ࡱ> \^[y ̠bjbjEE .''$33333GGGG+G&%%%%%%%&)%3%33%###33%#%###jGRG#%%0&#'*'*#'*3#<0""#D`%%!>&'* : A Primer on Summation Notation George H Olson, Ph. D. Doctoral Program in Educational Leadership Appalachian State University Spring 2008 Summation Operator The summation operator (") {Greek letter, capital sigma} is an instruction to sum over a series of values. For instance, if we have the set of values for the variable, X = {X1, X2, X3, X4, X5}, then  EMBED Equation.3 = X1+ X2+X3+ X4+ X5 Literally, the expression, EMBED Equation.3 , says: beginning with i=1 and ending with i=5, sum over the variables Xi. As an example, let Xi = 8, X2 = 10, X3 = 11, X4 = =15, X5 = 16. Then n = 5 {the number of cases}, and  EMBED Equation.3  = 8 + 10 + 11 + 15 + 16 = 60. In many contexts, it is clear that the summation is over all cases and we do not need the superscript over the summation operator. Furthermore, in most contexts it is assumed that the summation begins with i = 1. Hence, the notation, EMBED Equation.3 is taken to imply EMBED Equation.3 . In most situations, where the variable has only one subscript, as in Xi, the subscript can be omitted. In these situations,  EMBED Equation.3  implies EMBED Equation.3 . In other contexts, the variable X may have more than one subscript, e.g., Xij. This occurs, for instance, when individuals belong to two or more subgroupings or cross-classifications. We might have a situation as shown below in Table 1. Table 1 Group 1Group 2Group 3X11, X21, X31, X41X12, X22, X32, X42, X52, X62X13, X23, X33, X43, X53 Here we have three groups, each with a different number of cases. We denote the ith case in the jth group with the symbol, Xij. To sum all the cases, over all three groups, we would use the following, double summation operator,  EMBED Equation.3 , which instructs us to sum over the three groups (j=1, 2, and 3) and, within each group, sum over the number of cases in the group (i=1, 2, 3, 4 for Group 1; i=1, 2, 3, 4, 5, 6 for Group 2; i=1, 2, 3, 4, 5 for Group 3). For simplicity, we often write the summation expression as,  EMBED Equation.3 , where it is assumed that we are to sum over all groups and all cases within each group. For example, lets substitute the following numbers for the symbolic values given above. Table 2 Group 1Group 2Group 310, 8, 12, 136, 11, 8, 10, 8, 1214, 6, 6, 10, 9 Then,  EMBED Equation.3 = EMBED Equation.3  = [10+8+12+13] +[6+11+8+10+8+12] +[14+6+6+10+9] = 43+55+45 = 143. A more complex situation occurs when cases are grouped into cross-classifications. Table 3 represents a situation where cases are cross-classified by sex and age-category. Table 3 Age CategoryChildAdolescentAdultSexMale X111, X211, X311, X411 X112, X212, X312, X412 X113, X213, X313, X413 Female X121, X221, X321, X421 X122, X222, X322, X422 X123, X223, X323, X423  In the table the notation, Xijk, denotes the ith individual in the jth row (Sex category) and kth column (Age Category). Hence, X423 is the last case in the Adult-Female cell. To indicate summation over all the cases in the table, we would use the notation,  EMBED Equation.3 , where it is assumed that the summation is over all N cases, i, over all J rows, j, and all K columns, k. Dot Notation It is often easier to denote aggregates using dot notation. In an expression such as  EMBED Equation.3 , the dot (") represents aggregation or summation over the missing (dotted) subscript. For instance, in the example given earlier, where we had EMBED Equation.3 = 60, we could simply