*anoue I. I?eiyoea ?aoaiee a oneiaeyo iaii?aaeaeaiiinoe.
Aa?eaio 15.
( 0 , 1/2 ) ( 6 , 1/4 ) ( 5 , 1/5 ) ( 2 , 1/20 )
( 6 , 1/2 ) ( 2 , 1/4 ) ( 8 , 1/5 ) ( 22 , 1/20 )
( 9 , 1/2 ) ( 4 , 1/4 ) ( 3 , 1/8 ) ( 32 , 1/8 )
( -6 , 1/2 ) ( -4 , 1/4 ) ( -12 , 1/8 ) ( 10 , 1/8 )
A yoeo no?ieao iioneaai ae?iae:
( 0 6 5 2 )
( 6 2 8 22)
( 9 4 3 32)
( -6 -4 -12 10)
Iieo/aiiua no?iee iauaaeeiyai a iao?eoeo:
0 6 5 2
6 2 8 22
9 4 3 32
-6 -4 -12 10
?j = ( 1/2 1/4 1/5 1/20 )
?oeiaiaeeoaeue, iaiaaeaea?, iaycai ?ac?aoaoue i?iaeaiu, anoathuea ia?aae
iei, ia?aae eieeaeoeaii, eioi?ui ii ?oeiaiaeeo. Ii iaycai i?eieiaoue
?aoaiey. A oai?ee i?eiyoey ?aoaiee anoue niaoeeaeueiue oa?iei: EI? —
Eeoei, I?eieiathuaa ?aoaiey. Ieaea ii oaenoo aoaeai eniieueciaaoue yoio
oa?iei.
I?eiyoue ?aoaiea — yoi ?aoeoue iaeioi?oth yeno?aiaeueioth caaea/o, o.a.
iaeoe yeno?aioi iaeioi?ie ooieoeee, eioi?oth iacuaatho oeaeaaie, i?e
iaeioi?uo ia?aie/aieyo. Iai?eia?, eeiaeiia i?ia?aiie?iaaiea
i?aaenoaaeyao oeaeue eeann oaeeo yeno?aiaeueiuo caaea/. Iaoiaeu oai?ee
aa?iyoiinoae e iaoaiaoe/aneie noaoenoeee iiiiaatho i?eieiaoue ?aoaiey a
oneiaeyo iaii?aaeaeaiiinoe.
Ia ana neo/aeiia iiaeii “ecia?eoue” aa?iyoiinoueth. Iaii?aaeaeaiiinoue —
aieaa oe?ieia iiiyoea. Iaii?aaeaeaiiinoue oiai, eaeie oeeo?ie aaa?o
eyaeao ea?aeueiue eoaee, ioee/aaony io iaii?aaeaeaiiinoe oiai, eaeiai
aoaeao ninoiyiea ?inneeneie yeiiiieee /a?ac 15 eao. E?aoei aiai?y,
oieeaeueiua aaeeie/iua neo/aeiua yaeaiey naycaiu n iaii?aaeaeaiiinoueth,
ianniaua neo/aeiua yaeaiey iaycaoaeueii aeiioneatho iaeioi?ua
caeiiiia?iinoe aa?iyoiinoiiai oa?aeoa?a.
I?aaeiieiaeei, /oi EI? ?anniao?eaaao ianeieueei aiciiaeiuo ?aoaiee i =
1,…, m. Neooaoeey ia ii?aaeaeaia, iiiyoii eeoue, /oi iaee/anoaoao
eaeie-oi ec aa?eaioia 1/4 = 1,…, n. Anee aoaeao i?eiyoi i-a ?aoaiea,
a neooaoeey anoue j-y, oi oe?ia, aicaeaaeyaiay EI?, iieo/eo aeioiae qij.
Iao?eoea Q = (qij) iacuaaaony iao?eoeae iineaaenoaee (aiciiaeiuo
?aoaiee). Eaeia aea ?aoaiea ioaeii i?eiyoue EI?? A yoie neooaoeee iieiie
iaii?aaeaeaiiinoe iiaoo auoue auneacaiu eeoue iaeioi?ua ?aeiiaiaeaoeee
i?aaeaa?eoaeueiiai oa?aeoa?a. Iie ia iaycaoaeueii aoaeoo i?eiyou EI?.
Iiiaia aoaeao caaenaoue io aai neeiiiinoe e ?eneo. Ii eae ioeaieoue ?ene
a aeaiiie noaia?
Aeiionoei, iu oioei ioeaieoue ?ene, eioi?ue ianao i-a ?aoaiea. Iai
iaecaanoia ?aaeueiay neooaoeey. Ii anee au aa ciaee, oi aua?aee au
iaeeo/oaa ?aoaiea, o.a. i?eiinyuaa iaeaieueoee aeioiae. Eia/a aiai?y,
anee neooaoeey anoue j-y, oi auei au i?eiyoi ?aoaiea, aeathuaa aeioiae
qj = max qij. Cia/eo, i?eieiay i-a ?aoaiea, i
iu ?eneoai
iieo/eoue ia qj, a oieueei qij, cia/eo, i?eiyoea i-ai ?aoaiey ianao ?ene
iaaeia?aoue rij = qj – qij. Iao?eoea R = (rij) iacuaaaony iao?eoeae
?eneia.
