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STAT0005: Complete Summary Notes (Useful formulas and tricks/derivations) £19.99   Add to cart

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STAT0005: Complete Summary Notes (Useful formulas and tricks/derivations)

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These summary notes cover the 5 units of the course, where each unit is only a page or two long. The point is to provide clear, structured colour-coded information regarding formulas and concepts without going too deep into the material. Topics include: Moment generating functions, Multivariate...

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  • November 27, 2023
  • 9
  • 2021/2022
  • Summary
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SqueakyCleanNotes
Joint Probability Distributions
1I BasicProbability I Samplespace A Event A Eventspace W outomeselementaryevents
tot t
WER
E Elementof c subset of
OEmptyset
y
gypper
IIs
bag
shopping
PA 0,1 0,13
probability measuremapstheeventspacetotherealnumbers


Evetts are subsets elements
of 1 outcomes ware
elements
of are ofeventspaced
PCÉ
É A pB planB Independencebetween
PCARB PCAPCB PCBnc PCBPcc
ABandomustsatisfyPlano PCAIPCC PCanBnc PCAPCBPcc
Multiplication Rule PlanB PCABPCB pB a pea p Bna Conditionalindependence planBlc PCA c PCB c
Bayes Theorem formutually
if E
exclusive
PCAB PCB
PCB PlanB By Bewith OTP PCA IIPCAnB EIPCAIB PCB
PCB A pg PAIB PCB PCA o

1.2 UnivariateRVs are
genes
continuous
right


did X S 7112is a RV Fye PCwÉÉÉsx P Xa Pla XEb Fcb Fca
eotanlitneomegassuantna YCw
probability
xmapsomegatotnereais qq.I.e.se
0 XCw OXCwl

Discrete OIVarCX E XEET
Continuous CDF PCXsxj Fxcoy fgf.ludu Expectation FIX Ixippas Exc pea

1.3 JointDistributions Jogging Fxday PXcxYay Fantasy Fxy so x O Fxy xx 1

Plates y Yeys FxGaya Fridays Facesy Fxy a y
Bivariatecase



I3 S
Fx2 PCXEX.TO FxyGGX jointpmfPxycociyi P x ci y y Marginal


Independence
gl e.y
meansmultiply Pxysciy P Xxi
PxCxi fPx.ycxi.si
ii.emaaEE.aea Pmf
pyly P Y y EPCxxi.Yy7 E.px.ycxi.es
IfXHYthetheentriesinthe2 waytableare
y pxcxilpy.ly productsofthemarginalprobabilities
Conditional
XHYÉIfPx1y2,9 Px6i
probability
distributions
Pxiykily
fig or pylxcypxijpy.ly

wouldbe
Conditional Expectation
Ex y X Y y E xipxmcxi.gs if X thentheconditionalexpectation
thesameacross all y
Expansion
Iterated
conditional FIX Ey ExiyX1Y 014,4 É 06in Pxcxiy multiplicationrule
expectation Exleypacy EyEx ocx.ggyyduse
Continuouscase Fxy ay S.ISfxilusuldudw Fye Fxy x x fj ftp.cu.viduldu
where region
Geometrically fx oshoulabearectangle
Marginaldensity
floc Sfx.tl ldeandfycy SEfx.xCe.slde Independence fyoofycyjfactorie.int
fyycayj and
onlyyparts

Conditional
density fxiycxlyy fx.tk butif X then fix
fycy

, 1.4Furtherresultsonexpectations Covariance
Coulxx Var x Variance
VarX X2 Efx
com Etx Eaa eexs X FIX
Ii IÉÉIIÉÉÉÉÉÉÉI FIXYJ EEXJEEY Var varatva.cm
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IfXHY CovCXY 0 nottheotherwayaround X 14 FIXY
Correlation
cord Y XY
CorraY corrCax 4 corr X Y 1ECorr
jjjjjj



Et ygyyy
varcxjva.ca

um um w
1.5Standard multivariatedistributions ConditionallyIndependent Px.xzlx64azas Px.ix.Cx.lxspx.ix
independent
xnare cxaix.
Discrete
Px
IfX apostasy
Pxnxa.xsksxa.es
g my conditional
for ru xax px.ca

Multinomialdist nsamplesize success isfallinginto
oneofthe mtl categories pindividualisoftypei pi Niisthe oftype individualsinthesample

pmf p Nin Nmnm nm Phi if Nmt n ZIN
g otherwise Ifme wegetNinBinenp 2variance
Marginal
mean no
np.ci pi
Trinomialdist 3tupes i andkenotion it i pi p dist NinBincripi

I
thing
nFPiPip Nin n n 1 pip COV NiN npip
n
Pmf PNi ni Nj nj Pimp If piislarge
PF
minin pan nitnjen CorrNishi p cannotbelarge
o otherwise i pi l P
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Ptype Papein types
nottype
PCnottypej Pip go.gg
t1h NilNj nnBinChni PYp
Thebivariatenormaldistributionpdf for xcx.ycx nscmx.nycxox.sn o Iii
then


fi É 4 7 21311,4 1595 71
e action

notation

faecal deets exr E ME ex mi 7 Y m114 2 151.5 1,5155
remember this
multivariate
easilyextendto with soca pk of
dimensionK
conditionaldensity ofbivariate
normal
of normal
Marginal bivariate


Ifylosyldy oh exp gypsy
fxiykly ftp.EI.exp IpyIxlluxtpEcya i
zovariances mean


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mum

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