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Cacecoe oepoae Meoece yaa o oe aopaopx pao yaoo aa o aeaeco cace Toc 2004 Beee ye eoc pax cacecx poeyp eoxoo e aoe aop opo paoo oea c ec aoa pacpeee. Opaaa aoe ae o aoy-o aopy, ee oooc cpa pecaa eop c ec apaepa aoa.

eepa op c aa aoo pacpeee ocaoo ye coaa aoe-y oo "caapoe" pacpeeee, a ae opa ay y o eo, oopa ea peye ao pacpeee.

B aece cxooo aepaa oepoa pacpeee pe cyay ey, paoepo pacpeeey a opee [0,1]. ae ao cyao e aa cyau ucau.

Cyecye oo cocoo oye cyax ce a BM, a cpoee eepaop, a ae ceae cee aop [1,2]. Bo cex aopecx ax popapoa ceax cex cpeax peycope y, opaae ae cyax ce. B cpee MathCAD ec y rnd(x), opaaa aee cyao e, paoepo pacpeeeo a opee [0, x] [6].

1. Moepoae cpex cyax e 1.1 Cyae ucmau c y ucxoau O c y cxoa A, A ooc opeeec aae epooc p co A. coy oe eoepeco epooc, eo oaa, o co {A} {rnd(1) < p} paoc (pc.1).

A A p 0 1 Pcyo 1. Paococ co {A} {rnd(1) < p} 2 ocea ca oe co A cep n ca ee aop co J(p) cyay ey, pay 1, ec coe pooo pay 0 poo cyae1.

(1.1) J p :=< rnd 1 p, 1, 0, if (1.2) J p := p - rnd 1.

ec x y Xecaa, paa y p x < 0 paa ee poo cyae. B cpee MathCAD (v. 2.54) y Xecaa aec coeae a Alt -H.

1.2 ucpema cyaa euua c aa aoo pacpeeeu yc ao pacpeee cpeo cyao e aa ae.

Taa X x1 x2 xk P p1 p2 pk Moe ao cyao e ec o c k cxoa. Paoe opeo [0, 1] a k opeo oa u0, u1, , uk a, o oc yco:

k ui - u = pi, (i = 1 k ), pi =1 ( 1.3) i- i=p oepoa eoo ca ye oaa, o cyaa ea pa aee xi, ec cyaoe co oao i-m epa. Oe, o oceoaeoc oe u0, u1, , uk ec e o oe, a oceoaeoc ae opoo yuu pacnpeeeu.

p popapoa cpee MathCad eco aopa oo cooa oece oepa, opaae ey, ec peya oepa ca o ec oo.

popaa oepoa oa c ep cxoa. aae ae cyao e x epooc:

p1 := 0.1 p2 := 0.3 p3 := 0.4 p4 := 0.2 X1 :=1X2 := 3 X3 := 4.5 X := aae eop aopo co:

0. 0.K := 4 i :=1..K j := 2..K -1 pi = 1U1 := p1 U := U + pj U = j j- i 0. 10 0 0 1 0 x x x M x := if < U1,,if < U2,,if < U3,, 001 0 0 0 Coae oceoaeoc N aopo:

N :== k = 1..N L k = M rnd 1 J : LT X 50 : :

Beop J coep opy ooaoo pacpeee . Beop aco B cy oeo eopa epooce P:

k L k B := BT = 0.04 0.30.46 0.B = N Jk X1 XXX0 10 20 30 40 k Pcyo 2. Bopa ooaoo pacpeee 1.2.1 Moeupoaue cyaoo yau amepuao acmu no noco ceme Paccop aepay acy, cocoy cpee oe pee cya opao epeeac oo epex apae o ya apao oco ce.

Bee y cyaoo cee -1 0 11 3 D p := if p <,,if p <,,if p <,, 0 -1 42 0 aa eop aaoo cee:

R 0 := aa co cee eyee ooee ac:

N :=1000; i:=1..N; R i := R i-1 + D rnd ae aaoe oeoe ooee ac:

K 0 R 0 K 1 R N R,i K 0 j, 20 R K 1,i,,j Pcyo 3. Peaa cyaoo ya aepao ac.

e oaao peypyee ceee.

p oo ce cxoo oa ocpoe aopo co oyac pooe opy. Bo cyae oo cooa pyo aop.

