t I U PYP T TH Xvi I wz i i Pr I Then X A random variable - - PDF document

SLIDE 1

Distributions

July 27g2020

1

Geometric

Distribution

Application

The Copoun Collector's problem

2

Poisson Distribution

Question what

is the expected

number of times

we have

to

roll

a fair

6 sided die until

we roll

a

prew

63 55

65

bhim F 7

• Li

SLIDE 2

1 Geometric Distribution

Flip

a

coin

with Pr HJsP

until

we get H

Is

F r

X

the numberof

flips

until

the first

t

Heads appears

pr 9

Pr for instance

was H P

k

WITH

u p p

I

wz

T TH

U PYP

Xvi

I

t

i

i

Then

Pr X

I

Definition

A random variable

forwhich

Pr Asif

Ci pyo

p

Is

said

to have

the geometricdistributionwith

parameter P

X

GeometricLP

Sanity check

Pr xszT.si

Eod Fa

10

IEChigi

p P u Ps

PEELE

Px

p

prqtyyz.jp

Pxp

I

SLIDE 3

If

X

Geometric P

E

Me

Varix

Ect

a PrExsa

iprcxsigs.fi gPf

Theorem

For An Geometric

i

i

y P tuPisa.p

ECD II PREMD g g

p

N

E

E I PREF

Ei

Phx

i

prcxz.iq

jst

me

j I

E

SE

zprcxzi3

EiprEx3ieDsE9iPrcxs.i3

E7ci DPrExs.i

EI prcxzizli IED

zfprcx.si

I I

E

l P

Alternatively

p

E EX

Pe 2Pa

P

3pct PP 4pct Ppe

z

C P ECX3

O

Pu

P

2pct Pfe3pct PP

PEED

P

pet P

Pu Pfe

Pll Ps3

5

Ice pg

P

PEExJ

l

EEx3

SLIDE 4

Remember EEXJsaq.ua PrEXIST for V.V X

Then

Y

LOTUS

Define V.v Y Guy

EEY3

E gcxD

a agcaPrCX a3

function

Vara ECM

EEx

2

EEE

7 i2prcx i3 Efiki p

p

EEE

P e I U P P

9 et PTP

1611PPPs

a P EEN

• x

l P P

4

l PIP

9

l PPPe

g

tr v

PEEXT

P 3G Pj Pe 5

l PFP

7

l PPPs

w w

z

l2i 1

l p I

p

2 AE ics pyMp zf

u pJi p

2

EEK

I

I

PEER

f

i

ECXT z

tofpgsovarlN

eepg.EE

epa Ip tpz

pz pt

f

SLIDE 5

The

coupon

collectors Problem There

are

in different types of baseball cards

we get

the cards by buying boxes of cereal

Each

bon

contains exactly

• ne eard

This

card

is equallylikely to be any

• f the

n cards

Sh

The numberof boxes

we need to buy in

• rder to

collect all

n cards

What

is

Eton

Define Xis

• f cards

we buy before

WE get

the ith

new card

Then

Sn

X

Xz

Xn

so

E Csn

EE.EEtiJsE.ECXiI

what

is

the distribution for

Xi

SLIDE 6

Xi

Prc x

D

I

E

XD

lxPr AED

a

Xz

Pr

• ld

3

pre newEid Ten

Xz

Geometric

nz

CEXT

I

h

l

X38

pr

• ld card

2h

Proven

card

nz

3

Geomteric frenzy E EX

37 I

n

Z

i

Xi 8

Pr

• ld card

Pre new card I

fxinGeometrianizing EExi

th

n d D

E Csis

En Ect i3 EI

i

h E'T

9

a exercise

SLIDE 7

2 Poisson Distribution

Assume

The average number of

passing cars

Passing through

a tunnel per aint time is X

Define X The number of

cars Passing per

unit time

Question what is the

probability distributionofX

Poissondistribution

Definition

A random variable

for

which

prEx

e

Iso

• n

is said

to

have

Poisson distribution

An Poisson

X

Sanity

check

SPREED L

is 0

Eyrexsis.ES xeI E.E I

Exe

w

U

I

e

SLIDE 8

what

is

EEN

S the average

eext.EE

e

t

ei.E.EI

Is

e Ei

ie EiI.a

de txe

d

What

about

Var

X

var X

EEN

C Ex

2

e em Efi

EE iI

Ei uzEn

et.Eii i

e

es EI

eiEEn.te iE

e in.E

i.e i.ie

eit.EE I

etaE

i

e

e

SLIDE 9

e

He

Ethel

14h

EEx9 Varcx EEP

EExisted

432

1

varcxj.FI

Theorem

Let

X Poisoned and Yu Poisson beindependent

Poisson random variables

Tay

Then

Z Hey

Poisson hey

PIE

K

eHer

Proof

Pr

Zsk

Pr

x

Ysk

total Probabeity

n

E Pr

f I

t

K I

Iso

Jindefenden

Eko PVEX izxprcys.LI

n

E

iExI

e

K i

K

e UN

z

1

disk

i

Iso i

CK D

I

SLIDE 10

der

k

e

Iii

Cher

K

e

E

air

i

e Her

GK

Poisson

as

a Limit

• f Binomial

Theorem

Let

Xu BinomialLn In

where

A

is

a fixed constant Then

n

a

Paris

h

E

proof

• f

Pr

if

f piCi Psi

Pstn

The

idea

is

that

we

do

a testing

• n

many sample Points

n

and

we get

A success

for

the

• ccurrence
• f

the

n

events

so

X

is defined

sample points

nRss rateT

where

we

don't

know

p

but

we can get

an

estimate

• f

it by testing many Crusoe

SLIDE 11

Points

and counting the successful event

as A

so

Pstn

Xu Binomial hi

now

we

take

the

limit

• f

vi

• premise 7 LET u

5

h

u Inna Es

i

ni

• in

a

n

n

bi

hm

n

Ct ET

Et

Ct ET

Ct d Isi

prcxsidsk.atxe dxi

prcxsiss.IE