8G5A:0O >H81:0 :>MDD8F85=B0 M:AF5AA0. EQ sEx = \r(\f(24n(n-2)(n-3);(n+1)2(n+3)(n+5))) A;8 >B=>H5=85 Ex/sEx > 3, B> >B:;>=5=85 >B =>@<0;L=>3> @0A?@545;5=8O AG8B05BAO ACI5AB25==K<. EQ sEx = \r(\f(24 * 30(30-2)(30-3);(30+1)2(30+3)(30+5))) = 0.7 >A:>;L:C sEx < 3, B> >B:;>=5=85 >B =>@<0;L=>3> @0A?@545;5=8O AG8B05BAO =5 ACI5AB25==K<. !B5?5=L C4>2;5B2>@5==>AB8 1@0:><@0:B8G5A:8 ?>;=0O C4>2;5B2>@5==>ABL=0G8B5;L=0O C4>2;5B2>@5==>ABL'0AB8G=0O C4>2;5B2>@5==>ABL 8 =5C4>2;5B2>@5==>ABL!:>@55 =5C4>2;5B2>@5==>ABL, G5< C4>2;5B2>@5==>ABL=0G8B5;L=0O =5C4>2;5B2>@5==>ABL6,6%53,3%40%0%0%16,6%33,3%36,6%6,6%3,3% !;54C5B >B<5B8BL, GB> @57C;LB0BK B5AB0 2KO28;8, GB> A@548 8A?KBC5<KE 76,6% C4>2;5B2>@5==KE 8;8 G0AB8G=> C4>2;5B2>@5==KE A2>8< 1@0:><, 8<5NB ?@5B5=788 : A2>8< 2B>@K< ?>;>28=0<. A5 MB> 3>2>@8B > =0;8G88 :>=D;8:B=KE A8BC0F89 2 A5<LOE, ?@8=O2H8E CG0AB85 2 8AA;54>20=88. AA;54>20=85 ?@54@0A?>;>65==>AB8 : :>=D;8:B0< "5AB !B8;L ?>2545=8O 2 :>=D;8:B=>9 A8BC0F88 ;O >?@545;5=8O AB8;O ?>2545=8O <>;>4KE AC?@C3>2 2 :>=D;8:B=>9 A8BC0F88 8 ?>4B25@645=8O ?@54K4CI8E <5B>48: 1K;0 ?@8<5=5=0 040?B8@>20==0O .. @8H8=>9 <5B>48:0 0<5@8:0=A:>3> A>F80;L=>3> ?A8E>;>30 .. "><0A0 (1973) [43]. =0 ?><>305B 2KO28BL B8?8G=K5 A?>A>1K @5038@>20=8O @5A?>=45=B>2 =0 B5 8;8 8=K5 :>=D;8:B=K5 A8BC0F88. ! 55 ?><>ILN <>6=> >?@545;8BL, =0A:>;L:> G5;>25: A:;>=5= : A>?5@=8G5AB2C 8 A>B@C4=8G5AB2C 2 A5<L5, AB@5<8BAO ;8 >= : :><?@><8AA0<, 8715305B ;8 :>=D;8:B>2 8, =0>1>@>B, AB0@05BAO >1>AB@8BL 8E. &5;L 40==>9 <5B>48:8: ?>;CG8BL ?@54AB02;5=85 > 2K@065==>AB8 B5=45=F88 : ?@>O2;5=8N A>>B25BAB2CNI8E D>@< ?>2545=8O 2 :>=D;8:B=>9 A8BC0F88. ;O ?@>2545=8O <5B>48:8 8 >1@01>B:8, ?>;CG5==KE 40==KE <K 8A?>;L7>20;8 B5:AB 8 :;NG, :>B>@K5 ?@54AB02;5=K 2 @8;>65=88 3. ;O 87CG5=8O A?>A>1>2 @5038@>20=8O =0 :>=D;8:B=K5 A8BC0F88 2 48045 AC?@C3-AC?@C30 2 <>;>4>9 A5<L5 2 ?5@28G=>9 040?B0F88 <>6=> 8A?>;L7>20BL B01;8FC ?5@28G=KE @57C;LB0B>2, :>B>@0O 1K;0 A>AB02;5=0 =0<8 =0 >A=>20=88 ?>;CG5==KE 40==KE 8 ?@54AB02;5=0 2 @8;>65=88 3. 57C;LB0BK, ?>;CG5==K5 ?>A;5 ?>4AG5B0 >G:>2, =01@0==KE 8A?KBC5<K<8, ?@54AB02;5=K =865, 2 B01;8F5 6. ;O 8B>3>2>9 B01;8F5 ?> 2A5< B5AB0< 1K;> @5H5=> 8A?>;L7>20BL ?>:070B5;L "!>?5@=8G5AB2>". >;>4K5 ;N48 1K;8 @0745;5=K =0 425 3@C??K =0 >A=>20=88 C@>2=59 A?>A>1>2 @5038@>20=8O 2 B>9 8;8 8=>9 :>=D;8:B=>9 A8BC0F88: AC<<0 >B 0 4> 6 10;;>2  =87:89; AC<<0 >B 6 4> 12 10;;>2  2KA>:89. !>?5@=8G5AB2>!>B@C4=8G5AB2>><?@><8AA71530=85@8A?>A>1;5=85KA>:89 ?>:070B5;L46,6 %43,3 % 46,6 %33,3 %20 %60 %36,6 % 43,3 %33,3 %26,6 %87:89 ?>:070B5;L53,3 %56,6 %53,3 %66,6 %80 %40 %63,3 %56,6 %66,6 %73,3 % >;8G5AB2> 10;;>2 40QB ?@54AB02;5=85 > 2K@065==>AB8 B5=45=F88 : ?@>O2;5=8N A>>B25BAB2CNI8E D>@< ?>2545=8O 2 :>=D;8:B=>9 A8BC0F88. 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A. 264.  0?CAB8= !. . @0=8FK 2>7<>6=>AB59 ?A8E>;>38G5A:>3> :>=AC;LB8@>20=8O // >?@>AK ?A8E>;>388. - 1993. - A. 87-91.  0@15@ ..  ?@8@>45 ?A8E8:8. 17 C@>:>2 ?> ?A8E>;>388. >A>185 4;O ?@5?>4020B5;59 ?A8E>;>388, H:>;L=KE ?A8E>;>3>2, ABC45=B>2..: (:>;0-@5AA, 2009-135-145AB@  09F520 ". . "5>@8O ?A8E>;>38G5A:>3> B@5=8=30 2010-269-305AB@  5;;5@<0=, 8B5@ A8E>4@0<0 :@C?=K< ?;0=><. =0;87 B5@0?52B8G5A:8E <5E0=87<>2 2010-175-186AB@  >A:0;5=:> .. !>7028A8<>ABL: E0@0:B5@8AB8:8 8 ?@0:B8:0 ?@5>4>;5=8O,2009-65-69AB@      PAGE \* MERGEFORMAT 79 v BbZ0V"H~     D$dh$Ifa$gd+$dh`a$gd"3$dh`a$gd"3 dh`gd"3                   $ ( * , . : < B F J L N P X Z b d l n v x ~                                    ( * 0 hIh"3CJH*aJmH sH hIh"3CJH*aJmH sH hIh"3CJaJhIh"3CJaJmH sH N    . 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B0:8E A5<LOE :064K9 87 AC?@C3>2 45;8BAO B5<, GB> 53> BO3>B8B, 8 ?0@B=5@ 53> 2=8<0B5;L=> 2KA;CH8205B. # =8E =5B ?>B@51=>AB8 45;8BLAO A :5<-B> 4@C38<: 4@C7LO<8, @>4AB25==8:0<8, ?>B><C GB> 2 AC?@C65A:>< 4><5 8E ?>=8<0NB ;CGH5. !>3;0A=> @57C;LB0B0< B5AB0 50% <C6G8= 8 40% 65=I8=, ?@82;5G5==KE : >?