Ionoue iao?eoea iineaaenoaee anoue Q.
max
0 6 5 2 5
Q = 6 2 8 22 22
9 4 3 32 32
-6 -4 -12 10 10
Ninoaaei iao?eoeo ?eneia R. Eiaai q1 = 5, q2 = 22, q3 = 32, q4 = 10.
Neaaeiaaoaeueii, iao?eoea ?eneia anoue R.
9 0 3 30
R = 3 4 0 10
0 2 5 0
15 10 20 22
Caeanue iu aia?aua ano?aoeeenue n eiee/anoaaiiie ioeaieie ?enea.
Ianiiiaiii, /oi ?ene — iaeia ec aaaeiaeoeo eaoaai?ee
i?aaei?eieiaoaeueneie aeayoaeueiinoe, iaiouaieaiay /a?oa yoie
aeayoaeueiinoe. Eae ecaanoii, i?aaei?eieiaoaee aeeaoo a n?aaeiai eo/oa,
/ai inoaeueiay /anoue /aeiaa/anoaa. Yoi — iaa?aaea ei ca ?ene a iaeei
ian/anoiue aeaiue ieacaoueny ?aci?aiiui. ?ene — iiiyoea iiiaia?aiiia e
iu aua ia ?ac ano?aoeiny n iei.
I?eiyoea ?aoaiee a oneiaeyo iieiie iaii?aaeaeaiiinoe.
I?e i?eiyoee ?aoaiee a oneiaeyo iieiie iaii?aaeaeaiiinoe iaeioi?uie
i?eaioe?aie iiaoo neoaeeoue neaaeothuea i?aaeea-?aeiiaiaeaoeee.
I?aaeei Aaeueaea (i?aaeei e?aeiaai ianneiecia). ?anniao?eaay i-a
?aoaiea, aoaeai iieaaaoue, /oi ia naiii aeaea neooaoeey neeaaeuaaaony
naiay ieioay, o.a. i?eiinyuay naiue iaeue aeioiae ai = min qij. Ii
oaia?ue oaea auaa?ai ?aoaiea i0 n iaeaieueoei j ai0. Eoae,
i?aaeei Aaeueaea ?aeiiaiaeoao i?eiyoue ?aoaiea i0 oaeia, /oi ai0 = max =
max (min qij).
i j
min
0 6 5 2 0
Q = 6 2 8 22 2
9 4 3 32 3
-6 -4 -12 10 -12
Oae, a auoaoeacaiiii i?eia?a eiaai a1 = 0, a2 =2, a3 = 3, a4 = -12.
Oaia?ue ec /enae 0, 2, 3, -12 iaoiaeei iaeneiaeueiia. Yoi — 3. Cia/eo,
i?aaeea Aaeueaea ?aeiiaiaeoao i?eiyoue 3-a ?aoaiea. Aeaiiiio i?aaeeo
neaaeoao /aeiaae, aiyueeny ?enea.
I?aaeei Nyaeaeaea (i?aaeei ieieiaeueiiai ?enea). Aeaiiiio i?aaeeo
neaaeoao /aeiaae, aiyueeny ?enea. I?e i?eiaiaiee yoiai i?aaeea
aiaeece?oaony iao?eoea ?eneia R = (rij). ?anniao?eaay i-a ?aoaiea,
aoaeai iieaaaoue, /oi ia naiii aeaea neeaaeuaaaony neooaoeey
iaeneiaeueiiai ?enea bi = max rij. Ii oaia?ue oaea auaa?ai ?aoaiea i0
j
n iaeiaiueoei bi0. Eoae, i?aaeei Nyaeaeaea ?aeiiaiaeoao i?eiyoue ?aoaiea
i0 oaeia, /oi bi0 = min bi = min (max rij).
max
9 0 3 30 30
R = 3 4 0 10 10
0 2 5 0 5
15 10 20 22 22
Oae, a auoaoeacaiiii i?eia?a eiaai b1 = 30, b2 =10, b3 = 5, b4 = 22.
Oaia?ue ec /enae 30, 10, 5, 22 iaoiaeei ieieiaeueiia. Yoi — 5. Cia/eo,
i?aaeei Nyaeaeaea ?aeiiaiaeoao i?eiyoue 3-a ?aoaiea.
I?aaeei “?iciaiai iioeiecia”. EI? n/eoaao, /oi aeey iaai neiaeeony naiay
aeaaii?eyoiay neooaoeey, o.a. ii iieo/eo naiue aieueoie aeioiae a
?acoeueoaoa naiae aeayoaeueiinoe ci = max qij. Oaia?ue
j
auaa?ai ?aoaiea i0 n iaeaieueoei ci0. Eoae, i?aaeei “?iciaiai iioeiecia
?aeiiaiaeoao i?eiyoue ?aoaiea i0 oaeia, /oi ci0 =
max (max qij).
i j
max
0 6 5 2 6
Q = 6 2 8 22 22
9 4 3 32 32
-6 -4 -12 10 10
Oae, a auoaoeacaiiii i?eia?a eiaai n1 = 6, n2 = 22, n3 = 32, n4 = 10.
Oaia?ue ec /enae 6, 22, 32, 10 aa?ai iaeneiaeueiia. Yoi — 32. Cia/eo,
i?aaeei “?iciaiai iioeiecia” ?aeiiaiaeoao 3-a ?aoaiea.
