Session - Purdue Universitymrb/resources/LecturesF/Session_5.pdf · 76 Session 5 Recall: Last time we introduced the Borel field of the real line BCR). BUR) is the o-fieldgenerated

7 6

S e s s i o n 5

Recal l : L a s t t i m e w e i n t r o d u c e d-

t h e B o r e l f i e l d o f t h e

r e a l l i n e B C R ) .

B U R ) i s t h e o - f i e l d generated byt h e open intervals

o f t h e rea l l i n e

( i .e . , t he sma l les t o-f ie ld containing t he openintervals.)

n.be Sometimes w e n e e d t o 7 7

d e a l w i t h a sample spaceI = A C R

. ( e .g [ o i l ] )

s o o u r e v e n t space should b e

t h e B o r e l f i e l d o f A .

W e c a n get ① CAI by"cutting down"t h e Bo re l f i e l d o f R :

① ( A ) = { F ? A : I F c - BCR)}

Probabi l i t ies: 7 8

Intuitively: Assigns a n u m b e r

between O a n d I t h a t me a s u r e s

t h e certainty o r "likelihood"t h a t a n e v e n t w i l l o c c u r .

Mathematically-

i A s e t func t i on

P : F → I R

satisfying t h e axioms o f probability.

A x i ro bab i l i t y 7 9

I . ① (A) 2 0 , F A E F .

2 . P C S ) = I

3 . I f A , , A z , . . . , A n E F and

a r e d is jo in t , t h e n

P C II. A i ) = II, P I A ; )4 . I f A , , A z , . . . , Am, . . . E F

a n d a r e disjoint ,t h e n

① ( I . A i ) = ¥÷PCAi).

O t h e r properties o f P f . ) 8 0

t h a t f o l l o w f r o m t h e a x i o m s :

I . P(d) = O

Z . ①CFI) - l - P I A )

and o t h e r s . . .

Defy: A sequence o f s e t s 8 1

A, ,A z , . . . ,

A n . . .

i s s a i d t o b e increasing i f

A, C A z c As c . . .

and decreasing i f

A , > A - s A s o . ' ' .

Fa c t : I f A , , . . . , An, . . . 8 2

i s a n increasing sequenceo f s e t s

,t h e n

a s

1 i n An = U Ain o s i s t

( An = ¥1 A n )

I f A , , . . . , A n , . . . i s a decreasing

sequence o f s e t s,t h e n

Ling-An = :[Ai( A n = II,A i )

f a c t : I f A , , . . . , A n , . . . 8 3

i s e i t h e r a n increasing sequenceo f s e t s o f a decreasing

sequence o f s e t s,t h e n

① (Lings A n ) -l;izsPlAn).r e

-

l i m i t o f sets l im i t o f numbers

#

sequential continuity o f{theprobabilitymeasi

ExamplesofprobabilitySpacei 8 4

E x . I l e t S b e a f i n i t e sample-

space a n d l e t FCS ) b e

t h e power s e t o f S .

Suppose w e have a funct ion

p ( w ) :S → I R s i t

( i ) plan 2 0 ,V w E S

e i n I , pews = LW E S i .

Th i s f unc t i on i s ca l l ed a probabilitym a s s f u n c t i o n# 1p.m.f . )

I c a n u s e t h i s p.m.f. 8 5

t o d e f i n e a probability m e a s u r e

P : F - R

" s

P ( A ) = I i p i n , -VA E F .W E A

T h i s i s a va l i d probability measu re

n.hu pew)=P({w3), A W E S .

E x e s : A u n i f o ¥ 8 6

I = { w , , W z , . . . , W , 3 ( f i n i t e )

F f s ) = P ( s ) ,1 F f s ) ) = z "

pm f : pea) = In ,V w E S .

P I A , a ) = Iweia.P ' " = Ea. ( t )

classical

= th ¥a.'t = IAklothprobability

=/ A k / ( n .b . , IA,-I = n o . o f elements- i n A i a

)1 1 1

E x i t s : B i n o m i 8 7

S = { o , l , 2 , . . . , n }F C S ) = P ( S )

,1 F c s ) 1=2 " '

p ¥ : p ( K )= ( I ) ak ( i - a )

" -k, a c - [ o i l ]

K = O , l , Z , . . . , n

where (ya) = IK ! Cn-KI!

P CA) = ¥a put = ¥1 ( 1 ) a " h-a.)"-k.

I s t h i s pfk) a v a l i d p o u f ? 8 8

( i ) clearly, plk) - (Ya) aka-as"-kzo.- i n -> O t o Z O

n

( I i ) M u s t s h ow t h a t I ! (Ya) aka-a ,"- ' I l

k = '

fE×e¥)

* Binomial t h e o rem- :n

( a t b )"

= I i ( 1 ) a " b"- k

1<=0

f o r any t w o n u m b e r a a n d b .

¥ 4 T h e G e o m e t r i c pouf: 8 9

S = { 0 , 1 , 2 , . . . } ( S : { I , 2,3,...})F - P t s )

poufy." pCk ) = ( I - a ) a "

,a c - 1 0 , 1 )

K = O , l , Z , . . .

PCA) = Iziapfk) -E.'ace-a)a " ,AEF's?

I s t h i s a v a l i d pm f ?( i ) pck) = ( I - a ) a " z u

,H k = o , 1 , 2 , . . .

( i i ) P C S ) = I I .oci-a)a k a l . (exercise)

* t i n t : II.oak = # , l a l a ' t

E x . 5 T h e P o i s s o n pouf 9 0- - i

S = { 0 , 1 , 2 , . . . }F = P C s ) o ! = I

- : p Ck )=7 " E "

pouf - 1<=0 , 1 , 2 , . . .K ! 7 > 0

PCA ) = I , p a l ,H A E F ,

K E A

I S T h i s a v a l i d pouf?( i ) p ( k ) 2 0 ,

F k = D , 1 , 2 , . . .

( i i i P c t ) = E i I K E "

* t i n t ex-EI,¥,"'k i

= / ( e x e r c i s e )

Le t ' s l ook a t a n uncountable 9 1

sample space

E g . S - RF = ① ( I R )

P l . ) - H o w d o w e specifyt h i s ?

W e introduce t h e probability densityfunction fp.d.f.to#probabifes① FAI

,where A c - ① ( R ) .

Properties o f a pdf 9 2µ( i ) f o r ) z o , F r c - I R

c o

f i ) f f e r ) d r = l- c o

C - e v e n av a l i d pd f , w e get a

v a l i d probability m e a s u r e PC. )

f o r a n y A E ① CRI by integrating.

P ( A ) = {fer) d r =[far). Iacr) d r

where 9 3D

' tar) = { l , r e A

0 , R I Ai s c a l l e d t h e i n d i c a t i o no f t h e s e t A .