A Simple Representation of the Weighted Non-Central Chi-Square Distribution

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1 SSN: raoa Joura o ovav Rarch Scc grg a Tchoogy (A S 97: 7 Cr rgaao) Vo u 9 Sbr A S Rrao o h Wgh No-Cra Ch-Squar Drbuo Dr ay A hry Dr Sahar A brah Dr Ya Y Aba Proor D o Mahaca Sac u o Saca Su a Rarch Caro Uvry Caro gy cur D o Mahaca Sac u o Saca Su a RarchCaro Uvry Caro gy Abrac: h ar w cor h robaby y uco () o h wgh o-cra ch-quar rbuo wh gr o ro Th r h raur a a u Hr w r a arav ro o h whch ay b uu h uur u Th ru o Ka (6) a Ara a Barc (8) ca b oba a a ca ca ro our Kywor: Wgh o-cra ch-quar rbuo Mo B uco Mo Sruv uco Ho orua NTRDUCTN Grau (9) rouc h wgh o-cra ch-quar rbuo wh gr o ro whch ha acao a w cru o quay coro ara Th robaby y uco () o uch rbuo r a a u o cra ch-quar rbuo a oow: Whr a ar wo ov ubr h o-cray arar h cra ch-quar rbuo wh a o by Whr a () h gaa uco gr o ro Now () ubu or o w g or v a o by () Coyrgh o JRST wwwrco 977

2 SSN: raoa Joura o ovav Rarch Scc grg a Tchoogy (A S 97: 7 Cr rgaao) Vo u 9 Sbr Coyrgh o JRST wwwrco 978 ) ( a () Wh () ruc o h o h o-cra ch-quar rbuo wh gr o ro a o-cray arar Wh a () ruc o h o h cra chquar rbuo wh gr o ro gv by (5) Ka (6) r h o h o-cra ch-quar rbuo or o gr o ro r o h h -rvav o h hyrboc co uco Ara a Barc (8) go h a ru o Ka (6) bu hrough a arav roo ao hy uc h corrog orua or v gr o ro Thr roo ar ba o ror o o B uco o h r a o Ho orua or h h - rvav o a coo uco A h o-cra ch-quar rbuo ar rquy varou acao cug ac ao hory co hory a r aay h wgh o-cra ch-quar rbuo ar quay coro a aca aay hry a () rov ha h o h wgh o-cra ch-quar rbuo wh υ gr o ro ha a ro ha o o cy vov a u Th ro gv by ; ; ; ; F F a (7) (6)

3 SSN: raoa Joura o ovav Rarch Scc Whr a o gav gr uco o h r o ro orr a grg a Tchoogy (A S 97: 7 Cr rgaao) Vo u 9 Sbr F h gra hyrgorc uco h hyrboc co uco h o B h ar or urca vauao uro w r o y h u h o h wgh o-cra ch-quar rbuo wh υ gr o ro Sco w r o rqur ca uco a oao o aca urag h ar Sco w rov ha h o h rbuo wh υ gr o ro ca b rr a h u o o a v r ach ha a or ro ur roo ba o h o Sruv a B uco For h v r w oow h a argu a ha gv Ara a Barc (8) Tha our roo ba o o ror o o B uco o h r a o Ho orua or h h -rvav o a coo uco SM SPCA FUNCTNS AND NTATNS Th oowg ar o ca uco ha w b u our rvao - h gaa ucao uco a a - J () h B uco o h r o orr a a J R a - () h o B uco o h r o orr a a arcuar R a () R - H h Sruv uco o h r o orr whch va or a a a H R () () () 5- h o Sruv uco o h r o orr whch va or a a a R 6- Th rvav orua o h o Sruv uco a R (5) (6) Coyrgh o JRST wwwrco 979

4 SSN: raoa Joura o ovav Rarch Scc 7-Th rao bw h o Sruv uco gr gv by grg a Tchoogy (A S 97: 7 Cr rgaao) Vo u 9 Sbr a h o B uco () or o-gav R h oub acora uco a a 8- o v h ara hyrboc co uco a a 9- h v o For or a abou h abov ca uco Arw (99) A SMP FRM F TH PDF F h co w rov ha h o h wgh o-cra ch-quar rbuo wh υ gr o ro ha a or ro Th ro w b gv h oowg hor Th Ma Thor Th robaby y uco o h wgh o-cra ch-quar rbuo wh gr o ro gv by: whr h A ( ) (7) (8) (9) () () Coyrgh o JRST wwwrco 98