write X" = 60 {read X-dot = 60}. Using dot notation we would represent the cell aggregates in the Sex by Age Category table above as shown in Table 4. Table 4 Age CategoryChildAdolescentAdultSexMaleX"11X"12X"13FemaleX"21X"22X"23where, for the Male-Adolescent cell, X"12 =  EMBED Equation.3 . If we wanted to denote the sum of the values for all the males, we would write, X"1" ; for all females, X"2" Similarly, the sum of the values for all children is, X""1; all adolescents, X""2; and all adults, X""3 And the sum of the values for all the cases, X""" Note that X"1" =  EMBED Equation.3 , X""1=  EMBED Equation.3 , and X""" =  EMBED Equation.3 . Rules of summation Summation of a constant. Let c be some constant value. Then,  EMBED Equation.3 . In other words, summing a constant N times is the same as multiplying the constant by N. Hence, if c = 5, then  EMBED Equation.3 . This rule can be extended to double summation. Thus,  EMBED Equation.3 = EMBED Equation.3 = EMBED Equation.3 . As an example, consider the situation involving the three groups given earlier in Table 2. If all cases, in all groups, have the constant value, 10, then  EMBED Equation.3 = EMBED Equation.3  = [10+10+10+10] + [10+10+10+10+10+10] + [10+10+10+10+10] = (410) + (610) + (510) = 1510 = 150. Multiplication by a constant. If all the values of a variable, X, are multiplied by the same constant, c, then,  EMBED Equation.3 . For example, let the set of 5 values of the variable X be {3, 9, 5, 7, 10}, assume that each is multiplied by the constant, 2. Then,  EMBED Equation.3 = 2(3)+2(9)+2(5)+2(7)+2(10) = 2(3+9+5+7+10) = 34 =  EMBED Equation.3  = cX"" Again, this can be expanded to multiple summations:  EMBED Equation.3 = cX"",  EMBED Equation.3 = cX""", and so on. As an example, again consider the three group situation given earlier in Table 2. If all cases were multiplied by 5, then  EMBED Equation.3  = [(510) + (58) + (512) + (513)] +[(56) + (511) + (58) + (510) + (58) + (512)] +[(514) + (56) + (56) + (510) + (59)] = 5(43+55+45) =5 EMBED Equation.3  = 10(143) = 715. In some situations, the values in different groups are multiplied by different constants. For instance, suppose the values in Group 1 (in the example just given) were multiplied by the constant c1, the values in Group 2 by c2, and the values in group 3 by c3. Then, the sum of all the cases would be given by,  EMBED Equation.3 . In this situation, it is necessary that the constants remain within the summation operator. Now, suppose that in the Sex by Age Category example given earlier, we have the values as shown below in Table 5. Table 5 Age CategoryChildAdolescentAdultSexMale 2, 3, 5, 4  5, 1, 1, 3 2, 2, 3, 0 Female 1, 3, 5, 2 2, 0, 3, 1 5, 3, 4, 3  If all the male values are multiplied by the constant, 5, and all the female values are multiplied by the constant, 10. Then, letting c1 = 5 and c2 = 10, the sum over all cases, rows, and columns is given by,  EMBED Equation.3  =  EMBED Equation.3  = 5(14+10+7) + 10(11+6+15) = 5(31) + 10(32) = 475. Note that the right-hand side of the summation notation above could have been written as  EMBED Equation.3 , without the parentheses. Order of operations. The order of operations, when using summation operators, is the same as that for arithmetic. That is, operations involving the values in a summation operation are indicated by mathematical punctuation. When indicated by punctuation, operations are to be carried out on values prior to summation. Some examples should suffice.  