5.3 Law of Total Expectation

Agenda● Conditional Expectation● Law of Total Expectation (LTE)● Law of Total Probability (Continuous version)

Conditional Expectation

3

E(X|A) =X

x2Range(X)

xPr(X = x|A)

Law of Total Expectation

4

Pg Ain rF X

rxPrl

pennant

LET Pr X xIAi PrfAi

definite PrMi yxPrX x EHlAi Prati

bydefn E XfAiXc XL

Linearity of expectation appliesTo conditional expectation too!!

E(X+ Y | A) = E(X | A) + E(Y | A)

E(aX + b | A)= a E(X | A) + b

5

Law of total Expectation (RV version)

cRy Ry y ya

Y yY 73i

ProblemThe number of people who enter an elevator on the ground floor is a Poisson random variable with mean 10. If there are N floors above the ground floor, and if each person is equally likely to get off at any one of the N floors, independently of where the others get off, compute the expected number of stops that the elevator will make before discharging all the passengers.

7

Y stops elevator makeNN

I

s20

X peoplewhoenter elevator

X N poi IOoh

squid Suppose 20 people entery L

stopastronitufloor

offer elevatoro otherwise

yi KEEFFE a IO o w a f

F Yi _Pr elevatorstops at inflow 41 1157

y Liesel INDIgEaitanat

Prf somebody chooses ith floor whostop onintern31 person

l Pr nobody chooses that floorBin TN

2020 people

L I fN torrs

late E Yi Prfeelwstpsatihf6orI l tn s

X peepkenter.npo.ggY stops elev

F Y oE YX k Prcx

LTE j loTcl

simplifying ru Y

F Y tht tYw x k Yi lofting B

fq.ggqqqzqgg awz

FEMI KINI E ip

Solution

9

X number of people who enter

Y number of stops

E(Yi|X = k) =�1� (1� 1/N)k

�Yi indicates a stop on floor i

E(Y |X = k) = E(Y1 + . . .+ YN |X = k)

Pr(X = k) = e�10 10k

k!

E(Y ) =1X

k=0

E(Y |X = k)P (X = k)

Law of total probability (Cont version)LTPgdispYfe

y yprfEHPrHpyt

fPrfE S pref't g fyls dy

oo la

Prfyeyeytdyj.TTXyXa n XkiE.d.den iyfcDFI

IPrfxi minfxi.mxPrfX aXa XfXz IihXk

PraLY K i fGdxg

Pr Xp x Xp x X ex IXEx

I i

i iIf III II

duet

II Ep

h g iable

change of variable Idz F Ja IIE ft

as EE E t iu goes from

FCxl o I

BEI'T IT A

IHIfiuwf guk tdu jouwduYIljprfx.am

y zI

prfxwmmfxr.im fz

Multivariate: From Discrete to Continuous

END PIC

Alex TsunJoshua Fan

Multivariate: From Discrete to Continuous

Law of Total Expectation (Example from last time)

(LTE)

Law of Total Expectation (Example from last time)

(LTE)