I : continuous Random Variables last variable X :D - sik , - - PowerPoint PPT Presentation

Chapter

I

:

Discrete

&

continuous Random

Variables

last

For

random

variable

X :D -

sik ,

• get

a

distribution

µ :B -sik

by

MB)

IPLXEB)

• LPC X
• ' ( B ))
• get

a

cdf Fx

• HR

kg Fx ( x )

IP ( X E x)

(

C - A. x])

HE ( f-( X))

=

1Er ( f) "changeofvavibCe

NawSHffWT

① Describe

some

"

basic

"

random variables

in order

to characterize

discrete

random variables

② Gie

a

way

to characterize continuous

random variables

via

their

distributions

③ Continuous

version et

Law

• f

lazy

Statistician

I

let

CHR

Suppose

PlX=c)

• I

Then if

Xnm

and

if

BeBe

,

Then

µ (B)

=

IP ( x

• ' ( Bl)
• { to

CEB

CEB

• I, (c)

Defy

( Point - mass distribution)

For ER

,

define

Sc

: 93 → IR

by

8dB)

• Ipcc?

Observe

:

For

any

Borel

measurable f :IR -YR, we

have

§

, Htt Sddt)

= IES .

( f )

= Eplf Ix))

= IE,p(fk)) = fk)

"ch¥arib

we

know

from

before

that

we defined

discrete

random variables

as

random variables

which took

• n

a

countable number

• f

valves 443 , with

IP ( X

• xi)
• pi

satisfying ? pi

• I

Our

new

definition only

insists

that

"all the probability

"

for

X is bound up

in a countable

number et

values

Defy ( Discrete

Random Variable )

A random

variable

X

is

called discrete it

then

it

a countable collection

• x. , xz,

c- IR

so that

IP ( X

• ki)
• pi

with ? pi

• I

particular,

we

have

IP ( X

c- Ex . ,xz,

• 5)

= O

Then ( Distribution

chunotcnzahin of discrete

r

A

random

variable X

with

distribution µ

is

discrete

if

there

exist

x.pk

,

and

Epi

• I

so

That

µ

= E

pi Sci

PI The

" ⇒

"

direction

comes

from

• bserving
• ut

µ

LB)

= Pl X

• ' ( B))

= §

Pj

jB

= ?

pi ftp.lki)

= Epi 8, CB)

( you do the other half)

④

What

about

continuous

random

variables ?

Exe

( Nl 0,11) Suppose

X - NCO , l)

" old

" definition

was

Fx ( x)

= fo

"

'¥

e

• t% at

what

is the

distribution

n

.

93 → IR ?

"i'iii. "

"

s

. irani

Define

( Absolutely Continuous

random

variables

)

A

random variable X

is

called

absolutely

centners

( with

respect to

X )

if

the

exists som

Borel

faction

filth

• HR

so

that

the

distribution µ

• f

X

is

given

by

µ (B)

= fr Ht) Holt) Hdt)

The function

f

is

called the

density

function for

µ

Non

Suppose

that

X

• Sc

we claim

X

is

not

absolutely

continuous.

we need to show

that

There

is

no

measurable f : IR -1112

so

that

£ ( B)

=

§ ,

Htt 1pct) Hdt)

Nele :

Sc CB)

• Hplc)

particular

: Sella)

=

I

Observe

that

{ Ht) Health Htt)

= Hc) HKD to HMH) - o

• {

Hc)

if t

• c
• if the

thou ( continuous

version

• f

law of lazy

statistician )

suppose

that X yr

is

continuous

with density f

Then

for

any Borel

measurable

g

: IR -7112

,we get

IEM g)

'

Et fr GH) Mdt)

= § gltsflt) Mdt)

Tchany

• f

variables

IEplgcx))

Pf

we'll

again

duck this for indicate

functions

It

Then

follows

for

simple functions

lie , function

,

with

finite

image ) by

lcwmty A

expectation, then

fer

nen

• negation

Anakin,

by

MCT, and then

for

gaunt

functions

by

g

• gt
• g-

So :

let

g

• Hp

for

B

a

Beret

set .

Then

1Er ( HB)

= ! I , It) on Cdt )

= I

• pl

B)

t O

• pl

B

')

= µ

( B)

= !

f- It) ht, It) Ndt)

[def

'

n)

Daa

for ( old

• fashioned

defa

• f expected

value)

Suppose

XY

is

continuous

with density

f

Then

Epl X)

= fptflt

) Hdt)

HI

IE

( X)

=

Ep ( idk))

= Eyelid)

[

"

chge of variables

=

Ip fit) idk) Htt) ( last result)

T⑤