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 . On the other hand,  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 . Note that these rules apply even when we have multiple summations. For example, letting capital J represent the number of groups,  EMBED Equation.3 . In words, within each group, j: sum the values over cases, i, then square the sum. After squaring the sums within all of the J groups, sum the squared sums over groups. For example, consider the data given in Table 2, earlier.  EMBED Equation.3  = 432 + 552 + 452 = 6899. Distributive rule of summation>Zj    < @ p r  H I M N O R S V W Z [ ^ _ f g i j } ~  ŲŪřńh#L"h#L"6jh0h#L"EHU!jUTG h#L"CJUV^JaJjh#L"Uhdah#L"h#L"6H* h#L"6h#L" h06 h0^Jh0hT5hT5hT55hah^ h^h^h^h^5CJaJ2>l  < > g h % & X Y gd1gd#L"^gd1$a$gd^     $ % ' ( ) , 0 1 2 5 : ; < ? D E F I O P Q T V W Z [ _ ` t y z  jh0Uh0h06hkh06H*h h6H* h06h0 h#L"h#L"hkh#L"6H* h#L"6jdh0h#L"EHU!jUTG h#L"CJUV^JaJjh#L"Uh#L"h13 U ] , - . b ļķīĚtg_Uh K&h K&6H*h K&h K&6jb h0h K&EHU!jUTG h K&CJUV^JaJjh K&Uj,h0h K&EHU!j9YG h K&CJUV^JaJjh#L"Uh h6h K&hth#L" hk>*hkh0h1jh0Ujh0hkEHU!jUTG hkCJUV^JaJ b c d w x y z p햒햊zhX hOH*hX h:H*hYRhOh1h:h K&h K&6H*h K&h K&6 h K&6j h0h K&EHU!jUTG h K&CJUV^JaJj h K&h K&EHU!jZG h K&CJUV^JaJh K&jh K&U h K&h K&0ekdI$$IflF $ \   t!    44 la<ytX $$Ifa$gdX l 8:FGVWqrt()z{   Ͱ͖͖͖͒͞ɒɒjhpUhphShS6 hS6jUhShSEHU!j^G hSCJUV^JaJjhSUhhSh:h:6H*h:h:6 h:6h1hX hOH*hX h:H*h:6*+^gd1gdp^gd1ekd$$IflF $ \   t!    44 la<ytX $%&')+ !348<=PQRSTUhijkl{|֘ҌҌhP}Zh hShyCOj4hphpEHU!jbG hpCJUV^JaJjhphpEHU hphphYRh1jhpUjhphpEHU!j`G hpCJUV^JaJhp7 !1ekd$$IflF $ \   t!    44 la<ytX $$Ifa$gdX l 1234:;|^gd1gdpekdx$$IflF $ \   t!    44 la<ytX 6>   #$%(+.147:;<=@CFILORSTUVWX[\^abdghjklhX heLH*hX hOH*hX hP}ZH*heLhYRh1hphP}ZTpikd$$IfTlF t    44 laytX T$$Ifa$gdX l gdYR$a$gdeL YF$IfgdeLl kd1$$IfTlrW  t44 laytX T$$Ifa$gdX l #$XE$IfgdeLl kd$$IfTl4rW  t44 laytX T$$Ifa$gdX l $;<STklmnoXPP$a$gdeLkdw$$IfTl4rW   t44 laytX T$$Ifa$gdX l ln|pst ķy!j G h CJUV^JaJ hT55^JjhT55U^Jh hT5hT55 hT56hT5 h16jheLhT5EHU!jgG hT5CJUV^JaJjheLU heL6heLheL6H*heLheL6heLh1hP}Z/ors n$$Ifa$gdX l gda ^gd gdT5^gd1gdeL0.024lnprtvxzlxokkkgkkgckhah hkhk5H*^Jh5CJ H*^JaJ h h 5CJ H*^JaJ hT56hj"h0hT5EHU!jUTG hT5CJUV^JaJhT5jhT5U hT56^J hT5^J h 5^J hT55^JjhT55U^J jm hT5h 5EHU^J"$$Ifa$gdX l ikd/%$$IfTlF t    44 laytX T",oYFYYY$IfgdyCOl $$Ifa$gdX l kd%$$IfTlrW  t44 laytX T $&(*@BFJLPTVZʺ䐃wmcwhyCO56H*^JhyCOhyCO6H*hyCOhyCO6hyCOj'hkhkEHU!jlG hkCJUV^JaJjhkU hkhkhyCOhk6H*h ha5CJ H*^JaJ hyCOhk6hX hkH*^JhX hOH*hkhX hkH*hX h 5CJ H*^JaJ &,.0>HR\nXEXXX$IfgdyCOl $$Ifa$gdX l kdW&$$IfTl4rW  t44 laytX T\^TABng\\\ggWWgdd- ^gdagdT5kd&$$IfTl4rW   t44 laytX T 0268^`df "HJLNRTVZ\^`b𷮥uqiqjh4fUh4fhahyCO6H*j)hyCOhEHU!jG hCJUV^JaJjhyCOUha56^JhyCO56^Jha5CJ H*^JaJ hyCOhyCO6H*hahyCOhyCOH*hyCOhyCO6hyCOhyCO56H*^Jh ha5CJ H*^JaJ ) !>?@BCVWXɹՑՀ{vqq`Sj0h4fh4fEHU!jQG h4fCJUV^JaJ h4f6 h*0&6 h4f5hkj.heLh4fEHU!jgG h4fCJUV^JaJ ha5^J h4f5^Jha5H*^Jh ha5CJ H*^JaJ h4fh4f6hah4fjh4fUj3,h4fhEHU!jG hCJUV^JaJ XY[] !456789LMNOPQde|k!jbG hd-CJUV^JaJj7hd-hd-EHU!j G hd-CJUV^JaJj5hphd-EHU!jG hd-CJUV^JaJjhd-Uhd-j@3h4fh4fEHU!j G h4fCJUV^JaJ h4f6h4fh4f6hah4fjh4fU$B\] jk  H c d | } (!)!^gd gdd-gda^gdaefghik         3 4 5 6 = F K L M O R a g h i k l { ὰ្ٌٌՌ h ^Jj?hph EHU!j0G h CJUV^JaJj<hd-h EHU!j(G h CJUV^JaJ hd-hd-hph hahd- hd-6jhd-Ujw:hd-hd-EHU4 ! !&!)!*!=!>!?!@!B!D!y!z!!!!!!!!!"$"&","."T"V"µ䤗䓋z!jؿG hr'_CJUV^JaJjhr'_Uhr'_jgDh*0&h*0&EHU!j6G h*0&CJUV^JaJjAhphpEHU!jaG hpCJUV^JaJjh*0&U h*0&6hah*0&hd-hph  h ^J hp^J*)!C!D!!!"