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variance and covariance July 23,2020 f Applications of of Expectation Linearity Variance 2 Covariance and correlation 3 Last time Distributions p if Xsl Bernoulli CP x PREM ipso p i f pin pj I Xu PHX Bincngp Linearity of


slide-1
SLIDE 1

variance and

covariance

July 23,2020

f Applications of

Linearity

  • f Expectation

2

Variance 3 Covariance and correlation

Last time

Distributions

p

if Xsl

x Bernoulli CP

PREM

p

ipso

Xu

Bincngp

PHX

I

f pin pj

i

Linearity of

Expectation

ECC

Xi

Cnxn

e E Exif

EWE

slide-2
SLIDE 2

1 Applications of Linearity of

Expectation

Example

Hand back assignments

to n students

at random

Expected number of students who get

back theirhw

Xn

the number of students who get back their hw

tn

use indicator

r.r.fi

ttsIhnedirn

tbEs gmaE

  • thers

PrEIj IT In

Xn

I

e

In

EE Ide EE In

c Xn

E

Iie

In

EECTS ETI Mtn

linearity

I

E

I

E a Prodi

a

s

lxprEIEDeoxprcq.ro

achoo't

e

I

n

The expectednumberof students who get their own

home works

in

a class size he

is

slide-3
SLIDE 3

Example Throw

me balls

in

n

bins

Be

The expected number of

empty

bins

X number of empty bins

Help Indicators help

Pr

2

ith bin is empty

n m

Etsi

  • therwis

Presist

B

n

Xs

EI linearity

Ist t

n

EEN EEE Ii

EE.CIIS.EE 5senInT.ED

is

E IITs

Eaxprosi ajslxprczisD DXPRET.es

aC 0,12

n

I

m

s

m

For

her and

n

EG

n

E

C EX

h hut

n l 1a

hero

37N 37

  • f

tinamou n1

Ie

bins

will

be empty

slide-4
SLIDE 4

2J Variance.se

Examle

Random Walk

Particle 4

p

p

p

p

g

r r

a

r B I N

N

J X G

S

4 3

Z

l

I

z

3

4 5

6

a The Particle

moves

in

  • ne

dimension

a At each

Zime Step Particle moves

to the right or

left

with equal P.robability

Sn

The Position of

the Particle after

n

moves

Xi

the 2th movement

I

PRED E

pre'T

L

n

Z

Sn

O

X

1

Xze

Xn

Eti

E

n

CCsn

E j Xi

EEN

  • gootuseful

E xD

IaxPrcai

t.xprexi D

CDxprcxz.se73

AGH R

s 1 12

C 1 Nz

D

what is the expecteddistance from 0

Isnt

slide-5
SLIDE 5

we

work SnI

ttiiiiii.axn5

E.xiezEjxixoEES.n

C

Xie

Xn

EE.fi Ee2Ey.xixi3

E ECxE3e

2 Eg.Efxixi3 Ei

n

E xE

EGxprfdisaJ

lfprcxi

D

Clfprcxz.si

aef

119

1 12

1 12

I

CEXiXj3s E

a bxPr

XI

a Xjsb

m

a betel

pr Xi

a xPrCXj b

1 4

pr Xist PVCxjsD

I

preexist PrCXj

I

4

53

sq

GD

prcxis DPrcxjsD fyc yprcxis.BPrfl.jo D

variance

1k

late

1 1,2

1 14

  • f

Sn

slide-6
SLIDE 6

Definition

For

a

r v

X

with expectationEcosef

the variance of X is defined

to

be

Varys EUx

w T

what does variance measure

The deviation from

mean

value

The square

root

Cx

FarTX

is

the standard deviation

  • f

variable X

Theoreme

Var

Ms

EE h J EExFEEEff

proof

Vaux

E

X e5

linearity

E

K2

2 1 V2

ECE

ECD

EM

EE N

2N

r

12

E Ext

v2

Some properties of

Variance

Var

Xtc

Varun

