given a measure space wspu - ucsd mathematics | home
TRANSCRIPT
L2 [ § 10.5in Driver ]
Given a measure space Wspu) ,the
,Ssp) .- { f : IR measurable stuff'
dries } .
Note : for any real numbers 1g ,
O s (H - Igf = ft - Zlfgltgi . lfgl s { If↳of)
Thus, if f. get, then flfgl dm s ftlftgdu
Ie- Prop : If f. go.la then
= 's fftfuttfg.HU do .
fg GLt.
corpse ;:finite again:*, Infestation .
Cauchy -SchwarzFor f.genius , define :
" the :'-( ffidu)"
,
Lf,g >Li.
-
- ffgderr
Theorem : If f.gc-LYBF.pe) , thenKf,g>Els flfgldusllflkltg.lk .
Pf . lfftglsflfglt
µgµgy%2t"".
Pgt- SHH -tlgitdus.co#--fLf2-2tlfgltt2g4du-4llfHEkglk=fg2efe-2flfg1ef
- t tfgdu - t
⇐ O'
= t.flk-zflfgkfr.tt light
Cer : Etr,8, p) is a vector space , and HAE -
- ICHIis a norm on it . Iggy = fffefu t
Pf . If AHR , fat ,Hafile = fcaffdui = offYue 241 fth
i. Hdf the = klllflkz
If Iget, llftglh =
flftgfq-flftzfgtgoqr-fftzffgtfg.tnIf Hf HE
-
- o sff-tznfk.bg/ktfg2then ffkfu -
-o -
- pftlptzllflhllglhitllglk⇒ f'-
-o as . go = ( t.net/lgHef Los
⇒ fee as . yes . H
Fact : L2 is actually a Hilbert space : it is (Cauchy ) complete .
Ie. If fn E K s
.
t.
It fn - fm He → o as n,m→ es
,
then 3 ! ft L'et
. Hfn - f the → 0.
( The same holds true in Lt , and in general inLP for Is p sos . )
Weill return to prove this important fact in thenear future .
This will be very important when we make a
serious study of conditional probability .
Covariance
In a probability space left, IP) , L's Lt . In fact" X Hu = flxldp = fslx-lldpcs.gl/xlkalllplk2-- It K
r
Ie. LEWIS Elko suits fedD= t
Def. For X,Yet
,
let K - X- LECH,
I -- Y- EW " centered"
Their covariance is#ODIE I a
Cov NY El III-
- LECHE KY - ENTe Ef XT - X#W - HIGH HIPPEN)
For X th'
,its variance is
.
= TECH - ECXI #HI .
Var CX ) -
-
= Covlx,X) = #IM = TECH - CENT 30
Lemma : If XELYB.BR) and Varlx)=o,
then X E const. a.s .
Pf . O = Var CX ) = E ( 921 ⇒ I 2=0 as.
⇒ ex - o as .
8230 XIE i. KEN a. s .
H
Eg.
X & Nla,t)
.
X 'd of -2 ta, ZENG
,D
Varix) = EIN- att -- El@Zf) -- t Egf- t
.
Eg.
Tsd Exp la) , EITI-
- ta es
Var LT) = Effigy = ! Lt - tf de-atdt = '
ga
Eg . N El Pais la),TEENS = a
Varin) -- EUN-NY = Ck -ate"
! = a
Def : X,yeL4r,§p) are uncorrelated if Corky ) -- o .
In general , their correlation is corrupt =
Prop : covlxtgyf-covlx.tt d) '- GUNK)= # [ yegg
↳
for any at IR .
As a result, ¥184151
if Xi,
.>Xn are all ( pairwise)
" Ith d
= standard Donathanuncorrelated
,then =k¥
Var (X,t - - - + XD '- Var t - - - therexD
.
Pf . (XoD'
= I
Varlxit . -1- XD - tell xit . - -thing
s.Ejsxi.fxif-E.ECxiilat.in?.Gvcxiitd--efj-tk=
,
any;yo.
-
-f. VONDA
What Does " Uncorrelated "
Mean ?E. g . Let A.Be 8 , and consider DA
,DB
coulda,BB) x
! GA PCH D= pop
= #HAD B) - Elba)#IDB)PH - o)
-
- pipe) =p - pop} Bernoulli
= # ( band - El LEND = PlanB) - MA) PIB) = O iff ABare independent
Eg.
Toss a fair coin n times. xj=ff If
the j" toss isnpegqls
s-
- { Iw ... . ,wn) : wits 9133 ↳ Xjlw) e Wj
8- 2h
play = #fdn Exercise : Gvlxj,XD = Sjk