I?aaeei Ao?aeoea (acaaoeaathuaa ianneienoe/aneee e iioeienoe/aneee
iiaeoiaeu e neooaoeee). I?eieiaaony ?aoaiea i, ia eioi?ii aeinoeaaaony
iaeneioi {l min qij + (1 – l) max qij}, aaea 0 F l F 1.
Cia/aiea l auae?aaony ec noauaeoeaiuo niia?aaeaiee. Anee l i?eaeeaeaaony
e aaeeieoea, oi i?aaeei Ao?aeoea i?eaeeaeaaony e i?aaeeo Aaeueaea, i?e
i?eaeeaeaiee l e ioeth i?aaeei Ao?aeoea i?eaeeaeaaony e i?aaeeo
“?iciaiai iioeiecia”.
Aicueiai l = 1/2.
max
min
0 6 5 2 6
0
Q = 6 2 8 22 22
2
9 4 3 32 32
3
-6 -4 -12 10 10
-12
i1 = 1/2 * 6 + ( 1- 1/2 ) * 0 = 3
i2 = 1/2 * 22 + ( 1 – 1/2 ) * 2 = 12
i3 = 1/2 * 32 + ( 1 – 1/2 ) * 3 = 17.5
i4 = 1/2 * 10 + ( 1 – 1/2 ) * ( -12 ) = -1
Eoae, iu eiaai i1 = 3, i2 = 12, i3 = 17.5, i4 = -1. Oaia?ue ec /enae 3,
12, 17.5, -1 aa?ai iaeneiaeueiia. Yoi — 17.5. Cia/eo, i?aaeei Ao?aeoea
?aeiiaiaeoao 3-a ?aoaiea.
I?eiyoea ?aoaiee a oneiaeyo /anoe/iie iaii?aaeaeaiiinoe.
I?aaeiieiaeei, /oi a ?anniao?eaaaiie noaia ecaanoiu aa?iyoiinoe pj oiai,
/oi ?aaeueiay neooaoeey ?acaeaaaony ii aa?eaioo j. Eiaiii oaeia
iieiaeaiea iacuaaaony /anoe/iie iaii?aaeaeaiiinoueth. Eae caeanue
i?eieiaoue ?aoaiea? Iiaeii aua?aoue iaeii ec neaaeothueo i?aaee.
I?aaeei iaeneiecaoeee n?aaeiaai iaeeaeaaiiai aeioiaea. Aeioiae,
iieo/aaiue oe?iie i?e ?aaeecaoeee i-ai ?aoaiey, yaeyaony
qi1 qin
neo/aeiie aaee/eiie Qi n ?yaeii ?ani?aaeaeaiey . . .
.
p1 pn
Iaoaiaoe/aneia iaeeaeaiea M[Qi] e anoue n?aaeiee iaeeaeaaiue aeioiae,
iaicia/aaiue oaeaea Qi. Eoae, i?aaeei ?aeiiaiaeoao i?eiyoue ?aoaiea,
i?eiinyuaa iaeneiaeueiue n?aaeiee iaeeaeaaiue aeioiae.
A i?eaaaeaiiii i?eia?a aa?iyoiinoe oaeea (1/2, 1/4, 1/5, 1/20).
0 6 5 2
Q = 6 2 8 22
9 4 3 32
-6 -4 -12 10
?j = ( 1/2 1/4 1/5 1/20 )
0 6 5 2
Q1 :
1/2 1/4 1/5 1/20
6 2 8 22
Q2 :
1/2 1/4 1/5 1/20
9 4 3 32
Q3 :
1/2 1/4 1/5 1/20
-6 -4 -12 10
Q4 :
1/2 1/4 1/5 1/20
Q1 = 6/4 + 5/5 + 2/20 = 1,5 + 1 +0,1 = 2,6
Q2 = 6/2 + 2/4 + 8/5 + 22/20 = (30+5+16+11)/10 = 62/10 = 6,2
Q3 = 9/2 + 4/4 + 3/5 + 32/20 = (45+10+6+16)/10 = 77/10 = 7,7
Q4 = – 6/2 – 4/4 – 12/5 + 10/20 = (-30-10-24+5)/10 = – 59/10 = -5,9
Iaeneiaeueiue n?aaeiee iaeeaeaaiue aeioiae ?aaai 7.7, /oi niioaaonoaoao
3-io ?aoaieth.
I?aaeei ieieiecaoeee n?aaeiaai iaeeaeaaiiai ?enea. ?ene oe?iu i?e
?aaeecaoeee i-ai ?aoaiey yaeyaony neo/aeiie aaee/eiie
ri1 rin
Ri n ?yaeii ?ani?aaeaeaiey . . . .
p1
pn
Iaoaiaoe/aneia iaeeaeaiea M[Ri] e anoue n?aaeiee iaeeaeaaiue ?ene,
iaicia/aaiue oaeaea Ri. I?aaeei ?aeiiaiaeoao i?eiyoue ?aoaiea, aeaeouaa
ieieiaeueiue n?aaeiee iaeeaeaaiue ?ene. Au/eneei n?aaeiea iaeeaeaaiua
?enee.