&"\"l""#T#j#0$1$n$$$$$$$%^gdgdQ*0gd ^gdr'_gda ^gd*0&gda^gdaV"X"Z"^"`"b"f"j"""""#### ####:#<#>#@#B#D#H#N#j#l##$0$1$2$E$F$⹬▉⅁恅yh!jG hpCJUV^JaJjhQ*0UhQ*0hjmKhr'_hr'_EHU!jG hr'_CJUV^JaJ hr'_6jHhr'_hr'_EHU!jjG hr'_CJUV^JaJh ha5CJ H*^JaJ hr'_hr'_6hr'_hpjhr'_UjFhr'_hr'_EHU$F$G$H$I$L$M$N$O$Q$S$T$V$W$X$Y$^$_$`$b$g$h$i$k$l$s$t$u$v$w$|$}$~$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ hQ*0^JhyvhQ*0hpjhQ*0UjNhd-hpEHUR$%%%%%%%%%%%%%&&&&N&O&Q&R&e&f&g&h&j&m& '':';'D'E'z''''''''''''''ŻŻŻշͫͷ͉͉yhX h|H*h|hYRh`jFShKhKEHU!jĘG hKCJUV^JaJjhKUhyvhKhKhQ*06H*hKhQ*06hhphQ*0jhQ*0UjPhQ*0hpEHU!jG hpCJUV^JaJ/%%%%%%P&Q&k&l&;'<'F'G'H'U'$$Ifa$gdX l gdYRgdK^gdQ*0p^pgd^gdU'V'W'X'^'i'o'$$Ifa$gdX l ikdU$$IfTlF t    44 laytX To'p't'y'z'''''oYFYYYYY$Ifgd?l $$Ifa$gdX l kdV$$IfTlrW  t44 laytX T''''''''XkdV$$IfTl4rW  t44 laytX T$$Ifa$gdX l ''''''''''''$$Ifa$gdX l $Ifgd?l ''''''''''''''''W(X(Y(b(c(e(((((((((((((((((((((( ) )r)s))轰蟒jh|UjgZhKh EHU!jG h CJUV^JaJjXhKhKEHU!jQƘG hKCJUV^JaJjhKUh`hK6H*h`hK6hKhyvhX h|H*h`h|/''''(((((( )nfaaaXXXXX^gdgdK$a$gd`kdbW$$IfTl4rW   t44 laytX T ) )))q)r)))))++++=+>+W+X+k+l+++++^gd ^gd?gd ^gd|gd^gdK^gd))))))))))* *+++++++++ +3+4+5+6+7+<+>+?+R+S+չըՓzm\!jљG h?CJUV^JaJj/bh?h?EHU!jG h?CJUV^JaJjh?Uh?hOj`_hyvhyvEHU!jG hyvCJUV^JaJjh Uhdh h 6h|hh jh|Uj\hKh|EHU!jƘG h|CJUV^JaJS+T+U+l+m++++++++++++++++++++, ,, ,A,D,E,X,Y,ɼګڍ|xpx_!jG hYRCJUV^JaJhdhd6hdhyvjFnhOhOEHU!jG hOCJUV^JaJjkhOhOEHU!j$G hOCJUV^JaJj-hhOhOEHU!jG hOCJUV^JaJhOjhOUh?jh?Ujseh?h?EHU +++C,D,],^,B-C-w-x-z{./GH^gdI^gdPNgdPN^gdYR^gd^gdYR^gdO^gdOgd^gd?Y,Z,[,\,],^,_,|,},,,,,,,@-A-C-D-W-X-Y-Z-[-\-]-^-`-a-b-f-g-h-l-m-n-v-w-y--mp{|泦Ģ梚梐բu!jG hPNCJUV^JaJjhPNUU hPN6 hPNH*hPNhPNH*hPNjuthYRhPNEHU!jG hPNCJUV^JaJjhYRU hd6hdhyv hYR6hhOhYRjhOUjqhOhYREHU.. The summation operator is distributive when the value being operated upon is itself a sum (or difference). For example,  EMBED Equation.3  ,  EMBED Equation.3  Note that the last step in this equation follows from the rule pertaining to summation over a constant, given earlier. Hence,  EMBED Equation.3  Summation involving two or more variables. If each case has values on two variables, Xi and Yi, say, then,  EMBED Equation.3  Instead,  EMBED Equation.3 = (X1Y1 + X2Y2 + """ + XNYN). These rules can be extended, quite logically, to situations involving more than two variables. Exercises In each of the following exercises, use symbols to extend the expression given in the summation operation. For example,  EMBED Equation.3  = (X3 + X4 + X5 + X6 + X7) 1.  EMBED Equation.3  2.  EMBED Equation.3  3.  EMBED Equation.3  4.  EMBED Equation.3  5.  EMBED Equation.3  6.  EMBED Equation.3  Reduce the following extended expressions to an expression using the summation operator. For example, (X3 + X4 + X5 + X6 + X7) would be written as  EMBED Equation.3  7. X1Y1 + X2Y2 + X3Y3 + X4Y4 + X5Y5 + X6Y6 8. X1Y1 + X2Y2 + '-./0CDEFHJsµ۱wmfwmbZjhLEUhd hvhvhvhv6H*hvhv6hhv6j*~hIhiEHU!jbG hiCJUV^JaJjhIUhIhvjyhhvEHU!j}G hvCJUV^JaJjh&Uh& hPNhPNhPNjhPNUjwhPNhPNEHU#HI΍ύٍڍ "*+,GHcdp^pgd[fgds]o^gds]o^gds]ogd^gdvgdvʍˍ͍̍΍ڍۍ "$ εΠΠ΁zvnhs]ohs]o5h hs]ohs]o hs]o^J hs]o6H*hs]ohs]oH*hs]ohs]o6H*hs]ohs]o6jhs]ohOEHU!jYG hOCJUV^JaJjhs]oUhs]ohvjhLEUjMhLEhs]oEHU!jPG hs]oCJUV^JaJhLE$  "#%'(*+/0CDEFGKL_`abgh{|ͼͼͼͼոէոtc!jG h[fCJUV^JaJjh[fh[fEHU!jfG h[fCJUV^JaJjh[fUjhs]oh[fEHU!jG h[fCJUV^JaJh[f hs]o6H*hs]ohs]o6H*hs]ohs]o6hs]ojhs]oUjDhs]ohs]oEHU!jG hs]oCJUV^JaJ%|}~ϐАѐҐӐBCDHIMNRSWXyldZdZdZdZdZhh~Yhh~Y6H*hh~Yhh~Y6jthh~Yhh~YEHU!j֯G hh~YCJUV^JaJhh~Yjhh~YUh$jhwIh$EHU!jXG h$CJUV^JaJjhwIUhwIjh$h$EHU!jG h$CJUV^JaJh[fhvjh[fUjwh[fh[fEHU#dӐԐՐ=>Z[qr̠gdh~YgdwIgd[fXuvđőƑǑȑ "$,.02<>Fννννννννννννννh^h^H* h^H*h^ hh~Y^JU hh~Y6H*hh~Yhh~Y6H*hh~Yhh~Y6jhh~Yhh~YEHU!jᰙG hh~YCJUV^JaJjhh~YUhh~Y@""" + XNYN 9. (X5 + Y5)2 + (X6 + Y6)2 + (X7 + Y7)2 + """ + (X10 + Y10)2 10.  