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...
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
gqgi
covCaXtbY aCovCX Y covcxin covey
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
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
Ifpipe's corr l
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
XIY y NCiuxtpE.lyago cpay
mum
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller SqueakyCleanNotes. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for £19.99. You're not tied to anything after your purchase.