9 0 3 30
R = 3 4 0 10
0 2 5 0
15 10 20 22
?j = ( 1/2 1/4 1/5 1/20 )
9 0 3 30
R1 :
1/2 1/4 1/5 1/20
3 4 0 10
R2 :
1/2 1/4 1/5 1/20
0 2 5 0
R3 :
1/2 1/4 1/5 1/20
15 10 20 22
R4 :
1/2 1/4 1/5 1/20
R1 = 9/2 + 3/5 + 30/20 = (45+6+15)/10 = 66/10 = 6.6
R2 = 3/2 + 4/4 +10/20 = 1.5 + 1 +0.5 = 3
R3 = 2/4 + 5/5 = 15/10 = 1.5
R4 = 15/2 + 10/4 + 20/5 + 22/20 = (150+50+80+22)/20 = 302/20 = 15.1
Ieieiaeueiue n?aaeiee iaeeaeaaiue ?ene ?aaai 1.5, /oi niioaaonoaoao 3-io
?aoaieth.
Eiiaaea a oneiaeyo iieiie iaii?aaeaeaiiinoe i?eiaiyaony neaaeothuaa
i?aaeei.
I?aaeei Eaieana ?aaiiaiciiaeiinoe, eiaaea ana aa?iyoiinoe p n/eoathony
?aaiuie. Iinea yoiai iiaeii aua?aoue eaeia-ieaoaeue ec aeaoo
i?eaaaeaiiuo auoa i?aaee-?aeiiaiaeaoeee i?eiyoey ?aoaiee.
I?aaeei iaeneiecaoeee n?aaeiaai iaeeaeaaiiai aeioiaea.
0 6 5 2
Q = 6 2 8 22
9 4 3 32
-6 -4 -12 10
?j = ( 1/4 1/4 1/4 1/4 )
0 6 5 2
Q1 :
1/4 1/4 1/4 1/4
6 2 8 22
Q2 :
1/4 1/4 1/4 1/4
9 4 3 32
Q3 :
1/4 1/4 1/4 1/4
-6 -4 -12 10
Q4 :
1/4 1/4 1/4 1/4
Q1 = (6+5+2)/4 = 13/4 = 3,25
Q2 = (6+2+8+22)/4 = 38/4 = 9,5
Q3 = (9+4+3+32)/4 = 48/4 =12
Q4 = (-6-4-12+10)/4 = -12/4 = -3
Iaeneiaeueiue n?aaeiee iaeeaeaaiue aeioiae ?aaai 12, /oi niioaaonoaoao
3-io ?aoaieth.
I?aaeei ieieiecaoeee n?aaeiaai iaeeaeaaiiai ?enea.
9 0 3 30
R = 3 4 0 10
0 2 5 0
15 10 20 22
?j = ( 1/2 1/4 1/5 1/20 )
9 0 3 30
R1 :
1/4 1/4 1/4 1/4
3 4 0 10
R2 :
1/4 1/4 1/4 1/4
0 2 5 0
R3 :
1/4 1/4 1/4 1/4
15 10 20 22
R4 :
1/4 1/4 1/4 1/4
R1 = (9+3+30)/4 = 42/4 = 10,5
R2 = (3+4+10)/4 = 17/4 = 4,25
R3 = (2+5)/4 = 7/4 = 1,75
R4 = (15+10+20+22)/4 = 67/4 = 16,75
Ieieiaeueiue n?aaeiee iaeeaeaaiue ?ene ?aaai 1.75, /oi niioaaonoaoao
3-io ?aoaieth.
I?e aeaiiuo aa?iyoiinoyo ninoiyiee oaia?ue o?aaoaony i?iaiaeece?iaaoue
naiaenoai ec 4-o iia?aoeee: eaaeaeay iia?aoeey eiaao aeaa oa?aeoa?enoeee
— n?aaeiee iaeeaeaaiue aeioiae e n?aaeiee iaeeaeaaiue ?ene. Oi/ea (q’,
r’) aeiieie?oao oi/eo (q, r), anee q’?q e r’Fr. Oi/ea, ia
aeiieie?oaiay ieeaeie ae?oaie, iacuaaaony iioeiaeueiie ii Ia?aoi.
Iaianai aeey eaaeaeie iia?aoeee yoe oa?aeoa?enoeee ia ieineoth nenoaio
eii?aeeiao aeey auyaeaiey iia?aoeee, iioeiaeueiie ii Ia?aoi, aeioiae ii
aa?oeeaee e ?ene ii ai?eciioaee.
q 2.6 6.2 7.7 -5.9
r 6.6 3 1.5 15.1
Iieo/ei /aou?a oi/ee. *ai auoa oi/ea (q, r), oai aeioiaeiaa iia?aoeey,
/ai i?aaaa oi/ea, oai aieaa iia ?eneiaay. Cia/eo, ioaeii auae?aoue auoa
e eaaaa. Yoi oi/ea Q3 (7.7, 1.5). Iia yaeyaony iioeiaeueiie ii Ia?aoi,
o.e. aeiieie?oao inoaeueiua oi/ee.
Caoai iaeaeai auioeeoth iaiei/eo iiiaeanoaa iieo/aiiuo oi/ae e aeaaeei
eioa?i?aoaoeeth oi/ae iieo/aiiie auioeeie iaiei/ee.
Oi/ea Q5 iaoiaeeony ia ?aaiuo ?annoiyieyo io oi/ae Q1 e Q4, e
niioaaonoaaiii eiaao eii?aeeiaou (10.9, -1.7). Aiaeiae/ii, oi/ea Q6
?aniieiaeaia iaaeaeo oi/eaie Q1 e Q2 e eiaao eii?aeeiaou (4.8,
4.4).
Aaeanianeee iiaeoiae e i?eiyoeth ?aoaiee.
