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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
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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
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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
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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
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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
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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
o 000075