MAT377HI LECT OCT 5 2020 NEO YINSection 14

Lemma Assume that I 1 1 L x Elly 1 a Dand X Y independent Rvs

Then F XY is well defined and in factEXY EX EY

lase 1 Xi Y Z O z O

LHS ECXY Is Xliv Yw 1MWsup ZepXowYuu IPWFERfinite

z IP Xy z2CrangeXY

I xylPCX x and y y

wo.EEtshY

M Ii7I F i m

I xy IPA x My yxeronyeXytrangey

mge

xPCX x ymy ygcrange'Y

EX EYAny exchangeof sum is allowed if terms in the som Z O

general ruleofthumb fOR Absolutelyconvergent

KeyInsight if g so then ajqq.ge jEgas

Hn

Iff 2 ai order independentit

order dependent

eg EE aid qq.fyf.ie lEaijMeptfn.in

ieElaf.ffeefEaiaiu 33

can replace

gzip f I aij Gi with Gi

GEIN TEI G afinite finite union

sup I IfuN N aj

ay

order is irrelevant

Now laijl L X absolute convergence

Then Vi laijl That means V j 0

I aij aij I Igaij I

A

I la I o5 9

Tail ofconvergent series

TEXEY Expat x y PLY y

ftp.nitexPCX x t xtPCX likewise

finite sumsmall error

Z xy1PCX x PCy y teryrorseFGfinite

XEFLEG use IEXI E ElXt to

control 1

Issue EXY EXEY assuming X Y independentAND EX EY are defined i e EIN ElYl e 6

Can have examples where FXY is defined but F X Ey aren'teg EXE can be undefined

FubiniSuppose Xi Y Are independent lourider the random variable

fully usefulformulaAssume Elf X g I e s

Then Efx y Igfix y IP X x and y yindependance

2gLex y IP X x IP y y

yLex y IPCX x pcy y

So X Y independent we can write

E flag Zyfcx.gs P X x IP Y y

provided El Lally I e ne

what if X Y not independentFfully Tgfcx.gs P Y y1X

x PCX x

Conditional probability1PCX x y y P X x lP Y y IX x

F fully X x conditional expectationTHEMOST APPLIED FORMULA IN PROBABILITY

Almut