Lecture IS

More on covariance and corelativenm

Recall that covariance is definedas

cook H o yEffy E EAD

Oxy measures linear relationship between X and

Y ayoxy

cooili

Eso

Xox

i m

The correlation of X and Y also

measures linear relationships between X and yCorrelative may be easier

to interpret

Defy Gruen CRU's X Y jointlydistributee Thenthe correlation is defined to be

sxyi YI.IT oEotyEtstandarddenofXady

Excuse Shaw that for all X YI E Sxy Et't

If f y is close to T1 then the possible

values of X andX tend to be near a line with

positul slope with high probability

If 8 4 1 then Y aXtb for ato

If By I ther Y aXt b for a 770

Linear Functions of RandomVariables

Let X Xz Xu be random vambles

For constants co C s en

Y Co 1 C X t Czk t ten Xu

is a a linear function of X Xn and Y is itselfa random arable

Fact ECY cot aECX.TTten ECW

we can verify tin m the case of 2 RUS

E Ca f b X t eYJ ffcatbxtcyjfxylx.usd dy

ffafxytxyldxoytffbxfxydxdytffcyfxydxdy

afffxydxdyy.is xIdgdyyeffyfxydedgy

EADa t bFIX t c f EY as required

In general if Yco T C X t t C n Xn then

the war wel 03

y civcxdtczvfxdto.tcikxnltz.E.ficiox.is

Note If X ve X Xu al independent

then for any i I Ray 0 andso

Vfeotax t.tcnx.DEciVlXi

As a special case if X Xu are Ruis then

the average B

I th x txt tx

If Xi are such that Effi Mi then

EFX is the averageof the ui's at

For X Xu independent UCxi oi

Vfx t.E.fiso it a 2 02 ti VCI I

ReproductivePropertyof the Normal Distribution

unmurrummmurmesupposethat X Xu are independent normat

RU's with ECXd ai and VCki oil then

Y co t c X t t cnXu

is Tom a normal RV with Ely co t c U t hurt tena

and V ly FEETSpearulcasei Xi Xn are all Nfu o

then ly X E ten N nm no and

I xit n.tl N er oneThis may be viewed as a special case of the Central

Limit theorem Let Xi Xe be a sequenceofRU's independent with the same dorshrbwhon They

X tht t Xu nuNfo 1

Mn or noo

Exaplei suppose that water bottles are faked to

an arenerage of 591ml with a 5mL Suppose the

Volume ofa bottles are independentnormal RVs

Goren to bottlewhat B the prob that the

average volume is less than 581 ml

averageLet Xi the volume of the ith bottle

So Xi N 591 8 25

Let I X t tko

Want PCI 585 Note X N 581 N 594N 591 2.5

SOPCIE 5857 Plz c 58ft

OI f 3.79 I Plz et't

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