EMBED Equation.3  FHJLVX`bdflv|~ĠƠȠʠ̠Ω h^h^juh^h^EHU!jHG h^CJUV^JaJjh^U h^H* hh~YH*h^h^H* h^^Jh^h^H*h^ h^H*,1h/ =!"#$% dDd b  c $A? ?3"`?2ruVdJ*=D`!ruVdJ*=@@2Pxcdd``ed``baV d,FYzP1n:&! KA?H1>17T obIFHeA*ݿ;aR& Ma`QaǒXk!4oB]j muɜi%<;*u y _ gVBCXa`ZZ{xnoa#I`'<=`~&ɉ‡<{2! )l4 -F&&\o @ ]` "XdDd b  c $A? ?3"`?2ruVdJ*=`!ruVdJ*=@@2Pxcdd``ed``baV d,FYzP1n:&! KA?H1>17T obIFHeA*ݿ;aR& Ma`QaǒXk!4oB]j muɜi%<;*u y _ gVBCXa`ZZ{xnoa#I`'<=`~&ɉ‡<{2! )l4 -F&&\o @ ]` "XdDd b  c $A? ?3"`?2ruVdJ*= `!ruVdJ*=@@2Pxcdd``ed``baV d,FYzP1n:&! KA?H1>17T obIFHeA*ݿ;aR& Ma`QaǒXk!4oB]j muɜi%<;*u y _ gVBCXa`ZZ{xnoa#I`'<=`~&ɉ‡<{2! )l4 -F&&\o @ ]` "X6Dd b  c $A? ?3"`?29Eau\p`!T9Eau`@2P"xcdd`` @c112BYL%bpu17T obIFHeA*ݿ;aR& Ma`QaǒXk!4oB]j muɜi%<;*u y _ gVBCXa`ZZ{xnoa#I`'<=`~&ɉ‡<{2! )l4 -F&&\o @ ]` "XDd b  c $A? ?3"`?2iFgsF)3E `!=FgsF)3 d(+ xcdd``> @c112BYL%bpubSdv dF\?@ZVBk3$4/$37X/\!(?71y LL "*l:.v0o8L+KRsvePdh,s;.Ff 1NcdDd b  c $A? ?3"`?2ruVdJ*=)`!ruVdJ*=@@2Pxcdd``ed``baV d,FYzP1n:&! KA?H1>17T obIFHeA*ݿ;aR& Ma`QaǒXk!4oB]j muɜi%<;*u y _ gVBCXa`ZZ{xnoa#I`'<=`~&ɉ‡  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRTUVWXYZ]`abdcegfhijlkmonpqrtsuwvxyz|{}~Root Entry FGR_@^Data SWordDocument.ObjectPoolTahGRGR_1201165397FahGRahGROle CompObjfObjInfo  #&),/258;>ADGJMPSVY\_behknqrsvy| FMicrosoft Equation 3.0 DS Equation Equation.39q7B X ii=1n=5 " FMicrosoft Equation 3.0 DS EqEquation Native ^_1201166649 FahGRahGROle CompObj fuation Equation.39q7.: X ii " FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo Equation Native  J_1201167004FahGRahGROle  CompObj fObjInfoEquation Native B_1201168071 FahGRahGR7&P$l X  " FMicrosoft Equation 3.0 DS Equation Equation.39q7\& X ijin j " j3 "Ole CompObjfObjInfoEquation Native x_1201168613FahGRahGROle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q7?)$ X ij "  " FMicrosoft Equation 3.0 DS EqEquation Native [_1201168914FahGRahGROle CompObj fuation Equation.39q7Oh X ij " []  " FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo!!Equation Native "k_1201170354@$FahGRahGROle $CompObj#%%fObjInfo&'Equation Native (y_1201258252)FahGRahGR7]l X ijki " j " k " FMicrosoft Equation 3.0 DS Equation Equation.39q6C8 4S X "  =X ii "Ole *CompObj(*+fObjInfo+-Equation Native .__1201171644;.FahGRahGROle 0CompObj-/1fObjInfo03 FMicrosoft Equation 3.0 DS Equation Equation.39q75P X i12i " FMicrosoft Equation 3.0 DS EqEquation Native 4Q_1201269130^3FahGRahGROle 6CompObj247fuation Equation.39q6K &  X i1ki " k " FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo59Equation Native :g_12012691468FahGRahGROle <CompObj79=fObjInfo:?Equation Native @g_1201190737=FahGRahGR6K؆: X ij1i " j " FMicrosoft Equation 3.0 DS Equation Equation.39q7:' c=Nc i=1N "Ole BCompObj<>CfObjInfo?EEquation Native FV_1201190923,JBFahGRahGROle HCompObjACIfObjInfoDK FMicrosoft Equation 3.0 DS Equation Equation.39q7R; c=125=60 i=112 " FMicrosoft Equation 3.0 DS EqEquation Native Ln_1201191164GFahGRahGROle NCompObjFHOfuation Equation.39q7/O c  "  " FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoIQEquation Native RK_1201191180EOLFahGRahGROle TCompObjKMUfObjInfoNWEquation Native Xx_1201191266QFahGRahGR7\Xlf c in j " [] jJ " FMicrosoft Equation 3.0 DS Equation Equation.39q77-g n jjJ " cOle ZCompObjPR[fObjInfoS]Equation Native ^S_1201191720"VFahGRahGROle `CompObjUWafObjInfoXc FMicrosoft Equation 3.0 