Y F totalexpL Ky fy g dy Ef ace Poisson y f yeidy I law of total - - PDF document

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Y F totalexpL Ky fy g dy Ef ace Poisson y f yeidy I law of total - - PDF document

Sep 15, 2023 326 likes •393 views

r.ve's Practice with continuous Total Probability Law of Discrete 2 M D Pray PrfEI ME fPrCE Y g fy g dy Pr E Law of Total Expectation Discrete Continuous k Pr Yek E XlY F x w y fyb dy fE X Y F X E PrCx xH D Prcx Y D xf

slide-1
SLIDE 1

Practice with

continuous

r.ve's

Law of

Total Probability

M

2

Discrete

D

Pr

E

fPrCE Y g fy g dy

ME

PrfEI

Pray

Law of Total

Expectation

Discrete

Continuous

F

x E XlY

k Pr Yek

w

F

X

fE X Y

y fyb dy

E PrCx xH D

Prcx

Y

D

xf

µ ylYdx

w

f

y

x y

slide-2
SLIDE 2

Example

wrandan

accidents

aperson has in ayear

is

Poisson

a

but 2

is

rn

A

exp l

instead

D

Bindooi

fraction

  • f

population that has

A

prfnoauidfa

kq.ie

is

a EEE

go

g F i

Prfrandompersonhas noaccidents

meg

1awM

Prfrandn

person

has

noact 2 9

Gd's

1

exppy Xnexpli

fxcxt.ae

X

Poussonly

e

e9dg

P4X k E

y

co

e4dg

s

Efx

EE XR

k

F

accidents forrandenpenseningeay

Y

k

  • Pr Kk

Ef

ace

Ky fyg dy

totalexpL

Poissony

f yeidy

I

law of total probe

Pr Xd

Pr XcY Y

fyGldy

Pr Xss

Fxls

slide-3
SLIDE 3

X nu

Y

g X

I

find

busing 4 2X

I

1

probdensityfn forY

2

4

X

f Cx Fx x

Recipe

for computingfyG

to

S 25

FyGy PrYes Find

CDF forY

Prf2XEg

Fyls

Pr Yes Pr g

as

Efx

nurse

Pr Xe

differentiateto get p.d.fr

FX

fy

dagFyG yFyG

IzFxl

2

lfxfaj.LT

w

X Fwiw

PrfXZw PrfrwsXerw

intuition

i e to

see this

makes sense

Prfa

X sates

Efx

a

Y 2X

farepsilon

small

E fy

a

Pr

a ageYeates

PrfaEze 2XeatE PrfazEye KITE ez.f

E

E

fx E

slide-4
SLIDE 4

Xv Nlp d

Y

aXtb Claim

Yn Nfaprtb

aha

Proof

Fyly

Pr Yey Pr

axtbey

prfxey.bz Fx Ia

dd

yfxfs.bz

aLfx Ia

taa

e lEs nI

Hae

9

which is densityof

N aptb aka

slide-5
SLIDE 5

Joint

Distributions

Joint CDF

F

y

x g

Pr X ex Yey

F

asb

I

f

dydx

jointdensity fu

f

x g

f zy Ey

x y

Lf

f

x y

dxdy

I

Pr

a

X Laida

b

Ytbtdb

fC y dydx a flap dadb Marginal densityfor

so

f

x

f f

x g dy

  • Independence

Conditional

prob mass fus

X

Y indep

fxycx.us 1 6 ly y

V xD

for

y with fyly

  • f

y yCx

fXYk

fy y conditional prob densityfn

f yay

fy g fxpy.gl

fxkHyµ

slide-6
SLIDE 6

Example

X

Unif0,10

fXl

Too

  • 3

10

ifX 3

ynunifo.MY

fyniffi yoe

  • w

F Y

E Y X x f lx dx

any total expectation

10

f Iz fo

dx

2.5

O

discreteselting

XRY discretenu s

fxylx.us

f Glfyµ

prfk x.to

w

Uniffoyo Uniffo x

PrfX

x Y y Pr

foot

PrfY y X x Pray