I?aaeiieiaeei, i?aaei?eieiaoaeue ?acaeoiuaaao iaae aua?inii ia ?uiie
iiaiai ia?niaeoeaiiai oiaa?a. Ii ii ia ciaao, “iieaeao” ee oiaa?. Aeey
ooi/iaiey neooaoeee ii i?iecaiaeeo i?iaioth ia?oeth e niio?eo, eae ii
?aneoiaaony. Iinea yoiai neooaoeey noaiiaeony aieaa ii?aaeaeaiiie, aieaa
i?iaiice?oaiie. Aeey ooi/iaiey yoie neooaoeee iiaeii auionoeoue aua
iaeio i?iaioth ia?oeth e i?iaiaeece?iaaoue eaeea-ieaoaeue ae?oaea
iiiaiou.
. Anee EI? ?aoeo, /oi i?e ooi/iaiee i?iaiay iia?aoeey ii?aaaeuaaaony
(iai?eia?, anee oaaee/aiea n?aaeiaai iaeeaeaaiiai aeioiaea i?aauoaao
cao?aou ia i?iaaaeaiea i?iaiie iia?aoeee), oi ii aa i?iaiaeeo.
0 6 5 2
Q = 6 2 8 22
9 4 3 32
-6 -4 -12 10
?j’ = ( 1/6 1/6 1/3 1/3 )
0 6 5 2
Q1’ :
1/6 1/6 1/3 1/3
6 2 8 22
Q2’ :
1/6 1/6 1/3 1/3
9 4 3 32
Q3’ :
1/6 1/6 1/3 1/3
-6 -4 -12 10
Q4’ :
1/6 1/6 1/3 1/3
Q1‘= 6/6 + 5/3 + 2/3 = 20/6
Q2‘ = 6/6 + 2/6 + 8/3+ 22/3 = 68/6
Q3‘ = 9/6 + 4/6 + 3/3 + 32/3 = 83/6
Q4‘ = – 6/2 – 4/4 – 12/5 + 10/20 = -14/6
Iaeaieueoee aeioiae i?e i?iaiie iia?aoeee aoaeao iieo/ai i?e 3-ai
?aoaiee. Oaia?ue auyniei, noieo ee i?iecaiaeeoue i?iaioth iia?aoeeth,
o.a. iaeaeai ?aciinoue iaaeaeo n?aaeiei iaeeaeaaiui aeioiaeii io
iniiaiie iia?aoeee (ni. I?aaeei iaeneiecaoeee n?aaeiaai iaeeaeaaiiai
aeioiaea) e iieo/aiiuie a ?acoeueoaoa i?iaiie iia?aoeee aeaiiuie, (83/6
– 7,7 = 184/30 = 92/15 @ 6,13). A eoiaa iiaeii neacaoue, /oi noieiinoue
i?iaiie iia?aoeee a aeaiiii i?eia?a ia aeieaeia i?aauoaoue @ 6,13.
Aeey iaoiaeaeaiey eo/oeo iia?aoeee eiiaaea i?eiaiytho iiaeoiaeyuoth
acaaoeaathuoth oi?ioeo, eioi?ay aeey ia? (Q, r) aeaao iaeia /enei, ii
eioi?iio e ii?aaeaeytho eo/ooth iia?aoeeth.
Aeey aiaeeca neooaoeee iiaeii i?eiaieoue acaaoeaathuoth oi?ioeo E(Q, r)
= 4Q – r. Aeaiiay oi?ioea aiai?eo, /oi aeioiae oeaieony a /aou?a ?aca
aieueoa, /ai ?ene, o.a. oaaee/aiea ?enea ia 4 eiiiaine?oaony oaaee/aieai
aeioiaea ia aaeeieoeo.
E1 = 4*2.6 – 6.6 = 3.8
E2 = 4*6.2 – 3 = 21.8
E3 = 4*7.7 – 1.5 = 29.3
E4 = 4*(-5.9) – 25.1 = -48.7
Niaeanii yoie oi?ioea eo/oae iia?aoeeae n/eoaaony iia?aoeey ? 3, a
ooaeoae — iia?aoeey ? 4.
*anoue I I. Aiaeec aeioiaeiinoe e ?eneiaaiiinoe oeiainiauo iia?aoeee.
( 10, 1/4 ) ( 8, 1/4 ) ( 2, 1/3 ) ( 4, 1/6 )
( -6, 1/4 ) ( -2, 1/4 ) ( 10, 1/3 ) ( -6, 1/6 )
( 10, 1/3 ) ( 2, 1/3 ) ( 4, 1/6 ) ( 16, 1/6 )
( -6, 1/3 ) ( 15, 1/3) ( -4, 1/6 ) ( 3, 1/6 )
Ninoaaei iao?eoeo Q.
10 8 2 4
Q = -6 -2 10 -6
10 2 4 16
-6 15 -4 3
pj = ( 1/4 1/4 1/3 1/6 )
?ene eae n?aaeiaa eaaae?aoe/aneia ioeeiiaiea.
— yoi ia?a ?aca?inaiiinoe aiciiaeiuo cia/aiee aeioiaea aie?oa
n?aaeiaai iaeeaeaaiiai aeioiaea. Iaiiiiei, /oi D[Q] = M[(Q – mQ)2].
Iaeaeai ?enee a eo iiaii ii?aaeaeaiee ri aeioiaeia Qi.