DS Equation Equation.39q7Px3" 10 in j " j3 " FMicrosoft Equation 3.0 DS EqEquation Native dl_1201191728[FahGRahGROle fCompObjZ\gfuation Equation.39q7`ya 10 in j " [] j3 " FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo]iEquation Native j|_12012693456|`FahGRahGROle lCompObj_amfObjInfoboEquation Native p_1201192758YheFahGRahGR6© cX iiN " =cX iiN " ()=cX iiN " FMicrosoft Equation 3.0 DS Equation Equation.39qOle tCompObjdfufObjInfogwEquation Native xM71H%" 2X ii " FMicrosoft Equation 3.0 DS Equation Equation.39q71ȳl 2X ii "_1201192920jFahGRahGROle zCompObjik{fObjInfol}Equation Native ~M_1201193066coFahGRahGROle CompObjnpf FMicrosoft Equation 3.0 DS Equation Equation.39q7ÎXz x cX iji " j " =cX iji " j "ObjInfoqEquation Native _1201193130tFahGRahGROle CompObjsufObjInfovEquation Native _1201269640yFahGRahGR FMicrosoft Equation 3.0 DS Equation Equation.39q7ú0l cX ijki " j " k " =cX iji " j " k "Ole CompObjxzfObjInfo{Equation Native  FMicrosoft Equation 3.0 DS Equation Equation.39q6j, 5(X ij ) in j " j3 " FMicrosoft Equation 3.0 DS Eq_1201269767w~FahGRahGROle CompObj}fObjInfouation Equation.39q6v(" (X ij ) in j " j3 " () FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native _1201194222rFahGRjGROle CompObjfObjInfoEquation Native [_1201194577FjGRjGROle 7?P c ii " X i FMicrosoft Equation 3.0 DS Equation Equation.39q7o& c jCompObjfObjInfoEquation Native _1201247624mFjGRjGR X ijki " j " k " FMicrosoft Equation 3.0 DS Equation Equation.39q7( c j X ijki " () Ole CompObjfObjInfoEquation Native j " k " FMicrosoft Equation 3.0 DS Equation Equation.39q7oXs,N c j X ijki " j " k "_1201194707FjGRjGROle CompObjfObjInfoEquation Native _1201270143FjGRjGROle CompObjf FMicrosoft Equation 3.0 DS Equation Equation.39q6«$, X i2i " =(X 12 +X 22 +X 32 +"X N2 )ObjInfoEquation Native _1201248445FjGRjGROle  FMicrosoft Equation 3.0 DS Equation Equation.39q7×`g  X ii " = X 1 + X 2 +"+ X NCompObjfObjInfoEquation Native _1201248721FjGRjGROle CompObjfObjInfoEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39q7ç@c logX ii " =logX 1 +logX 2 +"+logX N_1201248930FjGRjGROle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q7ê( X ii " () 2 =(X 1 +X 2 +"+X 3 ) 2Equation Native _1201249060FjGRjGROle CompObjf FMicrosoft Equation 3.0 DS Equation Equation.39q7ØXwT  X ii "  = X 1 + X 2 +"+ X N FMicrosoft Equation 3.0 DS EqObjInfoEquation Native _1201249189FjGRjGROle CompObjfObjInfoEquation Native _1201250312FjGRjGRuation Equation.39q7ã8+ logX ii " ()=log(X 1 +X 2 +"X N ) FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjfObjInfoEquation Native H7,@<d! X ii " () j=1J "  2 =X 1j +X 2j +"+X nj () j "  2 =X " j2j "_1201250335FjGRjGROle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q7lG> X ii " () j=1J "  2 FMicrosoft Equation 3.0 DS EqEquation Native _12012508341FjGRjGROle CompObjfuation Equation.39q7æ 8 X i +Y i () i " =X ii " +Y ii " FMicrosoft Equation 3.0 DS EqObjInfoEquation Native _1201252221FjGRjGROle CompObjfObjInfoEquation Native \_1201272930FjGRjGR     !$%&),/258;>ADEHKLORSVYZ]`cfghiklmnopqstuvxuation Equation.39q7@9 X ij "2X " "  () i " jJ "  2 =X ij2 "22X " "  X ij +2X " " 2 () i " jJ " =X ij2i " jJ " "22X " "  X iji " jJ " +2X " " 2i " jJ " =X ij2i " jJ " "22X " "  X iji " jJ " +2X " " 2 n jjJ "Ole CompObjfObjInfoEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39q6| X " " 2i " jJ " =X " " 2 + in 1 " X " " 2 + in 2 " "+X " " 2in J " ()=n 1 X " " 2 +n 2 X " " 2 +"+n J X " " 2 ()=n jjJ " X " " 2 =X " " 2 n jjJ "<{2! )l4 -F&&\o @ ]` "X$$If<!vh5 5\ 5 #v #v\ #v :V l t!5 5\ 5 / a<ytX $$If<!vh5 5\ 5 #v #v\ #v :V l t!5 5\ 5 / a<ytX gDd b  c $A? ?3"`?2<_=a Յ0`!