TRNG I HC S PHM THNH PH H CH MINHKHOA TON TIN HC****** ******BO
CO TI NGHIN CUPHN PHI STUDENTTp H Ch Minh, Thng 01 Nm 2010Sinh Vin
Thc Hin:Nguyn Th Tho HuynLp: Tin 2CGio Vin Hng Dn:Nguyn Ch LongMc
lc I.Gii thiu.trangII.S thnh lp....trangIII.c trng...trangPHN PHI
STUDENTI.Gii thiu:Gosset pht minh ra t-kim tra x l cc mu nh kim sot
cht lng trong bia. ng vit di ci tn sinh vin.Sinh: Ngy 13 thng 6 nm
1876 Canterbury, Anh.Mt: Ngy 16 thng 10 nm 1937 ti Beaconsfield,
Anh.William Gosset c gio dc ti Winchester, sau nhp vo New College
Oxford, ni ng hc ha hc v ton hc.Gosset thu c mt bi nh l mt nh ha hc
trong cc nh mybia.Guinness ti Dublin vo nm 1899 v lm vic quan trng
v thng k.Gosset pht hin ra cc hnh thc phn phi t bi mt s kt hp ca
ton hc v kinh nghim lm vic vi s ngu nhin, mt ng dng u ca phng php
Monte-Carlo.McMullen ni: nhiu trong th gii thng k sinh vin c coi l
mt c vn thng k nh my bia Guinness, nhng ngi khc anh xut hin l mt nh
sn xut bia dnh thi gian rnh ri ca mnh thng k mc d c mt s s tht
trong c nhng tng m h b l cc im trung tm, c kt ni thn mt gia cc
nghin cu thng k ca mnh v nhng vn thc t m ng tham giaSinh vin lm mt
s lng rt ln cc thi quen bnh thng cng nh cng tc thng k ca ng ti nh
my bia, v tt c, thm vo cng tc thng k v t vn chun b giy t khc nhau
ca ng c xut bn.T nm 1922, ng nhn mt tr l thng k cc nh my bia, v ng
t t xy dng c mt b phn thng k nh m ng chy cho n 1934.Gosset chc chn
khng lm vic trong s c lp. ng trao i th t vi mt s lng ln cc thng k v
ng thng ving thm cha mnh trong Watlington ti Anh v trn nhng dp ng s
ving thm i hc College, London v cc Rothamsted nng nghip Trm th
nghim. ng s tho lun v vn thng k vi Fisher, Neyman v Pearson.Nm 1934
Gosset c mt tai nn xe my.Trong thc t khi b gii hn ng cho ba thng
sau v tai nn, ng c th tp trung vo s liu thng k. l mt nm trc khi ng
phc hi nhng ng vn gi li cho mt ci nm cn li ca cuc i.Trong xc sut v
thng k, sinh vin ca t-phn phi l mt phn phi xc sut pht sanh trong cc
vn ca cc c tnh c ngha ca mt s dn bnh thng khi kch thc mu l nh. N l
c s ca t cc sinh vin ph bin ca cc xt nghim cho cc ngha thng k ca s
khc bit gia hai c ngha l mu, v cho khong tin cy cho s khc bit gia
hai c ngha l dn s. Ca sinh vin phn phi l mt trng hp c bit ca phn
phi hypebolic Qut.Sinh vin phn phi pht sinh khi s dn lch chun l
khng r v phi c c tnh t d liu. Kh thng xuyn, tuy nhin, vn sch gio
khoa s x l s dn lch chun nh th n c bit n v do trnh nhu cu s dng t
ca hc sinh lm bi kim tra. Nhng vn ny ni chung ca hai loi: (1) nhng
ngi trong c kch thc mu l rt ln m mnh c th x l mt d liu da trn d ton
ca cc phng sai s nh th n l nht nh, v (2) nhng minh ha l lun ton hc,
trong cc vn ca cc c tnh lch chun l tm thi b b qua bi v khng phi l
im m tc gi hoc hng dn sau c gii thch.II.S thnh lp:Mt phn phi thng k
c pht hin bi William S.Gosset nm 1908.Gosset lm vic ti cng ty
Guinness quy nh rng ng khng xut bn di tn ring ca mnh.V th, ng vit
di bt danh Student.+nh ngha: Gi s X l bin ngu nhin c phn phi chun
ha, Y l bin ngu nhin c lp vi X v c phn phi Chi bnh phng n bc t do.
