8 . I

s e s s i o n 8

IndependentExperiments 8 . 2

Sometimes,t h e outcomes o f t h e t w o

c o n s t i t u e n t experiments a r e unrelated:

A x s z I S , x B ,

H A E F ,

T t B E F zindependent

I n t h i s c a s e w e sayt h a t t h e t w o

experiments

( S , , F , , P , ) and C S - F a , Pz) a r e

Independentexperiments

F o r independent experiments, w e8 . 3

assign t h e probability P C . ) a s

P l a x B ) = P¢AxSz) A ( S ,XB))= P ( A t f ) . P ( b ,XB) indep.experiments

= P, (A) . Pz ( B ) consistency conditions

T h e axioms o f probability f i l l i n t h eprobabilities o f events t h a t c a n n o t b e

wr i t ten a s c a r t e s i a n products ( b u t c a n

b e w r i t t e n a s a u n i o n o f disjoint'ca r tes ian products.)

S o f o r t w o independent experiments 8 . 4

( S . , F , , P , ) a n d ( S a , F z , P z )

w e h ave a combined experiment

( S , F , P ) w i t h

1 = 1 × 1I = o f { A x B : F A E # , a n d V .B E E } )

P ( AT B ) = P, CA) - PzCB)

T h i s gives t h e probabilitieso f t h e

cylinder s e t s A X B , and t h e a x i om s

o f probability f i l l i n everything e l s e .

8 . 5÷÷÷÷:÷÷:ii÷÷÷÷÷÷i:÷.amb e i n r a n d o m experiments

• F o r m t h e combined experiment ( S , F , P )w i t h S = S i xS - x - s - i b n

F : O ({A,xAzx... × A n : VA ,EJ , ,VAzEFz , . . . -VAnEFn3)

I f t h e experiments a r e independent, t h e n

PIA,xAzx---+ An) =P,CA,)- Pz( A z ) . . - PnCAN)

VA,E F , , VA>EFz,...,YAnEFn

Then t h e axioms o f probability f i l l i n t h e

r e s t .

8 . 6ImportantspecialcaseiBernoullitrials

- Consider a simple experiment ( l o , Fo,Po).- N o w a s s u m e I independently performt h i s experiment n t i m e s :

( S , , F , , P, ) , IS, ,Fz,Pz), . . . , I sn .tn , Ph),where

I , = S , = . . . = S n E S o

I , = I z = . . . = I n E t o

P,C.) - P z (a ) = . . . = P , C.) = Po c . )

We w a n t t o fo rm t h e combined experimento f t h e s e n independent t r i a l s .

8 . 7O u r combined experiment:

I = S , X S - x . . - t h , = SoxSox . . -x So

I = o ( { a l l cylindersets})u - t . i e #

N ow supposeI h ave a n e v e n t A c - t o

having probability P o ( A ) = p ,o s p o i l .

When I repeat t h e simple experiment

n t i m e s independently, I w a n t t o know

t h e probability o f B e e f d e f i n e d a s

13k£ A o c c u r s exactly k t i m e s i n

n repetitious o f ( S o ,F o , Po )

Let ' s w r i t e t h e probability o f 8 . 8

B e a s

Pulk) E P ( B k ) .

theorem: pack)= (k ) pkc,-pin-k

where p = p ( A ) .

' '5=0' ' ' 2 . ' ' i n ,

Prooff: s e e Papoulis

w e s ke t c h t h e proof a s f o l l ow s . . .

8 . 9

Ideas: T h e r e a r e exactly (Ya)sequences

o f o u t c om e s o f t h e

simple experiment where A

o c c u r s exactly k t i m e s a n d

A- o c c u r s n - K t i m e s .

YI n - 4,k - 2 : (42)=z¥z,- "if = 6

A A A A ; AAA A-, A-A A A

A-F A A , A-A A-A , AA-A-A

8 . 1 0• E a c h sequence corresponds t o a n

e v e n t E j E F .

• E a c h e v e n t E j has probability

P ( E j ) = pkg-pg"-k•• T h e eve n t s E j a r e disjoint.i . pack) - P (Bk ) = P (E ,U E - U . . . UE,,q,)

= P I E , ) t P ( E z ) + i i . T P ( E x )

= ( 1 ) pkc,-pin-K •

n i b . T h i s i s a binomial pouf !

8 . 1 1Examplew: A b o x contains t w o

co ins , a f a i r c o i n

and a "strange" c o i n .

• T h e f a i r co in h a s a probability 'z o f

coming u p "H e a d s " w h e n t o s s e d .

• T h e "strange" c o i n h a s a probability 3=1o f coming up

"Heads" wh e n tossed .

• I se lec t a c o i n a t random f r om t h e box

a n d f l i p i t 4 t imes . I t c o m e s up "Heads"

t h r e e t i m e s .

Q : what i s t h e probability I selected t h e "strange"c o i n ?

¥ = { F , S }F = f a i r g . i zs = strange

G z = { H H H H , H H H T , . . . ,T T T T }

S = S , X S , C I select a c o i n , a n d t h e nI t o s s t h a t c o i n 4 times.)

