Chapter Taking The III : distribution 's of view Point is :* : : - - PowerPoint PPT Presentation

SLIDE 1

Chapter

III

:

Taking The

distribution's

Point

• f

view

SLIDE 2

"

"

is:*:::::

like

a

distribution

"

,

can

we find

a

random variable

so that

X -µ

② methods for creating

distributions

③ How

we

use this

" distribution "

perspective

SLIDE 3

what is

a distribution ?

A distribution p

: 93 -s IR

is

a

probability

function

• n

IR ( w/ r

• algebra 931

Recall

too :

The

information

• f µ

is

encoded

in

the

corresponding

Cdf

Fx

: IR -

SIR defined by

Fx ( x)

= y l l - o, sis)

what

properties

does a cdf

have ?

① ¥7

. . Fxcx)

• O

③ Fx

is

non

• decreasing

④ Fx

is

right

continuous

③ ¥7

. Fxlk)

• I

lie , if Exam a. the SF

,fknHFkD

SLIDE 4

Theorem ( Random variable

maker)

Ippon

that

F : IR -HR

satisfies

properties

①

• ④ of

a

chef .

Then

There exists

a random ravine

X

en

lo , D

so that

F

• Fx

. In

particular , it µ

is

a

probability

measure for

93

an

IR, then

there

exists

a

random

variable

X

so

that

X ya

.

II let

U

be

The

uniform

distribution

au lo , D

.

We

define

to :( Oil) → lo , D

by

flu)

= int { x

: FCK) > u} . we

claim

X

• flu)

has

Fx

= F .

SLIDE 5

Big

claim :

Glu) s x

iff

us FCK)

why?

Since

F

is

right

continuous,

we got

flu)

• in f { x : FCK) tu}

= min { x : FCK) > u}

So

flu)

is

the smallest

value

• f x

so

FCK) >u Hence

if

F- ( x) su

,Then

x > flu) For the

• ther

inequality

, since

F

is

non

• decreasing

we

have

pea) s x

then u EFC local) E FCK)

.

So : IP ( local Ex )

= BLUE FCK)) =

FCK )

.

DM

SLIDE 6

Core ( Independent Random

Variable

Maker)

let

y

, ,µ . . .

• be

any

sequence

• f probability

Anubis

• n

IR

.

Then

there

exist rhndom

variable,

Xi, Xz ,

. -

that

are

independent

so

that

Xi -mi

.

e-

• o -

So :

how

can

we

use this

fact

to produce

new

distributions

(and

hence

new random

variables ) ?

we'll

have

the

answers

SLIDE 7

therm

( Density

begets

distribution )

suppose that

f : IR

• i IR

is

a

Borel

mensurable

function

that

satisfies

f-30

and I

,

Ht) Hdt) =L

.

Then

the function

p

: 93-HR

by

pl

B)

= fpflt) 1pct) Hdt)

is

a valid probability

function

(and

hence

a distribution )

.

PI

check The properties of

a probability

function

.

k$1

SLIDE 8

How

else

can

we

create

distributions ?

theorem ( Adding Distributions)

Suppose

µ ,yo,

. . .

are

distributed

, and if

{pile IR> o

so that Epi

• 1 ,

Then f- ? pi Mi

is

a

distribution

too

.

PI

cheek properties

fer g

: 93→IR

to

be a

probability

measure

. ( use The fnot That Mi obey

relevant

properties to

confirm desired

axioms . )

SLIDE 9

How

are statistics

for µ

related

to

statistics for µ :*

if

p

• Z pi Mi ?

theorem (Expectation

is

linear in

distribution )

let y

: 93 -HR

is

given

by

p

• Epi µ

.

with

each

µ

a

distribution

. Then

for

ay

Borel

mensurable

filth → R

, m

have

Eff

)

= ? pi Eff )

• -

fpflt)Mdt)

? pi frflt) mildt)

SLIDE 10

PI we'll

cheek this

fer f

• IB

with Be 73. We

get

the

general

result

" in the

usual way

"

( extend to

smipkf

by limit, etc

. )

So :

we

need

Epl #B)

= ? pi

'Emil Hp)

.

LHS

:

IE, this)

= 1

• pl B) to

. MB

' )

• MLB)

RHS

: Epi Emil Hp) = ? pi Mi ( B)

• l ? piri ) ( B)

These sides

agree

some

µ

• { pipe

. .

by

hypothesis

.

DM

SLIDE 11

Ey ( see

how haiwitg of expectation

in distribution play , at)

Consider

p

• I 81

t t ft

t I

pinion,

let

X

be

a

random

variable

with

Xiyu

.

What

is

IE! X) ? By

chap at variables

:

IE (X)

• Ep ( id (x))
• Eyelid)

= Is Eg , lid) t t

les,lid) t I term. . ., lid)

go

= Is

. I

t 's .tt

's,{¥it%

SLIDE 12

What

is

Ep ( X

')

?

let

sq

: IR

• HR

be sgmnjfxn

.

Ep ( X')

=

Ep ( sq( x) )

=

1Er ( sq)

• t IES

, Isg) t f

IES, Csg) t Is Emma, (9)

• t

. I

' t's

• it 't Is §

e-t%

. t

'

. Hdt) = ¥ . I

' t ¥

Th

i ¥ . I .

Nak then :

N ( X)

=

Ep ( X')

• (Ep(X) )

'

=

⇐ t's t't E )

• ft

t E. it to )