@>AC, 2K18@0NB, > G5< @0AA:07K20BL, 0 > G5< A;54C5B C<>;G0BL. 5GL =5 845B > B><, GB> AC?@C38 =5 4>25@ONB 4@C3 4@C3C  ?@>AB> >=8 ?>4E>4OB 1>;55 45;8:0B=> : MB>9 ?@>1;5<5. !>3;0A=> @57C;LB0B0< B5AB0 80% <C6G8= 8 73,3 % 65=I8= GC2AB2CNB, GB> 2 8E A5<LOE =5 E20B05B 2708<>?>=8<0=8O. >2>;L=> G0AB> >=8 =5 ?>A2OI0NB AC?@C30 2 A2>8 ?@>1;5<K. ABL @O4 25I59, > :>B>@KE 8< =C6=> 8 <>6=> 1K;> 1K 3>2>@8BL, => MB>3> =5 ?@>8AE>48B. 0 >A=>20=88 ?>;O :>@@5;OF88 <>6=> 2K428=CBL 38?>B57C (4;O 35=5@0;L=>9 A>2>:C?=>AB8) > B><, GB> A2O7L <564C 2A5<8 2>7<>6=K<8 7=0G5=8O<8 X 8 Y =>A8B ;8=59=K9 E0@0:B5@. 8=59=>5 C@02=5=85 @53@5AA88 8<55B 284 y = bx + a +  !8AB5<0 =>@<0;L=KE C@02=5=89. a" n + b"x = "y a"x + b"x2 = "y" x ;O =0H8E 40==KE A8AB5<0 C@02=5=89 8<55B 284 30a + 243 b = 273 243 a + 2491 b = 2679 7 ?5@2>3> C@02=5=8O 2K@0605< 0 8 ?>4AB028< 2> 2B>@>5 C@02=5=85: >;CG05< M<?8@8G5A:85 :>MDD8F85=BK @53@5AA88: b = 0.8948, a = 1.8523 #@02=5=85 @53@5AA88 (M<?8@8G5A:>5 C@02=5=85 @53@5AA88): y = 0.8948 x + 1.8523 ;O @0AG5B0 ?0@0<5B@>2 @53@5AA88 ?>AB@>8< @0AG5B=CN B01;8FC (B01;. 2.1) "01;8F0 2.1 0AG5B=0O B01;8F0 xyx2y2x " y109100819012101441001201218144324216911811219956253630161725628927281264144964416161610131001691305825644056253630121214414414464361624911811219913516925655525252518183243243246836644810610036606123614472131516922519530900693681541111121121121000005325915813641691046536253006036010161002561602432732491320126790@0<5B@K C@02=5=8O @53@5AA88. K1>@>G=K5 A@54=85. EQ \x\to(x) = \f("xi;n) = \f(243;30) = 8.1 EQ \x\to(y) = \f("yi;n) = \f(273;30) = 9.1 EQ \x\to(xy) = \f("xiyi;n) = \f(2679;30) = 89.3 K1>@>G=K5 48A?5@A88: EQ S2(x) = \f("x2i;n) - \x\to(x)2 = \f(2491;30) - 8.12 = 17.42 EQ S2(y) = \f("y2i;n) - \x\to(y)2 = \f(3201;30) - 9.12 = 23.89 !@54=5:204@0B8G5A:>5 >B:;>=5=85 EQ S(x) = \r(S2(x)) = \r(17.42) = 4.17 EQ S(y) = \r(S2(y)) = \r(23.89) = 4.89 >MDD8F85=B :>@@5;OF88 >20@80F8O. EQ cov(x,y) = \x\to(x " y) - \x\to(x) " \x\to(y) = 89.3 - 8.1 " 9.1 = 15.59 0AAG8BK205< ?>:070B5;L B5A=>BK A2O78. "0:8< ?>:070B5;5< O2;O5BAO 2K1>@>G=K9 ;8=59=K9 :>MDD8F85=B :>@@5;OF88, :>B>@K9 @0AAG8BK205BAO ?> D>@<C;5: EQ rxy = \f(\x\to(x " y) -\x\to(x) " \x\to(y) ;S(x) " S(y)) = \f(89.3 - 8.1 " 9.1;4.17 " 4.89) = 0.76 8=59=K9 :>MDD8F85=B :>@@5;OF88 ?@8=8<05B 7=0G5=8O >B  1 4> +1. !2O78 <564C ?@87=0:0<8 <>3CB 1KBL A;01K<8 8 A8;L=K<8 (B5A=K<8). E :@8B5@88 >F5=820NBAO ?> H:0;5 '544>:0: 0.1 < rxy < 0.3: A;010O; 0.3 < rxy < 0.5: C<5@5==0O; 0.5 < rxy < 0.7: 70<5B=0O; 0.7 < rxy < 0.9: 2KA>:0O; 0.9 < rxy < 1: 25AL<0 2KA>:0O;  =0H5< ?@8<5@5 A2O7L <564C ?@87=0:>< Y D0:B>@>< X 2KA>:0O 8 ?@O<0O. @><5 B>3>, :>MDD8F85=B ;8=59=>9 ?0@=>9 :>@@5;OF88 <>65B 1KBL >?@545;5= G5@57 :>MDD8F85=B @53@5AA88 b: EQ rx,y = b\f(S(x);S(y)) #@02=5=85 @53@5AA88 (>F5=:0 C@02=5=8O @53@5AA88). 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;";J;L;N;P;hIh"3CJH*aJjhIh"3CJUaJhIh"35CJaJhIh"3CJaJhIh"3CJaJmH sH K 9999999{jjjjjD$dh$Ifa$gd+kd$$$IfTr] m4ayt+T99!9#9&9)9,9{jjjjjD$dh$Ifa$gd+kd%$$IfTr] m4ayt+T,9-9/919396989{jjjjjD$dh$Ifa$gd+kd%$$IfTr] m4ayt+T8999<9?9C9G9K9{jjjjjD$dh$Ifa$gd+kd &$$IfTr] m4ayt+TK9L9P9T9Y9^9c9{jjjjjD$dh$Ifa$gd+kd&$$IfTr] m4ayt+Tc9d9>:h::@1%1 dh`gd"3$dh`a$gd"3kd'$$IfTr] m 24ap2yt+T: ;;;8<<<R====>?~@AA BDB|BBBCPDDDEv dh`gd"3$dh`a$gd"3P;;;;;;;;;;;;"<$<4<6<8<:<B<D<Z<\<^<z<|<<<<<<===N=P=R=T=p=r=======>>????????z@|@AABBRBVBBBBB*D,DPDRDZDhIh"3CJNHaJhIh"35CJaJhIh"3CJH*aJhIh"3CJH*aJjhIh"3CJUaJhIh"3CJaJGZD`DDDDDDDDDDEEEE.0^`Z\ 0TVX`bjl@BLN $&24 hIh"3CJH*aJhIh"3CJNHaJUhIh"35CJaJjhIh"3CJUaJhIh"3CJaJhIh"3CJH*aJB8=59=>5 C@02=5=85 @53@5AA88 8<55B 284 y = 0.89 x + 1.85 >MDD8F85=B0< C@02=5=8O ;8=59=>9 @53@5AA88 <>6=> ?@840BL M:>=><8G5A:89 A<KA;. >MDD8F85=B @53@5AA88 b = 0.89 ?>:07K205B A@54=55 87<5=5=85 @57C;LB0B82=>3> ?>:070B5;O (2 548=8F0E 87<5@5=8O C) A ?>2KH5=85< 8;8 ?>=865=85< 25;8G8=K D0:B>@0 E =0 548=8FC 53> 87<5@5=8O.  40==>< ?