5 SSN: raoa Joura o ovav Rarch Scc grg a Tchoogy (A S 97: 7 Cr rgaao) Vo u 9 Sbr Coyrgh o JRST wwwrco 98 a () () ) ( Whr a a A C a a a o gav gr Proo: worh og hr ha wh ug h ( ) ( ) a () w g h o h o-cra ch-quar rbuo whch h a a ha gv Ka (6) bu wh coy r roo a a ha o Ara a Barc (8) Ao wh h ( ) ( ) a () w g h o h o-cra ch-quar rbuo a ha gv Ara a Barc (8) Fr or cor gv () Ug h gaa ucao uco () wh a ug (5) w g ) (5 Ug (5) o h o Sruv uco o h r o orr a wh h (5) bco:

6 SSN: raoa Joura o ovav Rarch Scc grg a Tchoogy (A S 97: 7 Cr rgaao) Vo u 9 Sbr Coyrgh o JRST wwwrco 98 ) (6 Ug h rvav orua o h o Sruv uco (6) wh a w g Hc quao (6) a h or ) (7 b o = (7) w g ) (8 Ug (7) wh a w g (9) Cor h oowg orua o h o B uco o h r o orr gv Ka (6) ) ( C Tag = a () w g C Subug (9) a h (8) w g h ru () b v = (7) w g h ru () For u (6) w g

7 SSN: raoa Joura o ovav Rarch Scc grg a Tchoogy (A S 97: 7 Cr rgaao) Vo u 9 Sbr ow ha h Tag w g h ru Sco or gv () b o a rca h w or (7) Ao cor h oowg Ho' orua or h h -rvav o a coo uco gv Ara a Barc (8) g cor ( ) g a g car ha Coyrgh o JRST wwwrco 98 C g g () () w g C () Ao () Coquy () bco Cor h oowg aro gv Ara a Barc (8) a a aura ubr Whr Th () bco: Subug (7) w g () (5) (6)

8 SSN: raoa Joura o ovav Rarch Scc grg a Tchoogy (A S 97: 7 Cr rgaao) Vo u 9 Sbr Coyrgh o JRST wwwrco 98 Chagg (6) wh ) ( w oba h oowg orua Whr Now rwrg a oow: A a coquy h a r h uao ovr ro Hc w g h ru () b v () a rca h corrog w or (6) Now rca Ho orua () wh a g (7) C car ha Ug () a (5) h (7) bco: ) ( Subug (6) w g ) (8 Chagg (8) wh ) ( w oba h oowg orua Whr Aga a a h a r h uao ovr ro Hc w g h ru () h ca oow ro (7)

9 SSN: raoa Joura o ovav Rarch Scc grg a Tchoogy (A S 97: 7 Cr rgaao) Vo u 9 Sbr Coyrgh o JRST wwwrco 985 RFRNCS [] Ara S a Barc A "Pror o h Probaby Dy Fuco o h No-Cra Ch-Squar Drbuo" Joura o Mahaca Aay a Acao [] Arw C "Sca Fuco o Mahac or gr" Sco o McGraw-H c Nw Yor 99 [] hry A brah S A a Ab-a Y Y"Corbuo o h Wgh No-Cra Ch-Squar Drbuo" Th gya Saca Joura Ju u o Saca Su a Rarch Caro Uvry gy [] Grau D " Mo o h Ubaac No-Cra Ch-Squar Drbuo" Sac a Probaby r [5] Ka H "Corbuo o h Thory o h No-Cra Ch-Squar Drbuo" raoa Corc o Scc Coug a Vga Nvaa 6 APPNDX Th roo o (6) a (7) gv hry a () a oow Ug h oowg orua gv Ara a Barc (8) () a ug ao (5) () w g ) ( Ug () wh a w g ) ( car ha wh () ruc o h orua o h o h o-cra ch-quar rbuo gv Ka (6) a Ara & Barc (8) To y () cor h ora o B uco o h r o orr : gv Ara a Barc (8) ) ( Tag a h () ca b wr a oow: ) (5 Thror Aga ug h a quao w oba h a orua gv Ara a Barc (8) Now b v (5) w g

10 SSN: raoa Joura o ovav Rarch Scc grg a Tchoogy (A S 97: 7 Cr rgaao) Vo u 9 Sbr Coyrgh o JRST wwwrco 986 Ug h rvav orua ( Ara a Barc (8)) (6) a a a or a a a Wh a a w g Ug () wh a w g h ru (6) Now b o (5) w g Ug (6) wh a a w g Ug () w g Ug () wh a w g Ug h oowg orua gv Ara a Barc (8) a W g h ru o (7)

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