<_=a Յ0L (+Sxcdd``ed``baV d,FYzP1n:, lB@?b y@0&dT20|_&,eBܤ7Iv,\~ NjmȠ T$@ZV\L>|H3M-VK-WMa>#NJ,c sI2j.#\}`sgf?& τEYAQqG@5O=YPsQ1L₦.pJG4 0y{iI)$5a#\E.fV6Dd b   c $A? ?3"`?2U*-۴R;U;0\`!TU*-۴R;U;0L (+"xcdd`` @c112BYL%bpu|0A,@C2sSRsp?a`ey=3 bn?e5w?f Y\P$#t] `p321)W2ԡ"|b@3X? d$$If<!vh5 5\ 5 #v #v\ #v :V l t!5 5\ 5 / a<ytX $$If<!vh5 5\ 5 #v #v\ #v :V l t!5 5\ 5 / a<ytX 6Dd b   c $A? ?3"`? 2U*-۴R;U;0\B`!TU*-۴R;U;0L (+"xcdd`` @c112BYL%bpu|0A,@C2sSRsp?a`ey=3 bn?e5w?f Y\P$#t] `p321)W2ԡ"|b@3X? djDd  b   c $A? ?3"`? 2c 0Mx`!c 0M`hn 8Vxcdd``dd``baV d,FYzP1n:&v! KA?H1Z Yzjx|K2B* RAvf: KP 27)?!C‚B ;|.? _gk dF\ O+a5L>̷TJm>#G d/=,o(?#^Wpkz%ԝ aZZ3 dӁy;+(2OdBD4616b;F+KRsA2u(2tA4T}b@3Xo?$$If!vh555#v#v#v:V l t555/ / ytX T$$If!vh55555#v#v#v:V l t555/ / ytX T$$If!vh55555#v#v#v:V l4 t+,555/ ytX T$$If!vh55555#v#v#v:V l4 t+,555/ ytX TSDd (0b   c $A? ?3"`? 2\QKQ_̤y^`!q\QKQ_̤<@ k?xcdd``cd``baV d,FYzP1n:f! KA?H1Zqu2@0&dT20|_&,eBܤ,IMv,\~ =d*`ќi%ńʿ CS$/$37X/\!(?71} ^~S`{+a`{V?g`z geB\<0@]uƋ " F=cdbR ,.Iex2C D|b@53X?Bm^Dd b   c $A? ?3"`? 2$ `H  `!|$ `H ``hn :Jxcdd``ed``baV d,FYzP1n:v! KA?H1Z |@=P5< %! vf: KP 27)?G!K>bi_X u9O O+!5e 7n12ps00;s'\F Lݎa.HhnHfnj_jBP~nbgv`Y g1.ك= "*a \ b;v+KRsaPdk1 dDd b  c $A? ?3"`? 2ruVdJ*=#`!ruVdJ*=@@2Pxcdd``ed``baV d,FYzP1n:&! KA?H1>17T obIFHeA*ݿ;aR& Ma`QaǒXk!4oB]j muɜi%<;*u y _ gVBCXa`ZZ{xnoa#I`'<=`~&ɉ‡<{2! )l4 -F&&\o @ ]` "X$$If!vh555#v#v#v:V l t555/ ytX T$$If!vh55555#v#v#v:V l t555/ / ytX T$$If!vh55555#v#v#v:V l4 t+,555/ ytX T$$If!vh55555#v#v#v:V l4 t+,555/ ytX TBDd b  c $A ? ?3"`?243Cѷh'`!`43CѷR``\:.xڥQ=KA}3{~ JED¤4‰"$p\ej+B /^ "z~$b{ f{; `,B 9D\A4fƼ* NְRH^H<`VIT5x4\|pMY''s`> q{3%,X:MEž}]ҧEHOzh%q':谝4_zԃm6)[||iяJj[}}LȺ0e;3Sb^n պe\~e|(s0TDd $b  c $A ? ?3"`?2.;ӈMz#*`!r.;ӈM`HD P@xcdd``vdd``baV d,FYzP1n:lB@?b ؀f@0&dT20<_&,eBܤS\\|bرsA V0Zn#4AHqd2gi}Z s1AM-VK-WMa>#؞J؞L( +<(+D⎆乃F;0c?va%41 6W `p`dbR ,.IeԡRY`{RDd b  c $A ? ?3"`?2Wljݝ`exw,`!pWljݝ`er` >xcdd``dd``baV d,FYzP1n:6! KA?H1Z l ,@=P5< %! vf: KP 27)?^+}G!K>b|i@t9O&s֧0>1_?|( br<32p{Yɪ7ɄEYAAy2@eڻ3Ȅ,t0Pɬ # Aи``㜑I)$5!d.P"CDHg!t?0ezSDd (0b  c $A? ?3"`?2\QKQ_̤y.`!q\QKQ_̤<@ k?xcdd``cd``baV d,FYzP1n:f! KA?H1Zqu2@0&dT20|_&,eBܤ,IMv,\~ =d*`ќi%ńʿ CS$/$37X/\!(?71} ^~S`{+a`{V?g`z geB\<0@]uƋ " F=cdbR ,.Iex2C D|b@53X?BmhDd b  c $A ? ?3"`?2 l `8@r1`! l `8@r`@ Txcdd``dd``baV d,FYzP1n:&6! KA?H1 l zjx|K2B* Rt/ @2@penR~C8ףv,\~ Ne*F\7@ZVBS˼8q.N12ps3k 5w>NPsE$47$37X/\!(?71Hb v_%`~&I,$ 3LLN=إ\Na`ciGNLLJ% s: @> 1,WqDd b  c $A ? ?3"`?2m  D@"T3`!m  D@"T @ txcdd``> @c112BYL%bpu 1X +|-gJd*F\7@ZVBS<#M12ps3k 5ws'\F LLsa; 8_^0 M-VK-WM#?:@Qgh8 {0g NJf0 6p 1R#aglM\&h``#RpeqIj.C%\E.~$Dd b  c $A? ?3"`?2nƥ'LsGJ 6`!Bƥ'LsG`\(+xcdd`` @c112BYL%bpu#g-f ɕ\Pp}6b;FLLJ% s: @> 1,[NDd  b  c $A? ?3"`?2e MD6 `h|`08`!e MD6 `h|`* ~ 8wxcdd`` @c112BYL%bpu/m>#Gٌd/=,o(7#^Wpkz%ԝ aZZ3 d3$8<||F>f?.gBݑ@;+q3^xjX@&$WrAS81M%UF&&\? @ ]` bg!t?0"SzFDd b  c $A? ?3"`?22}%Zpl:`!d2}%Zp@`P2xcdd``ed``baV d,FYzP1n:! KA?H10017T obIFHeA*ݿ0l+ @Hfnj_jBP~nbÎ%r3ɴ@t9WS03Hgr%UBWO 3|$/=YPL>a+ssKs,(_{d2 \fKF&&\ s:@ĞB ~dl|h^Dd pb  c $A? ?3"`?2X"LAqI$"@RK=`!|X"LAqI$"@RK(+Jxcdd``Ned``baV d,FYzP1n:, B@?b t/ @2@penR~mI.v,\~ ׋Ƞ T$@ZV\L>|H3M-VK-WMa>#NJ,c 3j.#LۘAAjh^Fܫ,1.y0M`~&2E5&\X p0A\NKLLJ% {@2u(2t5= cDd 8 b  c $A? ?3"`?2rkSD/jsj_?`!rkSD/jsjrv 8xcdd`` @c112BYL%bpu 1X +|-"f4AHqY23>0A2 32p{ﱂ bV  WῄS?;X M-VK-WMw2BxOzCt2R^`f@ʌ ~L>τڛ@f :)L =4Ĥ\Y\`2C 2i"yDd b  c $A? ?3"`?2g+U!-|'A`!g+U!