@8<5@5 A C25;8G5=85< =0 1 548=8FC y ?>2KH05BAO 2 A@54=5< =0 0.89. >MDD8F85=B a = 1.85 D>@<0;L=> ?>:07K205B ?@>3=>78@C5<K9 C@>25=L C, => B>;L:> 2 B>< A;CG05, 5A;8 E=0 =0E>48BAO 1;87:> A 2K1>@>G=K<8 7=0G5=8O<8. > 5A;8 E=0 =0E>48BAO 40;5:> >B 2K1>@>G=KE 7=0G5=89 E, B> 1C:20;L=0O 8=B5@?@5B0F8O <>65B ?@825AB8 : =525@=K< @57C;LB0B0<, 8 4065 5A;8 ;8=8O @53@5AA88 4>2>;L=> B>G=> >?8AK205B 7=0G5=8O =01;N405<>9 2K1>@:8, =5B 30@0=B89, GB> B0:65 1C45B ?@8 M:AB@0?>;OF88 2;52> 8;8 2?@02>. >4AB0282 2 C@02=5=85 @53@5AA88 A>>B25BAB2CNI85 7=0G5=8O E, <>6=> >?@545;8BL 2K@>2=5==K5 (?@54A:070==K5) 7=0G5=8O @57C;LB0B82=>3> ?>:070B5;O y(x) 4;O :064>3> =01;N45=8O. !2O7L <564C C 8 E >?@545;O5B 7=0: :>MDD8F85=B0 @53@5AA88 b (5A;8 > 0  ?@O<0O A2O7L, 8=0G5 - >1@0B=0O).  =0H5< ?@8<5@5 A2O7L ?@O<0O. >MDD8F85=B M;0AB8G=>AB8. >MDD8F85=B M;0AB8G=>AB8 =0E>48BAO ?> D>@<C;5: EQ E = \f("y;"x) \f(x;y) = b\f(\x\to(x);\x\to(y)) EQ E = 0.89\f(8.1;9.1) = 0.8 >MDD8F85=B M;0AB8G=>AB8 <5=LH5 1. !;54>20B5;L=>, ?@8 87<5=5=88 % =0 1%, Y 87<5=8BAO <5=55 G5< =0 1%. @C38<8 A;>20<8 - 2;8O=85 % =0 Y =5 ACI5AB25==>. 5B0  :>MDD8F85=B EQ j = bj\f(S(x);S(y)) = 0.89\f(4.17;4.89) = 0.76 ".5. C25;8G5=85 x =0 25;8G8=C A@54=5:204@0B8G5A:>3> >B:;>=5=8O Sx ?@82545B : C25;8G5=8N A@54=53> 7=0G5=8O Y =0 0.76 A@54=5:204@0B8G=>3> >B:;>=5=8O Sy. >MDD8F85=B 45B5@<8=0F88 R2= 0.762 = 0.5839 B.5. 2 58.39 % A;CG052 87<5=5=8O E ?@82>4OB : 87<5=5=8N y. @C38<8 A;>20<8 - B>G=>ABL ?>41>@0 C@02=5=8O @53@5AA88 - A@54=OO. AB0;L=K5 41.61 % 87<5=5=8O Y >1JOA=ONBAO D0:B>@0<8, =5 CGB5==K<8 2 <>45;8. ;O >F5=:8 :0G5AB20 ?0@0<5B@>2 @53@5AA88 ?>AB@>8< @0AG5B=CN B01;8FC (B01;. 2.2) "01;8F0 2.2 F5=:0 :0G5AB20 ?0@0<5B@>2 @53@5AA88 xyy(x)(yi-ycp)2(y-y(x))2(xi-xcp)2|y - yx|:y10910.80.013.243.610.2121012.590.816.7115.210.26121812.5979.2129.2715.210.39119.913.611.20.810.0995566.339.610.119.610.0544161716.1762.410.6962.410.04898129.018.418.940.010.25445.4326.012.0516.810.36101310.815.214.843.610.17586.331.212.89.610.21566.339.610.119.610.0544121212.598.410.3515.210.0491647.2226.0110.374.410.819119.913.611.20.810.099513513.4816.8171.9924.011.7556.3316.811.769.610.27181817.9679.210.0017498.010.00232687.221.210.614.410.097410610.89.6123.043.610.86127.228.4122.844.410.4131513.4834.812.324.010.1304.5482.8120.5826.010697.220.013.164.410.2111111.693.610.488.410.06320000000536.3337.2111.069.611.118139.0115.2115.920.010.31657.2216.814.934.410.44061.859.6117.265.610.69101610.847.6127.043.610.32243273271.15633.89294.78457.099.45F5=:0 ?0@0<5B@>2 C@02=5=8O @53@5AA88. =0G8<>ABL :>MDD8F85=B0 :>@@5;OF88. ;O B>3> GB>1K ?@8 C@>2=5 7=0G8<>AB8  ?@>25@8BL =C;52CN 38?>B57C > @025=AB25 =C;N 35=5@0;L=>3> :>MDD8F85=B0 :>@@5;OF88 =>@<0;L=>9 42C<5@=>9 A;CG09=>9 25;8G8=K ?@8 :>=:C@8@CNI59 38?>B575 H1 `" 0, =04> 2KG8A;8BL =01;N405<>5 7=0G5=85 :@8B5@8O EQ t=01; = rxy \f(\r(n-2);\r(1 - r2xy)) 8 ?> B01;8F5 :@8B8G5A:8E B>G5: @0A?@545;5=8O !BLN45=B0, ?> 7040==><C C@>2=N 7=0G8<>AB8  8 G8A;C AB5?5=59 A2>1>4K k = n - 2 =09B8 :@8B8G5A:CN B>G:C t:@8B 42CAB>@>==59 :@8B8G5A:>9 >1;0AB8. A;8 t=01; < t:@8B >A=>20=89 >B25@3=CBL =C;52CN 38?>B57C. A;8 |t=01;| > t:@8B  =C;52CN 38?>B57C >B25@30NB. EQ t=01; = 0.76 \f(\r(28);\r(1 - 0.76\s\up6(2))) = 8.33 > B01;8F5 !BLN45=B0 A C@>2=5< 7=0G8<>AB8 =0.05 8 AB5?5=O<8 A2>1>4K k=28 =0E>48< t:@8B: t:@8B (n-m-1;/2) = (28;0.025) = 2.048 345 m = 1 - :>;8G5AB2> >1JOA=ONI8E ?5@5<5==KE. A;8 t=01; > t:@8B8G, B> ?>;CG5==>5 7=0G5=85 :>MDD8F85=B0 :>@@5;OF88 ?@87=05BAO 7=0G8<K< (=C;520O 38?>B570, CB25@640NI0O @025=AB2> =C;N :>MDD8F85=B0 :>@@5;OF88, >B25@305BAO). >A:>;L:C t=01; > t:@8B, B> >B:;>=O5< 38?>B57C > @025=AB25 0 :>MDD8F85=B0 :>@@5;OF88. @C38<8 A;>20<8, :>MDD8F85=B :>@@5;OF88 AB0B8AB8G5A:8 - 7=0G8<  ?0@=>9 ;8=59=>9 @53@5AA88 t2r = t2b 8 B>340 ?@>25@:0 38?>B57 > 7=0G8<>AB8 :>MDD8F85=B>2 @53@5AA88 8 :>@@5;OF88 @02=>A8;L=0 ?@>25@:5 38?