L e t F = f a i r c o i n i s selected

s = strange c o i n i s selected

A s = 3 "Heads" i n 4 t o s s e s .

w h a t i s P l s l A s ) ?

PCA315)PCs)P ( s ) A s ) = P¥a¥ =

i f8 . 1 3

Assumption: P C s ) - P I F ) = ' z

PCA.IS) -(43) (F)31¥)'s 412¥11't,)= 26¥

PCA3 ) = P I A , A S ) = P (Asf ( SOF ) )-

= P ( ( A h s ) UCANF))3

µ §disjoint

= P (ALS) t PCA I F )= P (Assts) P ( s ) t PCA , I F ) PCF ) = . . .

8 . 14• o o = 26¥. Iz t (43)(k)3l§#{

= 2,7µg t Ig = 4,13¥ = P (Az )

① (s lo ts ) = PCA3ptsn.PH= l¥¥}¥

- -

= 2413 I 0 . 6 2 7 9

P ( F IA , ) =

P/A3¥a]¥[..= ¥,

- -

I 0.3721

Claussiaprobabinity 8 . 1 5

I n c l a s s i c a l i t y , w e h a ve CS,F , P)

i n which

• S i s f i n i t e : I S I = n

• I = P ( S ) : I F 1 = 2 "

• A l l outcomes a r e equally likely,which m e a n s w e h a v e p mf

p ( w )= In ,

t w o s

⇒ P I A ) - I , pews = 'Ant = Hy,W E A

S o i n classical probability, w e have 8 . 1 6

• P ( d ) = ¥1, = In = O⇐

• P l s ) = ¥ , = I = I

• I f A , B E F a r e disjoint ( i .e AhB=¢)t h e n

PCA0131=14%34 = latte=IASI, +113

£= PCA) t PCB).

Wha t about A, B E F t h a t a r e

n o t d i s j o i n t ?

① (AUB ) = 14¥34 = ¥ ¥ ¥ I 1= # + ¥ , ; ¥ ,

" ' "

= P f a ) + P C B ) - P (ARB)

T h i s extends t o general f i n i t e u n i o n s . . 8 ' ' 8

I f A , , Az , . . .,A n E F a n d they

a r e disjoint, t h e n

Ain A j =D , f i t ;

t h e nI#Ail = ¥41A i l

⇒ F o r classical probability, i f A, , , , , , A na r e disjoint, t h e n

i i i . A i l = i¥j¥=÷÷, 'fit,=÷¥MAi)

8 . 19N o t e also i n classical probabilityt h a t s i n c e

t hy = 1 1 1 - L A I

P I E ) = I A Ii s ,

= '41,1= I - 1µg,

= I - P I A )

S o t h e c l a s s i c a l probability m e a s u r e

behaves l i k e a general axiomatic prob. m e a s u r e .

I t c a n b e argued t h a t Kolmogorovselected

(the a x i o m s o f probability based o n h o w

c l a s s i c a l probability b e h a ve s .

8 . 2 0Exam't

Fiona

8 . 2 1

?I÷:÷÷÷÷.at.me...r a n d om experiment taking o n a numerical

v a l u e . ' '

Mappingsandfunctions.

Define: G i v e n t w o a b s t r a c t spacesS a n d A , a n A - v a l u e d

Sanction o r mapping

f : S → I t

i s a n assignment o f a specific elementf r o m A t o e a c h e l eme n t o f S .

8 . 2 2T h u s , given WES , few )EA ,

V w E S .

G w e n

f : S - A

S i s c a l l e d t h e t ox i n o f f ;

J t i s ca l l ed t h e r a n g e o f F .

Exampled: I n calculus w e study r e a l va luedf u n c t i o n s o f a rea l v a r i a b l e :

f : I R - I R .

S = R a n d A - R .

8 . 2 3

Defy: Given any t w o s e t s F C S

a n d G C A , w e d e f i n e t h e

image o f F u n d e r f a s

" f ( f ) " I { a c -A : a = f e w ) f o r s o m e WEE}.

a n d t h e pre-image o r inverseimageo f G C A u n d e r f i s defined a s

"f-'

(G)":{wes: f i n e G }• f ( F ) i s t h e s e t o f a l l points i n A obtained

bTynapping points i n F throughf .

• f -'(G) i s t h e s e t o f a l l points W E L t h a t map t o G .

Pictorially, t h e s i tua t ion appears8 . 2 4

a s fo l l ow s :

f c . )

O¥##±:¥OE¥¥÷"

RandomVarniables 8 . 2 5

• We o f t e n c h a r a c t e r i z e t h e o u t c omeo f a r a n d om experiment by a

number (perhaps a measurement.)

9 1 - I n d u c e d e lec t r i c c u r r e n t i n

a n a n t e n n a d u e t o random

therma l motion o f charges.

- T h e n u m b e r o f "H e a d s " i n a

sequence o f N independent

co in t o s s e s .

KeyIdeai E a c h outcome produces a n u m b e r .

I n t u i t i v e ! Given C S ,F . P ) a 8 . 2 6

randomvariable i s a mapping

f r o m S t o t h e rea l l i n e 1 1 3 :

X : S → I R .

X "

RQ.in#.Fe.....