10 8 2 4
Q = -6 -2 10 -6
10 2 4 16
-6 15 -4 3
pj = ( 1/4 1/4 1/3 1/6 )
10 8 2 4
Q1 :
1/4 1/4 1/3 1/6
-6 -2 10 -6
Q2 :
1/4 1/4 1/3 1/6
10 2 4 16
Q3 :
1/4 1/4 1/3 1/6
-6 15 -4 3
Q4 :
1/4 1/4 1/3 1/6
= 10/4+8/4+2/3+4/6 = 70/12 @ 5.83
= -6/6-2/4+10/3-6/6 = 4/12 @0.33
= 10/4+2/4+4/3+16/6 = 84/12 = 7
= -6/6+15/4-4/3+3/6 = 17/12 @ 1.42
D1 = 2384/144 @ 16.56 r1 @ 4.07
D2 = 443/9 @ 49.22 r2 @ 7.02
D3 = 25 r3 = 5
D4 = 10091/144 @ 70.08 r4 @ 8.37
ioeeaaeuaaai ii aa?oeeaee, a ?enee — ii ai?eciioaee.
Iieo/eee /aou?a oi/ee. *ai auoa oi/ea (q, r), oai aieaa aeioiaeiay
iia?aoeey, /ai oi/ea i?aaaa — oai aieaa iia ?eneiaay. Cia/eo, ioaeii
auae?aoue oi/eo eaaaa e auoa. Oi/ea (q’, r’) aeiieie?oao oi/eo (q, r),
anee q’?q e r’Fr. A aeaiiii i?eia?a oi/ea Q3 aeiieie?oao oi/ee Q2 e
Q4, oi/ea Q1 aeiieie?oao oi/ee Q2 e Q4. Oi/ee Q1 e Q3 ian?aaieiu —
aeioiaeiinoue 3-ae aieueoa, ii e ?ene aa oiaea aieueoa. Oi/ea, ia
aeiieie?oaiay ieeaeie ae?oaie, iacuaaaony iioeiaeueiie ii Ia?aoi, a
iiiaeanoai anao oaeeo oi/ae iacuaaaony iiiaeanoaii iioeiaeueiinoe ii
Ia?aoi. Eaaei aeaeaoue, /oi anee ec ?anniio?aiiuo iia?aoeee iaaei
aua?aoue eo/ooth, oi aa iaycaoaeueii iaaei auae?aoue ec iia?aoeee,
iioeiaeueiue ii Ia?aoi.
I?aaeiieiaeei, /oi ana iia?aoeee iacaaeneiu ae?oa io ae?oaa, oiaaea
iiaeii auynieoue, iao iia?aoeee, yaeythuaeny eeiaeiie eiiaeiaoeeae
iniiaiuo iia?aoeee, aieaa oi?ioae, /ai eiathueany.
oaeaea no?aieoueny e ieieioio.
. Yoio ?ene aoaeao ?aaai 3.38, a aeioiae niioaaonoaaiii 6,08.
Iieo/aiiay oi/ea Q’(6.08, 3.38) aeiieie?oao oi/eo Q1(5.83,4.07).
Aeey iaoiaeaeaiey eo/oeo iia?aoeee eiiaaea i?eiaiytho iiaeoiaeyuoth
acaaoeaathuoth oi?ioeo, eioi?ay aeey ia? (q, r) aeaao iaeii /enei, ii
eioi?iio e ii?aaeaeytho eo/ooth iia?aoeeth.
Aeey aiaeeca neooaoeee i?eiaiei acaaoeaathuoth oi?ioeo E(Q, r) = 4Q –
r. Aeaiiay oi?ioea aiai?eo, /oi aeioiae oeaieony a /aou?a ?aca aieueoa,
/ai ?ene, o.a. oaaee/aiea ?enea ia 4 eiiiaine?oaony oaaee/aieai aeioiaea
ia aaeeieoeo.
Oiaaea aeey 1-ie iia?aoeee A = 19.25, aeey 3-ae iia?aoeee A = 23. I?e
n?aaiaiee ?acoeueoaoia aiaeeca aeaeii, /oi i?e aeaiiii ioiioaiee e
?eneiaaiiinoe iia?aoeee eo/oae yaeyaony 3-y iia?aoeey.
*anoue III. Aiaeec aeaiaaeiuo iioieia.
Aiaeec iaeiiia?iuo aeaiaaeiuo iioieia.
Enoiaeiua aeaiiua: aaeaaeiaaiua noiia?iua ca/eneaiey ii n/aoai
th?eaee/aneeo eeoe ca ai?aeue ianyoe.
/enei ianyoea aeaiue iaaeaee noiia (oun. ?oa)
1 n? 47
2 /o 44
3 io 31
4 na 28
5 an
6 ii 42
7 ao 48
8 n? 39
9 /o 40
10 io 38
11 na 15
12 an
13 ii 45
14 ao 53
15 n? 41
16 /o 27
17 io 56
18 na 25
19 an
20 ii 51
21 ao 32
22 n? 49
23 /o 21
24 io 35
25 na 13
26 an
27 ii 58
28 ao 59
29 n? 29
30 /o 30
/eneiaie ?yae (oi) /anoioa
(mi) /anoinoue
13 1 0,04 0,04
15 1 0,04 0,08
21 1 0,04 0,12
25 1 0,04 0,15
27 1 0,04 0,19
28 1 0,04 0,23
29 1 0,04 0,27
30 1 0,04 0,31
31 1 0,04 0,35
32 1 0,04 0,38
35 1 0,04 0,42
38 1 0,04 0,46
39 1 0,04 0,50
40 1 0,04 0,54
41 1 0,04 0,58
42 1 0,04 0,62
44 1 0,04 0,65
45 1 0,04 0,69
47 1 0,04 0,73
48 1 0,04 0,77
49 1 0,04 0,81
51 1 0,04 0,85
53 1 0,04 0,88
56 1 0,04 0,92
58 1 0,04 0,96
59 1 0,04 1,00
.