>B57K > ACI5AB25==>AB8 ;8=59=>3> C@02=5=8O @53@5AA88. =B5@20;L=0O >F5=:0 4;O :>MDD8F85=B0 :>@@5;OF88 (4>25@8B5;L=K9 8=B5@20;). EQ (r - t:@8B \f(1-r2;\r(n)); r + t:@8B \f(1-r2;\r(n))) >25@8B5;L=K9 8=B5@20; 4;O :>MDD8F85=B0 :>@@5;OF88 EQ (0.76 - 2.048\f(1-0.762;\r(30)); 0.76 + 2.048\f(1-0.762;\r(30))) r(0.61;0.92) =0;87 B>G=>AB8 >?@545;5=8O >F5=>: :>MDD8F85=B>2 @53@5AA88. 5A<5I5==>9 >F5=:>9 48A?5@A88 2>7<CI5=89 O2;O5BAO 25;8G8=0: EQ S\s\up4(2)y = \f("(y\s\do4(i) - y\s\do4(x))2;n - m - 1) EQ S\s\up4(2)y = \f(294.78;28) = 10.53 S2y = 10.53 - =5>1JOA=5==0O 48A?5@A8O (<5@0 @071@>A0 7028A8<>9 ?5@5<5==>9 2>:@C3 ;8=88 @53@5AA88). EQ Sy = \r(S2 y ) = \r(10.53) = 3.24 Sy = 3.24 - AB0=40@B=0O >H81:0 >F5=:8 (AB0=40@B=0O >H81:0 @53@5AA88). Sa - AB0=40@B=>5 >B:;>=5=85 A;CG09=>9 25;8G8=K a. EQ Sa = Sy \f( \r( "x2);n S(x)) EQ Sa = 3.24 \f( \r(2491);30 " 4.17) = 1.29 Sb - AB0=40@B=>5 >B:;>=5=85 A;CG09=>9 25;8G8=K b. EQ Sb = \f( Sy; \r(n) S(x)) EQ Sb = \f( 3.24; \r(30) " 4.17) = 0.14 @>25@:0 38?>B57 >B=>A8B5;L=> :>MDD8F85=B>2 ;8=59=>3> C@02=5=8O @53@5AA88. 1) t-AB0B8AB8:0. @8B5@89 !BLN45=B0. t:@8B (n-m-1;/2) = (28;0.025) = 2.048 EQ tb = \f(b;Sb) EQ tb = \f(0.89;0.14) = 6.3 >A:>;L:C 6.3 > 2.048, B> AB0B8AB8G5A:0O 7=0G8<>ABL :>MDD8F85=B0 @53@5AA88 b ?>4B25@6405BAO (>B25@305< 38?>B57C > @025=AB25 =C;N MB>3> :>MDD8F85=B0). EQ ta = \f(a;Sa) EQ ta = \f(1.85;1.29) = 1.43 >A:>;L:C 1.43 < 2.048, B> AB0B8AB8G5A:0O 7=0G8<>ABL :>MDD8F85=B0 @53@5AA88 a =5 ?>4B25@6405BAO (?@8=8<05< 38?>B57C > @025=AB25 =C;N MB>3> :>MDD8F85=B0). -B> >7=0G05B, GB> 2 40==>< A;CG05 :>MDD8F85=B>< a <>6=> ?@5=51@5GL. >25@8B5;L=K9 8=B5@20; 4;O :>MDD8F85=B>2 C@02=5=8O @53@5AA88. ?@545;8< 4>25@8B5;L=K5 8=B5@20;K :>MDD8F85=B>2 @53@5AA88, :>B>@K5 A =0456=>ABL 95% 1C4CB A;54CNI8<8: (b - t:@8B Sb; b + t:@8B Sb) (0.89 - 2.048 " 0.14; 0.89 + 2.048 " 0.14) (0.6;1.19) ! 25@>OB=>ABLN 95% <>6=> CB25@640BL, GB> 7=0G5=85 40==>3> ?0@0<5B@0 1C4CB ;560BL 2 =0945==>< 8=B5@20;5. (a - t:@8B Sa; a + t:@8B Sa) (1.85 - 2.048 " 1.29; 1.85 + 2.048 " 1.29) (-0.8;4.5) ! 25@>OB=>ABLN 95% <>6=> CB25@640BL, GB> 7=0G5=85 40==>3> ?0@0<5B@0 1C4CB ;560BL 2 =0945==>< 8=B5@20;5. "0: :0: B>G:0 0 (=>;L) ;568B 2=CB@8 4>25@8B5;L=>3> 8=B5@20;0, B> 8=B5@20;L=0O >F5=:0 :>MDD8F85=B0 a AB0B8AB8G5A:8 =57=0G8<0. 2) F-AB0B8AB8:0. @8B5@89 $8H5@0. EQ R2 = 1 - \f("(yi - yx)2; "(yi - \x\to(y))2) = 1 - \f(294.78;633.89) = 0.535 EQ F = \f(R2;1 - R2)\f((n - m -1);m) EQ F = \f(0.535;1 - 0.535)\f((30-1-1);1) = 32.21 "01;8G=>5 7=0G5=85 :@8B5@8O A> AB5?5=O<8 A2>1>4K k1=1 8 k2=28, FB01; = 4.2 >A:>;L:C D0:B8G5A:>5 7=0G5=85 F > FB01;, B> :>MDD8F85=B 45B5@<8=0F88 AB0B8AB8G5A:8 7=0G8< (=0945==0O >F5=:0 C@02=5=8O @53@5AA88 AB0B8AB8G5A:8 =0456=0). !2O7L <564C F-:@8B5@85< $8H5@0 8 t-AB0B8AB8:>9 !BLN45=B0 2K@0605BAO @025=AB2><: EQ t2r = t2b = \r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xi>;-2>, fixi * fi0:>?;5==0O G0AB>B0, S|x - xA@|*f(x - xA@)2*f'0AB>B0, fi/n274108420.27102.680.13284112816.2766.150.132912993.079.40.033330390126.212.810.1314124164.274.550.1332396190.20.01330.1344136237.7314.950.1335135242.938.60.033337274269.8748.680.0667382762811.8770.410.066740140297.9362.940.033342142309.9398.670.0333 30962 100.53499.871;O >F5=:8 @O40 @0A?@545;5=8O =0945< A;54CNI85 ?>:070B5;8: >:070B5;8 F5=B@0 @0A?@545;5=8O. !@54=OO 2725H5==0O EQ \x\to(x) = \f( "x " f;"f) EQ \x\to(x) = \f(962;30) = 32.07 >40 >40 - =081>;55 G0AB> 2AB@5G0NI55AO 7=0G5=85 ?@87=0:0 C 548=8F 40==>9 A>2>:C?=>AB8. 0:A8<0;L=>5 7=0G5=85 ?>2B>@5=89 ?@8 x = 2 (f = 4). !;54>20B5;L=>, <>40 @02=0 2 5480=0 5480=>9 (Me) =07K205BAO 7=0G5=85 ?@87=0:0, ?@8E>4OI55AO =0 A5@548=C @0=68@>20==>9 (C?>@O4>G5==>9) A>2>:C?=>AB8. 0E>48< xi, ?@8 :>B>@>< =0:>?;5==0O G0AB>B0 S 1C45B 1>;LH5 "f/2 = 16. -B> 7=0G5=85 xi = 3. "0:8< >1@07><, <5480=0 @02=0 3 >:070B5;8 20@80F88. 1A>;NB=K5 ?>:070B5;8 20@80F88. 07<0E 20@80F88 - @07=>ABL <564C <0:A8<0;L=K< 8 <8=8<0;L=K< 7=0G5=8O<8 ?