exercise Var

CX

I Varexy

exercise

i

slide-7
SLIDE 7

More Examples

Uniform distribution

Xa

a uniform random variable

  • n

the set 91 non

Uniformly off

PIKE In

VIELE zu

Var X

ECE

E Ex

9

EEN E.fi n5aa3 aEa

n InaEa

tnnengDEEX23

Ea2 PrExsa

a

h

and

NII

aef's TM

In 2

2

44

4 42

s

DR

6

Varcx

h

I 2

slide-8
SLIDE 8

Example

Hand back assignments

to n students

at random

Xn

the number of students who get back their hw

pr

Use indicatorsf

I

ft

studenti gets

theirhw

yn

  • therwise

Ans Ii e Iz i In Es Ii

exercise

ECM

Ty

Hitt

in

Eis ZEE

Var X

Var X

EEN

E EXT's E EXT

I

EUR3

ECCEIit

fi

ECI

t zig Eui Ij

DE

Ii

l Pr ZED on

Pr Ii Of Ln

cza s2xPrkisDxPIEf on.Tjen

napairEEXT

th

t

2

  • f indecis

haha

2 E

L

Z heh 1

varCX

s2

slide-9
SLIDE 9

Independent Random Variables

Theorem

For independent random variables

X Y

we

have

ECHIT

EIN ELY

independent

Proofof

X andY

EEEYI

Eae.AE gbaPrExsagY b 2

ab PrasadPREYS

aaPrEx a3JxfbEbPrcxsbD

sEEXJrEEYT.D

Theorem

For independent

random variables

y

we have

varcxey

Varchevarly

proof

CExpect'D

Varulay

Elway53 f Exey

E

212

EEx5ezE

EEY7eEEyI

linearityGO.EEig zEqyyypfqy2J

EExI 2EexLEEYJ7EY7

CEEx9 EExt e ENT Eey5 t2 EEXY7ECx3ECyD

slide-10
SLIDE 10

Vav.LK eVaVCy

l2

EExyJ ECx3ECyT

If

x

and y

are independent

  • Varcxey

star

Vary

If

x and y

are

independent then

remember

Varun y

Varun Evan

Var

Atty

Varix

Harty

varCA

t e Dwarcy

Varix

Vary

Example

kn

is the numberof heads

in

n tosses

  • f

a biased coin with heads probability P

ithtoss In

Bin en p

A

fine

p

xn Eti

n n

E

Kris EEE Ii3s

E EE Io

E P hip

jet

inn

Is

slide-11
SLIDE 11

Varun

s Var

Ishii

EI var

Ii

E PHP

Var

Ii

C Ii23

EE zig

2

f

nPhD

ExpresisBeozptiso

PE p p's Pel p

3

covariance and correlation

Definition

The

covariance

  • f random variables

X andY

is

CoV XM EEK G LY kDs EExt

E

EEN

p

Remarks

f

L

If Xgy are independent

CoV X Y

O

Can

we say

if

corex y

so

Roy are independent

No

the converse

  • f iz

is

not

true

2

Cov

Xix

s E Ex XJ

EEXTECD C EXT ECB

ratex

slide-12
SLIDE 12

3

Covariance

is

bilinear

For any collection of random variables

A

In

fy

gym and fixed constants

ai

a

san

g

big skim

Then

Cov

aixigo.ES bjYjJs

EaibjCovLxisyj

4

About CoVCR y

1 The sign

  • f

Xy

determines how X and Y are

related

2

The

magnitude

Is

hard

to interpret

Definition Correlation

suppose X and Y

are random variables

with

GCN

  • and

CY

so The

correlation

  • f x and y is

corrcx.yjscorcx.yjo.CN

y

slide-13
SLIDE 13

Theorem

I

feorrcxgyjflX

ayebifaso ocorrcx.gs 1

if

a

correngyys l

Corre E

t

Corry

A

s

l

corrciy

ecgyg.gg gsECxY3

EExIEaD

Ysaxsb

647601

E xcaxebD ECx3EEaxeb

Gex lat

X

64

6caxebjsevaraf.bg

M

aEEXDibEXDaEEXJ.BE

Exsa2Varcx

Lal Vara

t

Ei

i

fairy

a

a

l ASO