1.2.2 popaa oeupoau onmo c ou uco ucxoo Moepyeoe pacpeeee oe aao o po, o ye pacpeee.

B aae coepye cxoe ae:

n := 20 i := 0..n -1xi := 2i + rnd 1 j := 0..n -ae epooce o e ooe eopa pi ( ece e oaao). o aaoy aoy pacpeee cpo eopeecy y pacpeee cpo ee pa (pc. 4):

efri := pj xi xj j aae oe op, oep eea aop cyax ce ak:

K := 300; k := 0..K - 1; a := r n d k Pcyo 4. y pacpeee oepyeo cpeo cyao e cxea ee oepoa.

Coa acc aopo co ye ocea acaoo oepa ae cyao e, yoeopeo yco ak > fri ocae acoy co, pocypoa aop co:

k D k Dak >efri,k := 1 S := s := 0..last S K - i 0.Ss 0.ps 0 5 10 s Pcyo 5. Moee cepeae pacpeee.



Paccope aop oepoa cpex pacpeee ec yepca, eo oo pe oepoa epepx pacpeee, ec x y pacpeee peapeo ae ycoo-ocoo ye.

1.2.3 uouaoe pacnpeeeue oaoe pacpeeee oo coepoa c oo paccopeoo e yepcaoo aopa, oao ao paee ceae o cxo epx po e peapeoo aa y pacpeee.

oaoe pacpeeee ocae co (m) oe co cep n ca, ao oopx coe oec c epooc p.

popaa ocea ca co cep:

n := 10 i := 0..n -1m := j p i ec J(p) opeee e aop acye co oeo ca.

popaa oepoa op.

aae oe op K :=100 j := 0..K -oeco ca cep n := 20 i := 0..n -Coae opy B:

Bi := p J j 1.2.4 Pacnpeeeue yaccoa coye aop oepoa ocoa a ceye ae: cyaa ea j = max j : e-, > 0 ( 1.4) Ui i ocaec pacpeeee yaccoa. ee op oo oy oceoaeo yea co eo ( j) poee o ex op, oa e apyc ycoe j ( 1.5) U e-, i i=acaoe aee ( j), yoeopee oy yco ec oepeoe aee cyao e.

popaa coau opu.

aae oe op oep eea op:

N := 10 k := 0..N -pye aae ae acca poee:

z0,k := rnd aa acaoe co eeo poee:

n :== i : 0..n := 4, exp - = 0.aae apaep :

Coae acc poeezi+1,k :=zi,k rnd :

k := zi,k - exp - oy eop-opy i poeypy oye op oo oc cey opao. ae ee acca poee Z cooecy aopa Qi,k := zi,k - exp - 0 1 2 3 4 5 6 7 8 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 2 1 1 0 1 1 1 1 0 1 3 1 1 0 1 1 0 1 0 0 4 1 0 0 1 1 0 1 0 0 Q 5 0 0 0 1 0 0 1 0 0 6 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 oeco e ao coe ec aee eea op.

0.Oea 0.Moe 0 5 10 Pcyo 6. Moeoe pacpeeee yaccoa eo opoa oea.

2. Moepoae epepx pacpeee 2.1 pueeue opamo yuu yc F x epepa cpoo oooa y pacpeee cyao e. Toa cyecye opaa e y F -1(y), opeeea a opee [0, 1]. yc U cyaa ea, paoepo pacpeeea a [0, 1]. Toa, a ocoa paococ yx co oe aca -P Ui F x = P F Ui x = P xi x = F x, -e xi = F Ui.

Ceoaeo, cyaa ea X ee pacpeeee F x. Ta opao, ec ece opao y F-1(y), o oyae cey ecece aop oepoa cyao e c aa aoo pacpeee.