@87=0:0 ?5@28G=>3> @O40. R = Xmax - Xmin R = 42 - 27 = 15 !@54=55 ;8=59=>5 >B:;>=5=85 - 2KG8A;ONB 4;O B>3>, GB>1K CG5ABL @07;8G8O 2A5E 548=8F 8AA;54C5<>9 A>2>:C?=>AB8. EQ d = \f("|xi - \x\to(x)| " f;"f) EQ d = \f(100.53;30) = 3.35 064>5 7=0G5=85 @O40 >B;8G05BAO >B 4@C3>3> =5 1>;55, G5< =0 3.35 8A?5@A8O - E0@0:B5@87C5B <5@C @071@>A0 >:>;> 55 A@54=53> 7=0G5=8O (<5@0 @0AA5820=8O, B.5. >B:;>=5=8O >B A@54=53>). EQ D = \f("(xi - \x\to(x))2 f;"f) EQ D = \f(499.87;30) = 16.66 !@54=55 :204@0B8G5A:>5 >B:;>=5=85 (A@54=OO >H81:0 2K1>@:8). EQ  = \r(D) = \r(16.662) = 4.08 064>5 7=0G5=85 @O40 >B;8G05BAO >B A@54=53> 7=0G5=8O 32.07 =5 1>;55, G5< =0 4.08 B=>A8B5;L=K5 ?>:070B5;8 20@80F88.  >B=>A8B5;L=K< ?>:070B5;O< 20@80F88 >B=>AOB: :>MDD8F85=B >AF8;;OF88, ;8=59=K9 :>MDD8F85=B 20@80F88, >B=>A8B5;L=>5 ;8=59=>5 >B:;>=5=85. >MDD8F85=B 20@80F88 - <5@0 >B=>A8B5;L=>3> @071@>A0 7=0G5=89 A>2>:C?=>AB8: ?>:07K205B, :0:CN 4>;N A@54=53> 7=0G5=8O MB>9 25;8G8=K A>AB02;O5B 55 A@54=89 @071@>A. EQ v = \f(;\x\to(x)) = \f(4.08;32.07)100% = 12.73% >A:>;L:C v d" 30%, B> A>2>:C?=>ABL >4=>@>4=0, 0 20@80F8O A;010O. >;CG5==K< @57C;LB0B0< <>6=> 4>25@OBL. 8=59=K9 :>MDD8F85=B 20@80F88 8;8 B=>A8B5;L=>5 ;8=59=>5 >B:;>=5=85 - E0@0:B5@87C5B 4>;N CA@54=5==>3> 7=0G5=8O ?@87=0:0 01A>;NB=KE >B:;>=5=89 >B A@54=59 25;8G8=K. EQ Kd = \f(d;\x\to(x)) = \f(3.35;32.07)100% = 10.45% >:070B5;8 D>@<K @0A?@545;5=8O. >MDD8F85=B >AF8;;OF88 - >B@0605B >B=>A8B5;L=CN :>;51;5<>ABL :@09=8E 7=0G5=89 ?@87=0:0 2>:@C3 A@54=59. EQ Kr = \f(R;\x\to(x)) = \f(15;32.07)100% = 46.78% !B5?5=L 0A8<<5B@88 !8<<5B@8G=K< O2;O5BAO @0A?@545;5=85, 2 :>B>@>< G0AB>BK ;N1KE 42CE 20@80=B>2, @02=>AB>OI8E 2 >15 AB>@>=K >B F5=B@0 @0A?@545;5=8O, @02=K <564C A>1>9. 081>;55 B>G=K< 8 @0A?@>AB@0=5==K< ?>:070B5;5< 0A8<<5B@88 O2;O5BAO <><5=B=K9 :>MDD8F85=B 0A8<<5B@88. As = M3/s3 345 M3 - F5=B@0;L=K9 <><5=B B@5BL53> ?>@O4:0. s - A@54=5:204@0B8G5A:>5 >B:;>=5=85. M3 = 1342.02/30 = 44.73 EQ As = \f(44.73;4.083) = 0.66 >;>68B5;L=0O 25;8G8=0 C:07K205B =0 =0;8G85 ?@02>AB>@>==59 0A8<<5B@88 F5=:0 ACI5AB25==>AB8 ?>:070B5;O 0A8<<5B@88 405BAO A ?><>ILN A@54=59 :204@0B8G5A:>9 >H81:8 :>MDD8F85=B0 0A8<<5B@88: EQ sAs = \r(\f(6(n-2);(n+1)(n+3))) A;8 2K?>;=O5BAO A>>B=>H5=85 |As|/sAs < 3, B> 0A8<<5B@8O =5ACI5AB25==0O, 55 =0;8G85 >1JOA=O5BAO 2;8O=85< @07;8G=KE A;CG09=KE >1AB>OB5;LAB2. A;8 8<55B <5AB> A>>B=>H5=85 |As|/sAs > 3, B> 0A8<<5B@8O ACI5AB25==0O 8 @0A?@545;5=85 ?@87=0:0 2 35=5@0;L=>9 A>2>:C?=>AB8 =5 O2;O5BAO A8<<5B@8G=K<. 0AG5B F5=B@0;L=KE <><5=B>2 ?@>2>48< 2 0=0;8B8G5A:>9 B01;8F5: xi(x - xA@)3*f(x - xA@)4*f27-520.272636.0228-269.011093.9929-28.8488.4430-26.4854.7331-4.855.1832-0.0008895.9E-53428.9155.883525.2474.0437240.131184.6538417.762478.7140499.313961.1642980.139735.99 1342.0221368.8EQ sAs = \r(\f(6(30-2);(30+1)(30+3))) = 0.41  0=0;878@C5<>< @O4C @0A?@545;5=8O =01;N405BAO ACI5AB25==0O ;52>AB>@>==OO 0A8<<5B@8O (0.66/1.63 = 0.41<3) @8<5=ONBAO B0:65 AB@C:BC@=K5 ?>:070B5;8 (:>MDD8F85=BK) 0A8<<5B@88, E0@0:B5@87CNI85 0A8<<5B@8N B>;L:> 2 F5=B@0;L=>9 G0AB8 @0A?@545;5=8O, B.5. >A=>2=>9 <0AAK 548=8F, 8 =57028AOI85 >B :@09=8E 7=0G5=89 ?@87=0:0. 0AAG8B05< AB@C:BC@=K9 :>MDD8F85=B 0A8<<5B@88 8@A>=0: EQ Asp = \f(\x\to(x) - Mo;) = \f(32.07-2;4.08) = 7.37 ;O A8<<5B@8G=KE @0A?@545;5=89 @0AAG8BK205BAO ?>:070B5;L M:AF5AA0 (>AB@>25@H8==>AB8). -:AF5AA ?@54AB02;O5B A>1>9 2K?04 25@H8=K M<?8@8G5A:>3> @0A?@545;5=8O 225@E 8;8 2=87 >B 25@H8=K :@82>9 =>@<0;L=>3> @0A?@545;5=8O. '0I5 2A53> M:AF5AA >F5=8205BAO A ?><>ILN ?>:070B5;O: EQ Ex = \f(M4;s4) - 3 ;O @0A?@545;5=89 1>;55 >AB@>25@H8==KE (2KBO=CBKE), G5< =>@<0;L=>5, ?>:070B5;L M:AF5AA0 ?>;>68B5;L=K9 (Ex > 0), 4;O 1>;55 ?;>A:>25@H8==KE (A?;NA=CBKE) - >B@8F0B5;L=K9 (Ex < 0), B.:. 4;O =>@<0;L=>3> @0A?@545;5=8O M4/s4 = 3. M3 = 21368.8/30 = 712.29 EQ Ex = \f(712.29;4.084) - 3 = 2.5656 - 3 = -0.43 '8A;> 3 2KG8B05BAO 87 >B=>H5=8O 4/ 4 ?>B><C, GB> 4;O =>@<0;L=>3> 70:>=0 @0A?