-|'В xڕ;H@ǿMM"V⤈Nv ND,+PҩSwA.Vp1V/|{%H~w.B%0 N¶m(Fķ\I4Bl*pDǤ-R2+Yk2bK52^XSN{sjR aHRqkY*>ܯo^Sо6UN1'|JUMkA|o9~{2uS*:E(iv}u QGM'*1. L R[ū_T}z8u=(!grɐw:W0ꅜ*nn8bk@at x*W :t]@2`'PU/3H{5BDd b  c $A? ?3"`?2@L"&hD`!`@L"&R``\:.xcdd``dd``baV d,FYzP1n:,B@?b Xzjx|K2B* Rt/ @2@penR~.v,\~ h eF\?@ZVBS3$4/$37X/\!(?71y \~`s3+!|&bO?agG& 2P`s*!а``㔑I)$5d.P"CDHg!t?0egv!BDd b  c $A? ?3"`?2dz|OՒ!hF`!`dz|OՒ!R``\:.xڥ1KPwbCRHG?:VX0Vh !7qqEjAqфݽG.MgEʋlF\a֗;HZG×:IbXM 䪣n#xp>7G6ey/@qI:׸=QWdmy3C u <wo~֝GC該Wܲ.}!ј5:Ff.-{ ǚ^̝v[ (F-ܕ Z5  vDd 0b  c $A? ?3"`?2K=؉sЇH /I`!K=؉sЇH `Gknxcdd``.dd``baV d,FYzP1n:>! KA?H1Z |@@=P5< %! vf: KP 27)?]KÎ%r.,@@@ڈ',֧0~.*ߜCo"\|gdX d-\0sA,@sC2sSRsp b.#\~N Y0(7 30U$3 &,8ؿ0y'͝O?f[ {# bC"1F&&\ {:  z1e\Dd | 0b  c $A? ?3"`?2^.;${bK`!^.;${b$`h{kxcdd`` @c112BYL%bpub 1X +|-I+Ƞ T$@ZV\L>|H3M-VK-WMa>#NJ,c 3j.#LL #4+a>_ο b"|' j3c.8*aL(_ه Yw$t[3ۓ)+Ȅ,.hPptAӞ;+KRsA2u(2tA4T}b@3X)\Dd O b   c $A? ?3"`?2OҦL&WP`!OҦL&W 0, 8xcdd``feb``baV d,FYzP1n:&! KA?H1Z +Xzjx|K2B* RAvfjvL@(\UÎ%P4E @@0/gi^ o`Bw@?]J~=9\?*?_\w~FK`sV |= a`ZZ3 0Bblǯq%'\t?`bf%k`"GQϫɧP/(ffAܑ@;͸lOVz8RLf " K΋LLJ% s:@ĞB ~`GʃCDd b ! c $A? ?3"`? 2wƜGm$ 3iS`!awƜGm$ 3``\:/xcdd``ed``baV d,FYzP1n:,B@?b Xzjx|K2B* Rͤ `[YB2sSRsv,\~  e*F\ L, >?}d>F&b_?cd.`aQ 㳃Is/P@penR~[i&B8jOFsVrAC Ҍ`!v 0y{qĤ\Y\ˠ 2C D|b@3X? Hd$$If!vh555#v#v#v:V l t555/ / ytX T$$If!vh55555#v#v#v:V l t555/ / ytX T$$If!vh55555#v#v#v:V l4 t+,555/ ytX T$$If!vh55555#v#v#v:V l4 t+,555/ ytX TbDd 0b " c $A? ?3"`?!2&lӼ?^aIX`!&lӼ?^a\ (kNxڝJPƿ{o$E) ".RlZ#65Vhd"P&{8h<&b¹|wU2eI!L&dEUŒ-}9TCLz "݌)ew;*vFx3f'k ap=*xZ|@n Zǵ 5nPE./aS?X<7Jx!^x=wݠyO@߳[TO턉~dꋌf b)ge~<L&d#/jw%Kȳvk4X1(X[pDd b # c $A? ?3"`?"2E>.-Z+orSnIZ`!E>.-Z+orSnI  xڕJAݘdHXZ)6/Iaael\#$ b#XE 7lL] R80=g!.(!V8 "J}_FdL%¼$mHdx45BHFsG|w7yIb,y;6Е_--@RV Yfz sn ώ3>D0.X~]X{81gw3j*|ByA+~ ?h&YvZr-oIϔy-SqB\a NՎ3SV8қxa 0=/LjJ;s9ɢWo.zBb{~]1_8cDd 0b $ c $A? ?3"`?#2"o{s:A]`!"o{s:\ `kOxcdd``cd``baV d,FYzP1n:B@?b 1@0&dT20|_&,eBܤsB\v,\~ N;T T~23>P>>0@,@B2sSRsp?g>7?7 ο7g``S9"[X: .kG.cl ;Ȅ,3.p\1m-.#RpeqIj. = @ ]` "z0tDd b % c $A? ?3"`?$2$+>qR;9H[7_`!$+>qR;9H[7Z``:xڥO@]A4rVq~@'Fa0qcD#$tpw`Ёg?w jBֻki,BC"(M%%!l(l}v zP@F`@(7tNW.e#A2_j*>y p̒ jYIUeTizI2ZMEcnC7)͌/A;OKfRVp v A?]+9IN>99:IlrYea"r0a!ƥ7hŵ6"q 3v5^'':HZh&ZS(I` :mxF+W"X\cpأޔg,VճDDd \ 0b & c $A? ?3"`?%2%Xpb~+jsb`!b%Xpb~+*`"!k0xڥT=hA~vs=u/&@,$I\a<ĀsxnwUba! b!xZYDQO], a {>h,d,FQ;įmlٗ\ "g$H\ p̒1OKp%u=\lUl Hоci1Ԝ{ek>k4wm{KW5;7~ 75Z_OG;L2MlFk7 L3ERa#x"f-1/rd΃;WE΄ZBhO|?g0^"3o4q5ͅE~uVR $%yiO}Out~#{ב W}QkbZJVW(T'W$ɜ'|$ͣҀ큣suW1sJ8VU9CL mxDd b ' c $A ? ?3"`?&2TچZNG 7 -e`!TچZNG 7 -@`$':xڥ;KPZUPH">@[ 1E 5Y\"EqUAD=7I!"jBro. (*8@Wx=3K+v d;SBO"  Q'Ь&Bi-M^~+K떹cLecζ [c 7V1a/:wqR|_˷| >Uvd0<ց_he61blګf$+y5nB_hvc-ۛmF;>~;M+V[|zqMkf4ǩ1[. dBLښz4ygGG\ҥ A+?FDd p b ) c $A"? ?3"`?(2S49`X# %<lDk`!dS49`X# %<*S!2xڥTMhA~vtwn(aU(ZEHZJ7ҵ K{Cz(="zPVxВ3vf1톁}0u00o U(d4bIt9ch_J\\_ \7a$}uʼnu@mKSC_d̾=~\)>Ur Y_ ,aί{9a1ܓG U#v!+Ѧ|JhDGLO_=Hɐ۲Dd b * c $A#? ?3"`?)2" znWs n`! znWs N!xڥ;KAgNMhh@)B bI)l#0^b1ؤ#(bea T/DϝAb>ػ-3-/2@W^DȘڶ-q-Z<ԫ`G~Mɠs>1V H<+fKb G"T:3`prP.aL<FV\xÚA*)z%{(cLWf@3a͓\oP6$6+<#4k}/{s|"; qorvOٿp /Wdk5_HČFBi=~ؿo}^6{4ocuuׯUsxF!n7HR~|~v $sw}}4k6e~_ox)toLMX Ǧ t`!wYY>X Ǧ x `\xcdd``> @c112BYL%bpu%`UZ]HFAQЃA)&@bA Ex_ijKeswsk~~w !KMiKT p6B@9sq.=[VQeI0hVϴKEhҭiM^!jqh)Ϫ] HVejA|S@yz륍u/zdkvlyέ"{5IG %$=̟Z7yKHCI^gv9mJt~6 ˸ckP_kǫ}E3>C>nKDm^q>UsRՙA6Yl75$$[ ']O3rJ9.r>q/L2t:)+y݌9.*`|[/PNoHzLYwzNy|TQjz2U@^oίכc^?X ֫:`֛aoB}-8L"o|1wǃ0p9 B1,E C,B,B3? ?6_wyykwڤpZjQ_TB^_]bpO[fG!xCgIZ X ij2 +Y j () i2 " j3 " FMicrosoft Equation 3.0 DS EqObjInfoBEquation Native C_1201254172FjGRjGROle FCompObjGfObjInfoIEquation Native J_1201253976FjGRjGRuation Equation.39q7s`\ 5X i +Y i +Z i () i5 " FMicrosoft Equation 3.0 DS Equation Equation.39qOle MCompObj