Khi bin ngu nhin:T=Yn X(1)c gi l phn phi Student vi n bc t do.K hiu
T~Tn.Cc s c trng E(T)=0 ( bc t do n>1 ); V(T)=2 n n( vi
n>2).By gi ta i tm hm mt ca bin ngu nhin T~Tn..Bi v X v Y c lp
cho nn hm mt ng thi s l g(x,y) =g X(x)gY(y) vi g X(x)v gY(y) ln lt
l hm mt c lp ca cc bin ngu nhin X v Y.g(x,y)=( ) 1 2 /2 / ) 1 (22
21+
,_
nnynexp
,_
22y xChng ta s i xc nhhm mt ng thi f(t, ) ca T v W. Bng cch t
W=Y v T nh (1), ta c:'Y n XTY W tng ng'W TnXW Y1(2)Jacobian ca php
i bin t X v Y sang Tv W l:J=tx xty y =nnt 20 1=nHm mt xc sut ng thi
f(t,) ca T v W thu c t hmmt ng thi g(x,y) bng cch thay x v y (2) v
nhn viJ=n.Ta tm c f(t,) v i -+ < < tv 0 > nh sau:f(t,
)=
,_
+22 212 / ) 1 (nn( )en 1 2 / xp)'+ ) 1 (212nt(3)T (3) ta xc nh
hm mt l fT(t) ca bin ngu nhin T.f(t)=0) , ( d t fv ta tm cfT(t)=2 /
) 1 (21221+
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+
,_
,_
+ nntnnnvi -+ < < t(4)Trong l hm Gamma xc nh bi:( ) + 01)
( ) 1 ( , p p p dt t e pp t 2123,21, 1 ) 1 (
,_
,_
)! 1 ( ) ( k k.Nu p=k l s nguyn chn >0( )nnn21 2 ...... 5 . 3
. 121
,_
+ Ngoi ra fT(t) c vit di dng:fT(t)=+ < <
,_
+
,_
+ tntnnBn, 121,212 / ) 1 (2Trong B l hm Beta, c nh ngha bi:B(dx
x x 101 1) 1 ( ) , Do nu T c phn phi Student vi n bc t do th hm mt
xc sut ca T l fT(t) nh (4). th ca hm mt c phn phi Student c minh ha
hnh sau.Ging nh phn phi chun ha, hm mt ca bin ngu nhin c phn
phi.Student i xng qua trc tung T=0. th hnh chung tng t nh th ca phn
phi chun nhng c nh thp hn v hai phn ui cao hn so vi th ca phn phi
chun. Hn na l khi n cng ln th hm mt ca T, T~Tn cng ging vi hm mt
chun ha bi v:X2 212...nX X + + vi Xi,(i=1n), l cc bin ngu nhin c lp
cng phn phi chun ha.Theo nh l lut s ln th X1 /2 pn. Cng t nh l
Stutsky, th T=Xxn XF 2.Vy khi n ln ( trong thng k th n30 ) th phn
phi ca bin ngu nhin T~Tn c xp x bng phn phi ca bin ngu nhin X vi
X~N(0;1) thniiXnX121~tD[t]=+ 2) (2n ndt t f ttHm mt xc sut:Hm phn b
tch:III.c tnh:Phn phi student l phn phi xc sut ca t l:*Z l bnh thng
vi gi tr d kin 0 v phng sai 1.*V c mt phn phi-chi vung vi v ca t
do.*Z v V c c lp.Trong khi, i vi bt k hng nh, l mt bin ngu nhin ca
t noncentral-phn phi vi tham s noncentrality .1. Hm mt xc sut:-Phn
phi student c hm mt xc sutfT(t)=2 / ) 1 (21221+
,_
+
,_
,_
+ nntnnnvi -+ < < ttrong n l cc mc t do v l hm
Gamma.a.Derivation:(ngun gc)Gi s X1,,X2 l c lp bin ngu nhin l bnh
thng vi gi tr k vng v phng sai 2 choXn=(X1++Xn)/nc cc mu c ngha l,
vSn2=1/(n-1)L phng sai mu. C th thy rng cc bin ngu nhin(n-1)Sn2/2c
mt phn phi-chi vung vi n-1 bc t do. N l d dng cho thy s lngPht hnh
bnh thng vi ngha l 0 v phng sai 1, k t khi mu c ngha l Xn pht hnh
bnh thng vi ngha l v tiu chun li /sqrt(n). Hn na, n c th cho thy
rng hai bin ngu nhin-mt trong nhng hnh bnh thng v chi-square-phn
phi mt l c lp. Do s lng ch cht,M khc vi Z trong chnh xc lch chun c
thay th bi cc bin ngu nhin Sn, c t ca mt hc sinh phn phi theo nh
ngha trn. Thng bo rng dn s cha bit phng sai 2 s khng xut hn trong
T, v n c ci thin c hai trong t s v denominators, do , n b hy b. V
mt k thut, (n-1)Sn2/2 c mt Xn-1 2 phn phi bi nh l ca Cochran.
Gosset ca cng vic cho T thy c hm mt xc sut:fT(t)=2 / ) 1
(21221+
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+
,_
,_
+ nntnnnvi -+ < < tPhn phi ca T by gi gi l t-phn phi. Tham
s n c gi l s lng cc mc t do, phn phi ph thuc vo n, nhng khng hoc ;
thiu s ph thuc vo v l diieeuf lm cho cc t-phn phi quan trng trong c
l thuyt v thc hnh.b.Hm phn b tch:Cc chc nng phn b tch ly c cho bi
cc chc nng Beta regularized khng y ,Vi2.Confidence khong:Gi s s A l
la chn mPr(-A