, aaea a — aa?oiyy a?aieoea e b — ieaeiyy a?aieoea aeey eioa?aaeia, v
— eiee/anoai eioa?aaeia. Aeey aeaiiiai i?eia?a a = 59, b = 13, v = 6, a
h = 9.
eioa?-aaeu
[ai-ai+1) na?a-
aeeia eioa?-aaea
(yi) /anoioa
(mi) /anoinoue
) auai?i/iay ooieoeey ?ani?aaeaea-iey
auai?i/iay ieioiinoue
)
9-18 13,5 2 0,08 0,08 0,22
18-27 22,5 2 0,08 0,16 0,22
27-36 31,5 7 0,27 0,43 0,78
36-45 40,5 6 0,23 0,66 0,67
45-54 49,5 5 0,19 0,85 0,56
54-63 58,5 4 0,15 1 0,44
auaeyaeeo neaaeothuei ia?acii.
Iiiaioaieueiee eioa?aaeueiuo /anoinoae aeaao aieaa iaaeyaeiia
i?aaenoaaeaiea i caeiiiia?iinoe eciaiaiey aaeaaeiaaiuo aeaiaaeiuo
iioieia, o.e. noiiu ca/eneaiee a ?aciua aeie ?acee/iu e eo iiaeii
aiaeece?iaaoue oieueei ii eo aoiaeaeaieth a eaeie-eeai eioa?aae.
Auai?i/iia n?aaeiaa n/eoaaony neaaeothuei niiniaii:
.
ii aeene?aoiiio aa?eaoeeiiiiio ?yaeo
.
ii eioa?aaeueiiio aa?eaoeeiiiiio ?yaeo
.
Aiaeiaii aeenia?nee yaeyaony auai?i/iay aeenia?ney:
.
.
.
N?aaeiaa eaaae?aoe/aneia ioeeiiaiea ?ann/eouaaaony eae eaaae?aoiue
ei?aiue ec aeenia?nee.
Enneaaeoaiay iaie aieueoay niaieoiiinoue iacuaaaony aaia?aeueiie
niaieoiiinoueth. Oai?aoe/anee iiaeao auoue aaneiia/iie A aeaiiii i?eia?a
auai?ea ninoieo ec 26 yeaiaioia. Iiiyoey aaia?aeueiie niaieoiiinoe e
neo/aeiie aaee/eiu acaeiicaiaiyaiu.
Ethaay ooieoeey io auai?ee iacuaaaony noaoenoeeie.
, eioi?ay aeieaeia ioeaieaaoue, iiaeao auoue i?eaeeaeaiii, ia?aiao? Q.
.
Noaoenoeea aeieaeia oaeiaeaoai?youe neaaeothuei o?aaiaaieyi:
.
aeey anao aeinoaoi/ii aieueoeo n.
. Yoi e yaeyaony ianiauaiiie ioeaieie aaia?aeueiie aeenia?nee.
(ni. auoa).
.
, ?aaia aaeeieoea.
.
Ioiaoei iieo/aiiua oi/ee ia a?aoeea
Iieiaeaiea i ii?iaeueiii caeiia ?ani?aaeaeaiey ia i?ioeai?a/eo enoiaeiui
aeaiiui.
— 0,996.
Oaia?ue ?ann/eoaai, ca neieueei aeiae iaaei eiaoue eioi?iaoeeth, /oiau n
aa?iyoiinoueth ia iaiaa 0.9 iiaeii auei iaeeaeaoue, /oi au/eneaiiia ii
yoie eioi?iaoeee n?aaeiaa ca/eneaiea ioee/aaony io aaia?aeueiiai
n?aaeiaai ca/eneaiey ii aaniethoiie aaee/eia ia aieaa, /ai ia 10%
aaee/eiu n?aaeiaai ca/eneaiey.
Eniieuecoy ia?aaainoai *aauoaaa.
Eniieuecoy oeaio?aeueioth i?aaeaeueioth oai?aio.
Enoiaeiua aeaiiua — aaeaaeiaaiua noiia?iua nienaiey ni n/aoia
th?eaee/aneeo eeoe ca ai?aeue ianyoe.