@545;5=8O 4/ 4 = 3. "0:8< >1@07><, 4;O =>@<0;L=>3> @0A?@545;5=8O M:AF5AA @025= =C;N. AB@>25@H8==K5 :@82K5 >1;040NB ?>;>68B5;L=K< M:AF5AA><, :@82K5 1>;55 ?;>A:>25@H8==K5 - >B@8F0B5;L=K< M:AF5AA><. Ex < 0 - ?;>A:>25@H8==>5 @0A?@545;5=85 'B>1K >F5=8BL ACI5AB25==>ABL M:AF5AA0 @0AAG8BK20NB AB0B8AB8:C Ex/sEx 345 sEx - A@54=OO :204@0B8G5A:0O >H81:0 :>MDD8F85=B0 M:AF5AA0. EQ sEx = \r(\f(24n(n-2)(n-3);(n+1)2(n+3)(n+5))) A;8 >B=>H5=85 Ex/sEx > 3, B> >B:;>=5=85 >B =>@<0;L=>3> @0A?@545;5=8O AG8B05BAO ACI5AB25==K<. EQ sEx = \r(\f(24 * 30(30-2)(30-3);(30+1)2(30+3)(30+5))) = 0.7 >A:>;L:C sEx < 3, B> >B:;>=5=85 >B =>@<0;L=>3> @0A?@545;5=8O AG8B05BAO =5 ACI5AB25==K<. @>25@:0 38?>B57 > 2845 @0A?@545;5=8O. 2. @>25@8< 38?>B57C > B><, GB> % @0A?@545;5=> ?> 70:>=C C0AA>=0. EQ pi = \f(i;i!) e- 345 pi  25@>OB=>ABL ?>?040=8O 2 i-9 8=B5@20; A;CG09=>9 25;8G8=K, @0A?@545;5==>9 ?> 38?>B5B8G5A:><C 70:>=C. 0) 0E>48< ?> 7040==><C M<?8@8G5A:><C @0A?@545;5=8N 2K1>@>G=CN A@54=NN (x = 32.067). 1) @8=8<05< 2 :0G5AB25 >F5=:8 ?0@0<5B@0  @0A?@545;5=8O C0AA>=0 2K1>@>G=CN A@54=NN xA@ = 32.067. !;54>20B5;L=>, ?@54?>;0305<K9 70:>= C0AA>=0 8<55B 284: EQ pi = \f(32.067i;i!) e-32.067 2) 0945< ?> D>@<C;5 C0AA>=0 25@>OB=>AB8 Pi, ?>O2;5=8O @>2=> i A>1KB89 2 n 8A?KB0=8OE. 0E>48< B5>@5B8G5A:85 G0AB>BK ?> D>@<C;5 npi i = 0: p0 = 0, np0 = 0 i = 1: p1 = 0, np1 = 0 i = 2: p2 = 0, np2 = 0 i = 3: p3 = 0, np3 = 0 i = 4: p4 = 0, np4 = 0 i = 5: p5 = 0, np5 = 0 i = 6: p6 = 0, np6 = 0 i = 7: p7 = 0, np7 = 2.0E-6 i = 8: p8 = 0, np8 = 1.0E-5 i = 9: p9 = 1.0E-6, np9 = 3.5E-5 i = 10: p10 = 4.0E-6, np10 = 0.000113 i = 11: p11 = 1.1E-5, np11 = 0.000328 2) KG8A;O5< A;0305<K5 AB0B8AB8:8 8@A>=0 ?> D>@<C;5 (5 AB>;15F B01;8FK): EQ K = "\f((ni - n pi)2;n pi) i01;N405<0O G0AB>B0 nipi68405<0O G0AB>B0 npi!;0305<K5 AB0B8AB8:8 8@A>=0 Ki040001400021000330004400053000640053333325.337102.0E-6416664.678201.0E-5416662.67921.0E-63.5E-5113956.111014.0E-60.0001138886.891111.1E-50.0003283044.92 30  54292540.59?@545;8< 3@0=8FC :@8B8G5A:>9 >1;0AB8. "0: :0: AB0B8AB8:0 8@A>=0 87<5@O5B @07=8FC <564C M<?8@8G5A:8< 8 B5>@5B8G5A:8< @0A?@545;5=8O<8, B> G5< 1>;LH5 55 =01;N405<>5 7=0G5=85 K=01;, B5< A8;L=55 4>2>4 ?@>B82 >A=>2=>9 38?>B57K. >MB><C :@8B8G5A:0O >1;0ABL 4;O MB>9 AB0B8AB8:8 2A5340 ?@02>AB>@>==OO: [Kkp;+"). Q 3@0=8FC Kkp = 2(k-r-1;) =0E>48< ?> B01;8F0< @0A?@545;5=8O 2 8 7040==K< 7=0G5=8O< s, k (G8A;> 8=B5@20;>2), r=1 (?0@0<5B@ ). Kkp(0.05;10) = 18.30704; K=01; = 54292540.59 01;N405<>5 7=0G5=85 AB0B8AB8:8 8@A>=0 ?>?0405B 2 :@8B8G5A:CN >1;0ABL: =01; > Kkp, ?>MB><C 5ABL >A=>20=8O >B25@30BL >A=>2=CN 38?>B57C. 0==K5 2K1>@:8 @0A?@545;5=K =5 ?> 70:>=C C0AA>=0. ;O @57C;LB0B>2 2B>@>3> >?@>A0 ?@>2545< 0=0;>38G=K5 8AA;54>20=8O >AB@>8< 48A:@5B=K9 20@80F8>==K9 @O4. ;O MB>3> >BA>@B8@C5< @O4 ?> 2>7@0AB0=8N 8 ?>4AG8B05< :>;8G5AB2> ?>2B>@5=8O 4;O :064>3> M;5<5=B0 @O40. 182426272727282828292929293032323232333336373738394041434446"01;8F0 4;O @0AG5B0 ?>:070B5;59. xi>;-2>, fixi * fi0:>?;5==0O G0AB>B0, S|x - xA@|*f(x - xA@)2*f'0AB>B0, fi/n18118114.47209.280.03332412428.4771.680.03332612636.4741.820.033327381616.489.650.128384913.459.850.12941161313.8748.070.1330130142.476.080.0333324128181.870.870.1333266201.070.570.066736136213.5312.480.033337274239.0741.10.066738138245.5330.620.033339139256.5342.680.033340140267.5356.750.033341141278.5372.820.0333431432810.53110.950.0333441442911.53133.020.0333461463013.53183.150.0333 30974 154.81211.471;O >F5=:8 @O40 @0A?@545;5=8O =0945< A;54CNI85 ?>:070B5;8: >:070B5;8 F5=B@0 @0A?@545;5=8O. !@54=OO 2725H5==0O EQ \x\to(x) = \f( "x " f;"f) EQ \x\to(x) = \f(974;30) = 32.47 >40 >40 - =081>;55 G0AB> 2AB@5G0NI55AO 7=0G5=85 ?@87=0:0 C 548=8F 40==>9 A>2>:C?=>AB8. 0:A8<0;L=>5 7=0G5=85 ?>2B>@5=89 ?@8 x = 2 (f = 4). !;54>20B5;L=>, <>40 @02=0 2 5480=0 5480=>9 (Me) =07K205BAO 7=0G5=85 ?@87=0:0, ?@8E>4OI55AO =0 A5@548=C @0=68@>20==>9 (C?