1. eeppye oepeoe aee cyao e Ui.

-2. o opye xi = F Ui axo oepeoe aee cyao e X.

Ec opay y F(x) o e a e yaec, o oo cooa ee apoca ooea, paoa po e po [2].

yyooc pacpeef y = exp, y 0. (2.1) e yy pacpee-( 2,2) F(y) = 1- ee Maeaecoe (2.3) my = oae = 2 cepc (2.4) y Opaa y y = -2ln(1- u) ( 2.5) ooc pacpeee f ( y) = exp - y, y 0. ( 2.6) y pacpeee F(y) = 1- e- y ( 2.7) Maeaecoe oae cepc my == ( 2.6) y Opaa y y =- ln 1- u ( 2.7) ooc pacpeee f (y) = ( 2.8) y - a b 1+ b y pacpeee 1 y - a F(y) = arctg( ) + ( 2.9) b Maeaecoe oae cepc e opeee.

Opeee oa eaa:

mody = medy = a ( 2.10) Opaa y y = b tg u - + a ( 2.11) ooc pacpeee f (y) = ( 2.12) y - a b 1by pacpeee 1 y - a F(y) = arcsin + (2.13) b Maeaecoe oae cepc bmy == ( 2.14) a, y Opaa y y = b sin u - + a ( 2.15) aae apaep pacpeee oe op:

a :== =: 2= : 5000 i : 0..N 1b N Coae opy apao p:

Yi := a + b sin rnd 1 - 0.5 Z := sort Y oec a o pcyo pa eopeeco cepeao ooc pacpeee, ocpoeo o ao ope:

1.1.1.f 1.l tf(x) 0.0.0.0.1 0 1 2 x Pcyo 7. cepeaa (o) eopeeca (coa ) ooc pacpeee aoa apcyca.

2.2 Hopaoe pacpeeeue ox cocoo oepoa opao pacpeeex cyax e paccop oo e, oope poe ceo peaoa cpee MathCAD. ae, o epc MathCAD 6.0 e aoee yopex pacpeee coa y, eeppye ae cooecyx cyax e. Opay y opaoo pacpeee oo oy e ao-o apoca [2]. oce eo pe oca e aop. Oao poe ceo ocooac eoop cea coca opaoo pacpeee.

eco, o pacpeeee poee yx eacx cyax e, oa oopx ee peeecoe pacpeeee (6), a pya pacpeeea o aoy apcyca (19) c apaepa (0, 0.5), ec opa [3].





o ooe oppoa opay cyay ey c oo ceyeo peopaoa:

y = sin(2 U1) -2ln(U2) ( 2.16) e U1 U2 eace peaa cyax ce. apaep oyaeo opao cyao e yy (0, 1).

n := 100 k := 0..N -1s := 4.5m := Bk := m + s sin 2 rnd 1 -2ln rnd Coaco epao peeo eopee, pacpeeee cy eacx oaoo pacpeeex cyax e p eopaeo yee ca caaex cpec opaoy pacpeee.

yc U U , U eace cyae e, pao1, 2 n epo pacpeeee a [0, 1]. coe xapaepc Ui pa:

M[Ui] = 0.5; D[Ui] = 1/12. ( 2.17) ax cyax e eo coppoa cyay ey, ey pacpeeee, oe caapoy:

U - n i i yn = ( 2.18) n e p n = 12 opya (25) ae cyay ey c pacpeeee, caapoy. Bo cyae (25) ee ocoeo poco :

y12 = - 6 ( 2.19) U i i=Pacpeeee c poo apaepa (m, s2)eo oy caapoo c oo peopaoa:

y = m + s y12 ( 2.20) popaa oepoa caapoo pacpeee n := 10 i := 0..n -1N m, s := m + s rnd 1 - i K := 500 k :=1..K Bk := N 2,0.0.T l ) 0 dnorm( x 0.4 3 2 1 0 1 2 3 h dl x Pcyo 8. Bopa opaoo pacpeee 2.3 Pacpeeeu, cae c opa e eace op caapoo pacpeee, eo oy op - pacpeee, pacpeee Cea pacpeee Ceeopa.