/enei ianyoea aeaiue iaaeaee noiia (oun. ?oa)
1 n? 46
2 /o 54
3 io 42
4 na 28
5 an
6 ii 57
7 ao 26
8 n? 48
9 /o 45
10 io 32
11 na 29
12 an
13 ii 52
14 ao 33
15 n? 50
16 /o 22
17 io 36
18 na 14
19 an
20 ii 59
21 ao 49
22 n? 30
23 /o 31
24 io 43
25 na 16
26 an
27 ii 40
28 ao 41
29 n? 39
30 /o 62
Iino?iei eioa?aaeueiue aa?eaoeeiiiue ?yae e a?aoee auai?i/iie ooieoeee
ieioiinoe.
eioa?-aaeu
[ai-ai+1) na?a-
aeeia eioa?-aaea
(yi) /anoioa
(mi) /anoinoue
) auai?i/iay ooieoeey ?ani?aaeaea-iey
auai?i/iay ieioiinoue
)
8-16 12 1 0,04 0,04 0,005
16-24 20 2 0,08 0,12 0,010
24-32 28 5 0,19 0,31 0,024
32-40 36 4 0,15 0,46 0,019
40-48 44 6 0,23 0,69 0,029
48-56 52 5 0,19 0,88 0,024
56-64 60 3 0,12 1,00 0,014
Auai?i/iay ooieoeey ieioiinoe.
Iaeaeai ianiauaiiua auai?i/iua ioeaiee
.
.
Iaianai oi/ee ia a?aoee
I?aaeiieiaeaiea i ii?iaeueiii caeiia ?ani?aaeaeaiee ia i?ioeai?a/eo
enoiaeiui aeaiiui.
Aiaeec aeaoia?iuo aeaiaaeiuo iioieia.
Enoiaeiua aeaiiua: aaeaaeiaaiua noiia?iua ca/eneaiey e nienaiey ni
n/aoia th?eaee/aneeo eeoe ca ai?aeue ianyoe.
/enei ianyoea aeaiue iaaeaee noiia ca/eneaiee (oun. ?oa) noiia nienaiee
(oun. ?oa)
1 n? 47 46
2 /o 44 54
3 io 31 42
4 na 28 28
5 an
6 ii 42 57
7 ao 48 26
8 n? 39 48
9 /o 40 45
10 io 38 32
11 na 15 29
12 an
13 ii 45 52
14 ao 53 33
15 n? 41 50
16 /o 27 22
17 io 56 36
18 na 25 14
19 an
20 ii 51 59
21 ao 32 49
22 n? 49 30
23 /o 21 31
24 io 35 43
25 na 13 16
26 an
27 ii 58 40
28 ao 59 41
29 n? 29 39
30 /o 30 61
Iino?iei aeaoia?ioth ei??aeyoeeiiioth oaaeeoeo:
0 0 0,57 0,33 0,33 0,33
.
.
(?an/aou ni. auoa).
A?aoee iiey ei??aeyoeee e eeiey a?oiiiauo n?aaeieo eiiiiiaiou Y.
Y\X=12 13,5 22,5 31,5 40,5 49,5 58,5
0 1 0 0 0 0
M[Y/X=12] = 22,5
D[Y/X=12] = 0
Y\X=20 13,5 22,5 31,5 40,5 49,5 58,5
1/2 0 1/2 0 0 0
M[Y/X=20] = 22,5
D[Y/X=20] = 81
Y\X=28 13,5 22,5 31,5 40,5 49,5 58,5
1/5 1/5 1/5 0 2/5 0
M[Y/X=28] = 33,3
D[Y/X=28] = 207,36
Y\X=36 13,5 22,5 31,5 40,5 49,5 58,5
0 0 1/4 1/4 1/4 1/4
M[Y/X=36] = 45
D[Y/X=36] = 101,25
Y\X=44 13,5 22,5 31,5 40,5 49,5 58,5
0 0 2/6 1/6 1/6 2/6
M[Y/X=44] = 45
D[Y/X=44] = 128,25
Y\X=52 13,5 22,5 31,5 40,5 49,5 58,5
0 0 1/5 3/5 1/5 0
M[Y/X=52] = 40,5
D[Y/X=52] =32,4
Y\X=60 13,5 22,5 31,5 40,5 49,5 58,5
0 0 1/3 1/3 1/3 0
M[Y/X=60] = 40,5
D[Y/X=60] = 54
D[Y, ino] = 121,25
Eiyooeoeeaio aeaoa?ieiaoeee E = 1 – 121,25/169 = 0,28
(aeecinoue ei??aeyoeeiiiiai ioiioaiey e aaeeieoea oeacuaaao ia oi, /oi
caaeneiinoue Y io O aeecea e ooieoeeiiaeueiie).
Ei??aeyoeeiiiue iiiaio
. Iieacuaaao noaiaiue eeiaeiie caaeneiinoe iaaeaeo neo/aeiuie
aaee/eiaie.
, ?aaai 1,008.
, ?aaai 1,004.
Ioiioaiea eiyooeoeeaioa aeaoa?ieiaoeee e eiyooeoeeaioa ei??aeyoeee ?aaii
0,76.
.
Oaia?ue ioeaiei, ia neieueei i?ioeaioia (ii ioiioaieth e ?acia?o
n?aaeiaai aaeaaeiaaiiai ca/eneaiey) eciaieony iaeeaeaaiia cia/aiea
aaeaaeiaaiiai nienaiey i?e oaaee/aiee ia 1% (ii ioiioaieth e ?acia?o
aaeaaeiaaiiai nienaiey) aaeaaeiaaiiai ca/eneaiey.
y=0.37x+25.57
(0,37*40,4+25,57)/(0,37*40+25,57)=1,004
Cia/eo, i?e oaaee/aiee aaeaaeiaaiiai ca/eneaiey ia 1% iaeeaeaaiia
cia/aiea aaeaaeiaaiiai nienaiey oaaee/eony ia 0,4%.
Нашли опечатку? Выделите и нажмите CTRL+Enter