>@O4>G5==>9) A>2>:C?=>AB8. 0E>48< xi, ?@8 :>B>@>< =0:>?;5==0O G0AB>B0 S 1C45B 1>;LH5 "f/2 = 16. -B> 7=0G5=85 xi = 3. "0:8< >1@07><, <5480=0 @02=0 3 >:070B5;8 20@80F88. 1A>;NB=K5 ?>:070B5;8 20@80F88. 07<0E 20@80F88 - @07=>ABL <564C <0:A8<0;L=K< 8 <8=8<0;L=K< 7=0G5=8O<8 ?@87=0:0 ?5@28G=>3> @O40. R = Xmax - Xmin R = 46 - 18 = 28 !@54=55 ;8=59=>5 >B:;>=5=85 - 2KG8A;ONB 4;O B>3>, GB>1K CG5ABL @07;8G8O 2A5E 548=8F 8AA;54C5<>9 A>2>:C?=>AB8. EQ d = \f("|xi - \x\to(x)| " f;"f) EQ d = \f(154.8;30) = 5.16 064>5 7=0G5=85 @O40 >B;8G05BAO >B 4@C3>3> =5 1>;55, G5< =0 5.16 8A?5@A8O - E0@0:B5@87C5B <5@C @071@>A0 >:>;> 55 A@54=53> 7=0G5=8O (<5@0 @0AA5820=8O, B.5. >B:;>=5=8O >B A@54=53>). EQ D = \f("(xi - \x\to(x))2 f;"f) EQ D = \f(1211.47;30) = 40.38 !@54=55 :204@0B8G5A:>5 >B:;>=5=85 (A@54=OO >H81:0 2K1>@:8). EQ  = \r(D) = \r(40.382) = 6.35 064>5 7=0G5=85 @O40 >B;8G05BAO >B A@54=53> 7=0G5=8O 32.47 =5 1>;55, G5< =0 6.35 B=>A8B5;L=K5 ?>:070B5;8 20@80F88.  >B=>A8B5;L=K< ?>:070B5;O< 20@80F88 >B=>AOB: :>MDD8F85=B >AF8;;OF88, ;8=59=K9 :>MDD8F85=B 20@80F88, >B=>A8B5;L=>5 ;8=59=>5 >B:;>=5=85. >MDD8F85=B 20@80F88 - <5@0 >B=>A8B5;L=>3> @071@>A0 7=0G5=89 A>2>:C?=>AB8: ?>:07K205B, :0:CN 4>;N A@54=53> 7=0G5=8O MB>9 25;8G8=K A>AB02;O5B 55 A@54=89 @071@>A. EQ v = \f(;\x\to(x)) = \f(6.35;32.47)100% = 19.57% >A:>;L:C v d" 30%, B> A>2>:C?=>ABL >4=>@>4=0, 0 20@80F8O A;010O. >;CG5==K< @57C;LB0B0< <>6=> 4>25@OBL. 8=59=K9 :>MDD8F85=B 20@80F88 8;8 B=>A8B5;L=>5 ;8=59=>5 >B:;>=5=85 - E0@0:B5@87C5B 4>;N CA@54=5==>3> 7=0G5=8O ?@87=0:0 01A>;NB=KE >B:;>=5=89 >B A@54=59 25;8G8=K. EQ Kd = \f(d;\x\to(x)) = \f(5.16;32.47)100% = 15.89% >:070B5;8 D>@<K @0A?@545;5=8O. >MDD8F85=B >AF8;;OF88 - >B@0605B >B=>A8B5;L=CN :>;51;5<>ABL :@09=8E 7=0G5=89 ?@87=0:0 2>:@C3 A@54=59. EQ Kr = \f(R;\x\to(x)) = \f(28;32.47)100% = 86.24% !B5?5=L 0A8<<5B@88 !8<<5B@8G=K< O2;O5BAO @0A?@545;5=85, 2 :>B>@>< G0AB>BK ;N1KE 42CE 20@80=B>2, @02=>AB>OI8E 2 >15 AB>@>=K >B F5=B@0 @0A?@545;5=8O, @02=K <564C A>1>9. 081>;55 B>G=K< 8 @0A?@>AB@0=5==K< ?>:070B5;5< 0A8<<5B@88 O2;O5BAO <><5=B=K9 :>MDD8F85=B 0A8<<5B@88. As = M3/s3 345 M3 - F5=B@0;L=K9 <><5=B B@5BL53> ?>@O4:0. s - A@54=5:204@0B8G5A:>5 >B:;>=5=85. M3 = 2064.9/30 = 68.83 EQ As = \f(68.83;6.353) = 0.27 >;>68B5;L=0O 25;8G8=0 C:07K205B =0 =0;8G85 ?@02>AB>@>==59 0A8<<5B@88 F5=:0 ACI5AB25==>AB8 ?>:070B5;O 0A8<<5B@88 405BAO A ?><>ILN A@54=59 :204@0B8G5A:>9 >H81:8 :>MDD8F85=B0 0A8<<5B@88: EQ sAs = \r(\f(6(n-2);(n+1)(n+3))) A;8 2K?>;=O5BAO A>>B=>H5=85 |As|/sAs < 3, B> 0A8<<5B@8O =5ACI5AB25==0O, 55 =0;8G85 >1JOA=O5BAO 2;8O=85< @07;8G=KE A;CG09=KE >1AB>OB5;LAB2. A;8 8<55B <5AB> A>>B=>H5=85 |As|/sAs > 3, B> 0A8<<5B@8O ACI5AB25==0O 8 @0A?@545;5=85 ?@87=0:0 2 35=5@0;L=>9 A>2>:C?=>AB8 =5 O2;O5BAO A8<<5B@8G=K<. 0AG5B F5=B@0;L=KE <><5=B>2 ?@>2>48< 2 0=0;8B8G5A:>9 B01;8F5: xi(x - xA@)3*f(x - xA@)4*f18-3027.6543799.9824-606.935138.6626-270.421748.7327-490.12679.2428-267.341194.1429-166.65577.7130-15.0137.0232-0.410.19330.30.163644.11155.8637186.33844.738169.42937.4539278.871821.9640427.533220.6941621.385302.43431168.6912310.15441534.1417693.73462478.6533544.33 2064.9131007.12 EQ sAs = \r(\f(6(30-2);(30+1)(30+3))) = 0.41  0=0;878@C5<>< @O4C @0A?@545;5=8O =01;N405BAO ACI5AB25==0O ;52>AB>@>==OO 0A8<<5B@8O (0.27/0.67 = 0.41<3) @8<5=ONBAO B0:65 AB@C:BC@=K5 ?>:070B5;8 (:>MDD8F85=BK) 0A8<<5B@88, E0@0:B5@87CNI85 0A8<<5B@8N B>;L:> 2 F5=B@0;L=>9 G0AB8 @0A?@545;5=8O, B.5. >A=>2=>9 <0AAK 548=8F, 8 =57028AOI85 >B :@09=8E 7=0G5=89 ?@87=0:0. 0AAG8B05< AB@C:BC@=K9 :>MDD8F85=B 0A8<<5B@88 8@A>=0: EQ Asp = \f(\x\to(x) - Mo;) = \f(32.47-2;6.35) = 4.79 ;O A8<<5B@8G=KE @0A?@545;5=89 @0AAG8BK205BAO ?>:070B5;L M:AF5AA0 (>AB@>25@H8==>AB8). -:AF5AA ?@54AB02;O5B A>1>9 2K?04 25@H8=K M<?8@8G5A:>3> @0A?@545;5=8O 225@E 8;8 2=87 >B 25@H8=K :@82>9 =>@<0;L=>3> @0A?@545;5=8O. '0I5 2A53> M:AF5AA >F5=8205BAO A ?><>ILN ?>:070B5;O: EQ Ex = \f(M4;s4) - 3 ;O @0A?