tYff distribution of random a variable connection between with - - PowerPoint PPT Presentation

Chapter

I :

Distribution

• f

a

Random

Variable

Lattin Defy

( Identically

Distributed )

A

collection

• f

random

variables { Xilie#

identically

distributed it

for

all

seek

and

any

i.je I

have

IP ( Xi ex)

• IPCX; ex)

theorem

{Xilie #

identically distributed

iff

fer ay

measurable

f : IR -1112

have

CECHXI) )

• IEHCX

;))

for

all

i.j

tYff

① distribution of

random

variable

② connection

between

random

Vanities with

equal

d.striations

and

Dein

( Distribution)

let

X

be

a

random variable

• n

Cr , F, IP)

the

distribution of

X

( aka

is the

fuckin

µ

given

by

Borel

sets

put B)

IP ( X EB) =P ( X

• ' ( B))

We write

X - µ

to

indicate

that

X

has

ddkuhh.eu

µ

AHemaknotnh.is#M--Mx--LCx)--lPX

"

t

Danger

! also notation

for IECX)

Def ( cumulative

distribution function

If

X

is

variable

the

cdf

• f

X

is

the function

Fx

' IR

defined by

Fx ( x)

• IP ( X a- x)
• all
• a. xD

Nate : to

say that

X

and

Y

are identically

distributed

is

the

same

as saying

Ex

• Fy

lemme

( doppelganger)

② ① X

and Y

identically

distributed

iff

Fx

iff- ③LIX) =L ( Y)

PI ① ⇐ ②

we've already

mentioned

above

① ⇐③

given

in the

midst of the past that

identically

distributed implies

equal integrals

Cory ( Equal

distribution implies equal integrals)

If

X

and

Y

are

random

variables

with

LIX) =L ( Y) ,

and

if f : IR -

HR

is

Boel measurable

function , Then

⇐ ( f- (x ) )

PI

By the

last

result, hairy equal distributions implies

X

and

Y

are

identically distributed

TX

One

final

• bservation : if

X-p

1112,93

is

probability

space

( r,F, P )

(R ,

M)

1112,93

, m)

For

f :D-2112

a

Borel

measurable function

,

we

have

z random

variables ① f

• n

1112,93,µ) ②HX)

• n (AFP)

HOW

ARE

THEY

RELATED ?

Lemmy

If

X

is

random

variable

• n

( r , F, p) with

X

• µ

and

if

• f. IRAN

is

Borel measurable

function ,

then

the random

variables

f : IR → IR

( under y)

and

FIX)

• b -' IR

( under

IP)

have

Ff

• Fffx)

PI

let KEIR

We have

Fux, (x)

• IP ( f (X) EX)

=p ( (full

• '

( C - A,x3 )) =p ( X

• ' (f
• ' ( C
• six] )))

=µ( f-

• A,x3 ))

Ff ( x )

1%1

Cor ( change of

variables )

• with the

same

setup

as

before

have

IE ( FIX))

E- ( f)

TEK

KEE p

• INI

( not

under

Lebesgue)

Recall :

IEFHX))

is

denoted { flxldlp

Hxllw) Mdw)

HE(f)