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