- Pacpeeee x-apa ee cyaa ea n =, ( 2.21) ni i=e eace cyae e, ee caapoe i pacpeeee.

Bopy oea K oo oy, cypy eace caapo pacpeeee ca, oyaee o oca e aopa K :=1000 j := 0..K -1n := 10 i := 0..n -ti := [sin 2 rnd 1 ]2 ln -2ln rnd j.

.

0.0.T l dchisq( x n) 0.0 5 10 15 20 25 30 h d x l Pcyo 9. Moepoae pacpeee x-apa c 10- cee coo Pacpeeee Cea ee cyaa ea, paa ooe eacx caapo pacpeeeo e op apaoo cyao e HI, ee x-apa pacpeeee, eeo a co ee ceee coo n:

= ( 2.22 ) n HI n K :=100 j := 0..K -1n := 10 j := 0..n -sin 2 rnd 1 ] ln -2ln rnd ti := [sin 2 rnd 1 ]2 ln -2 ln rnd n j 0.0.T l dt( x n) 0.4 2 0 2 h d x l Pcyo 10. Moepoae pacpeee Cea c n cee coo aoe pacpeeee ocae ooee yx eacx cyax e, ex x-apa pacpeeee:

HI1 NF = :, ( 2.23) HI2 Ne N1, N2 cee coo pacpeee.

K :=1000 i := 0..K -N1 := 10 u := 0..N1-1N 2 := 20 := 0..N 2- sin 2 rnd 1 2 ln rnd 1 N u Fi := sin 2 rnd 1 2 ln rnd 1 N Fl 0.dF( x N1 N2) 0 1 2 3 4 5 h d x l Pcyo 11. Moepoae pacpeee epa-Ceeopa c N1 N2 cee coo 2.4 ompo emop paecx ee a ec oepoae opooo apae pexepo pocpace. Paccop payc-eop, coo oe oopoo oe axoc pooo oe a cepe eoo payca. opooc eopa peoaae, o epooc oaa coooo oa eopa pooy oac a cepe poopoaa oa (epe) o oac e ac o ee op.

B cepeco ccee oopa opea eopa aaec y ya: yo ey eopo oc OZ yo ey poee eopa a ococ XOY oc OX (pc.12).

Bepe oac D a cepe e cepecoo oca, apaeoo ococ XOY. Taa oac oaae aeae coco: ee oa e ac o ooe a cepe (a aope a oce), a opeeec oo co dZ SH = R h ( 2.24) e R payc cep. a co: { oe eopa oa oac D} {poe Rz po aoe aee} ae. Bepooc x co ocoa p ee poe a opee [-R, R]. Ceoaeo, oo yepa, o poe Rz eoo opooo eopa paoepo pacpeeea a opee [-1,1].

Pcyo 12. aae opea payc-eopa cepecx oopaax.

Bcy cep oac D yo ye paoepo pacpeee a opee [0, ].

C ey eapo cepec oopaa aec cey coooe:

Rx = R sin cos R = R sin sin ( 2.25 ) y R = R cos z Ceoaeo, oepoa opooo eopa eoxoo aae aa e paoepo pacpeeee cyae e: Rz , a ae o opya (1.32) pacca ocae poe eopa. p o ao yec, o sin = 1- Rz.

3. Meo Moe-apo Meoa Moe-apo aa cee eo pacea, coye cacecoe oepoae pacpeee cyax e.

Cy eoo Moe-apo coco o, o ea, oeaa ce, pecaec a aeaecoe oae eoopo y o cyao e c ec pacpeeee. Ta opao, pacea aaa coc oee aeaecoo oa cyao e c ec pacpeeee, oopa eo peayec paccope e eoa cacecoo oepoa.

Paccop aop ce oopaoo opeeeoo epaa b J = ( 3.1 ) h(x)dx a yc f x ooc pacpeee eoopo cyao e, aao a [a, b]. poee oeceoe peopaoae oepao pae.

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