@545;5=89 1>;55 >AB@>25@H8==KE (2KBO=CBKE), G5< =>@<0;L=>5, ?>:070B5;L M:AF5AA0 ?>;>68B5;L=K9 (Ex > 0), 4;O 1>;55 ?;>A:>25@H8==KE (A?;NA=CBKE) - >B@8F0B5;L=K9 (Ex < 0), B.:. 4;O =>@<0;L=>3> @0A?@545;5=8O M4/s4 = 3. M3 = 131007.12/30 = 4366.9 EQ Ex = \f(4366.9;6.354) - 3 = 2.6779 - 3 = -0.32 '8A;> 3 2KG8B05BAO 87 >B=>H5=8O 4/ 4 ?>B><C, GB> 4;O =>@<0;L=>3> 70:>=0 @0A?@545;5=8O 4/ 4 = 3. "0:8< >1@07><, 4;O =>@<0;L=>3> @0A?@545;5=8O M:AF5AA @025= =C;N. AB@>25@H8==K5 :@82K5 >1;040NB ?>;>68B5;L=K< M:AF5AA><, :@82K5 1>;55 ?;>A:>25@H8==K5 - >B@8F0B5;L=K< M:AF5AA><. Ex < 0 - ?;>A:>25@H8==>5 @0A?@545;5=85 'B>1K >F5=8BL ACI5AB25==>ABL M:AF5AA0 @0AAG8BK20NB AB0B8AB8:C Ex/sEx 345 sEx - A@54=OO :204@0B8G5A:0O >H81:0 :>MDD8F85=B0 M:AF5AA0. EQ sEx = \r(\f(24n(n-2)(n-3);(n+1)2(n+3)(n+5))) A;8 >B=>H5=85 Ex/sEx > 3, B> >B:;>=5=85 >B =>@<0;L=>3> @0A?@545;5=8O AG8B05BAO ACI5AB25==K<. EQ sEx = \r(\f(24 * 30(30-2)(30-3);(30+1)2(30+3)(30+5))) = 0.7 >A:>;L:C sEx < 3, B> >B:;>=5=85 >B =>@<0;L=>3> @0A?@545;5=8O AG8B05BAO =5 ACI5AB25==K<. !B5?5=L C4>2;5B2>@5==>AB8 1@0:><@0:B8G5A:8 ?>;=0O C4>2;5B2>@5==>ABL=0G8B5;L=0O C4>2;5B2>@5==>ABL'0AB8G=0O C4>2;5B2>@5==>ABL 8 =5C4>2;5B2>@5==>ABL!:>@55 =5C4>2;5B2>@5==>ABL, G5< C4>2;5B2>@5==>ABL=0G8B5;L=0O =5C4>2;5B2>@5==>ABL6,6%53,3%40%0%0%16,6%33,3%36,6%6,6%3,3% !;54C5B >B<5B8BL, GB> @57C;LB0BK B5AB0 2KO28;8, GB> A@548 8A?KBC5<KE 76,6% C4>2;5B2>@5==KE 8;8 G0AB8G=> C4>2;5B2>@5==KE A2>8< 1@0:><, 8<5NB ?@5B5=788 : A2>8< 2B>@K< ?>;>28=0<. A5 MB> 3>2>@8B > =0;8G88 :>=D;8:B=KE A8BC0F89 2 A5<LOE, ?@8=O2H8E CG0AB85 2 8AA;54>20=88. AA;54>20=85 ?@54@0A?>;>65==>AB8 : :>=D;8:B0< "5AB !B8;L ?>2545=8O 2 :>=D;8:B=>9 A8BC0F88 ;O >?@545;5=8O AB8;O ?>2545=8O <>;>4KE AC?@C3>2 2 :>=D;8:B=>9 A8BC0F88 8 ?>4B25@645=8O ?@54K4CI8E <5B>48: 1K;0 ?@8<5=5=0 040?B8@>20==0O .. @8H8=>9 <5B>48:0 0<5@8:0=A:>3> A>F80;L=>3> ?A8E>;>30 .. "><0A0 (1973) [43]. =0 ?><>305B 2KO28BL B8?8G=K5 A?>A>1K @5038@>20=8O @5A?>=45=B>2 =0 B5 8;8 8=K5 :>=D;8:B=K5 A8BC0F88. ! 55 ?><>ILN <>6=> >?@545;8BL, =0A:>;L:> G5;>25: A:;>=5= : A>?5@=8G5AB2C 8 A>B@C4=8G5AB2C 2 A5<L5, AB@5<8BAO ;8 >= : :><?@><8AA0<, 8715305B ;8 :>=D;8:B>2 8, =0>1>@>B, AB0@05BAO >1>AB@8BL 8E. &5;L 40==>9 <5B>48:8: ?>;CG8BL ?@54AB02;5=85 > 2K@065==>AB8 B5=45=F88 : ?@>O2;5=8N A>>B25BAB2CNI8E D>@< ?>2545=8O 2 :>=D;8:B=>9 A8BC0F88. ;O ?@>2545=8O <5B>48:8 8 >1@01>B:8, ?>;CG5==KE 40==KE <K 8A?>;L7>20;8 B5:AB 8 :;NG, :>B>@K5 ?@54AB02;5=K 2 @8;>65=88 3. ;O 87CG5=8O A?>A>1>2 @5038@>20=8O =0 :>=D;8:B=K5 A8BC0F88 2 48045 AC?@C3-AC?@C30 2 <>;>4>9 A5<L5 2 ?5@28G=>9 040?B0F88 <>6=> 8A?>;L7>20BL B01;8FC ?5@28G=KE @57C;LB0B>2, :>B>@0O 1K;0 A>AB02;5=0 =0<8 =0 >A=>20=88 ?>;CG5==KE 40==KE 8 ?@54AB02;5=0 2 @8;>65=88 3. 57C;LB0BK, ?>;CG5==K5 ?>A;5 ?>4AG5B0 >G:>2, =01@0==KE 8A?KBC5<K<8, ?@54AB02;5=K =865, 2 B01;8F5 6. ;O 8B>3>2>9 B01;8F5 ?> 2A5< B5AB0< 1K;> @5H5=> 8A?>;L7>20BL ?>:070B5;L "!>?5@=8G5AB2>". >;>4K5 ;N48 1K;8 @0745;5=K =0 425 3@C??K =0 >A=>20=88 C@>2=59 A?>A>1>2 @5038@>20=8O 2 B>9 8;8 8=>9 :>=D;8:B=>9 A8BC0F88: AC<<0 >B 0 4> 6 10;;>2  =87:89; AC<<0 >B 6 4> 12 10;;>2  2KA>:89. !>?5@=8G5AB2>!>B@C4=8G5AB2>><?@><8AA71530=85@8A?>A>1;5=85KA>:89 ?>:070B5;L46,6 %43,3 % 46,6 %33,3 %20 %60 %36,6 % 43,3 %33,3 %26,6 %87:89 ?>:070B5;L53,3 %56,6 %53,3 %66,6 %80 %40 %63,3 %56,6 %66,6 %73,3 % >;8G5AB2> 10;;>2 40QB ?@54AB02;5=85 > 2K@065==>AB8 B5=45=F88 : ?@>O2;5=8N A>>B25BAB2CNI8E D>@< ?>2545=8O 2 :>=D;8:B=>9 A8BC0F88. K2>4K ?> 3;025 @>2545==>5 8AA;54>20=85 ?>4B25@48;> 2K428=CBCN 38?>B57C 8 ?>72>;8;> 2KO28BL 8 >?8A0BL MB0?K AB0=>2;5=8O, 0 B0:65 >A>15==>AB8 @0728B8O AC?@C65A:8E >B=>H5=89 =0 @07;8G=KE C@>2=OE. >;CG5==K9 =0<8 <0B5@80; ?>72>;O5B =0<5B8BL ?CB8 40;L=59H8E 45B0;L=KE 8AA;54>20=89 @0728B8O AC?@C65A:8E >B=>H5=89 =5 B>;L:> 2 <>;>4>< 1@0:5, => 8 =0 40;L=59H8E MB0?0E @0728B8O AC?@C65AB20. 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