Ses s i o n 1 9

Reca l l . . .

1 9 . I

÷::÷÷÷÷÷i÷÷.( S , F , P ) , why n o t two?

no:÷÷¥i÷÷÷÷÷: Ye . )→

Recall....

1 9 . 2Defy : T h e j o i n t e d o f t w o

R Vs def ined o n CS, F , P ) i s

t h e probability o f t h e event

{ H e x } n E t s y } :

F*µcx,y1EPl{Xex3n{Yey3)= P ({web:Xcw1sx3n{WES:,Ywky3)

W e specify F * , i x .y) f o r a l l i x .g)E R ?

Deff : T h e joint pdf o f t w o R V s 1 9 . 3

X a # f i n e d o n ( S ,F , P )

i s

f*µHy'I £§¥§I.

%5ffi.fi?toheiomtp#cii-fffIf*.wcx,yldxdy=l( i i i )

"f § f * µ Id.pl d o dp = E , ex ,y ' .- i s - i s

1 9 . 4( I v ) F o r any DEBC I R ? )

P ( { ( X ,Y ) e D 3 ) = 5 5 £ , any) did,D

= I f f * , cxiylotpccx.gl/dxdy1122

I,I X .YI E D

I s . I,,((x,Y)) = { O, Cx,yI¢D

15.16

Reca l l . . .

1 9 . 5StatisticallyIndependentRVs

De f n i T w o R V s X a n d Y def ined

o n I S , F , P ) a r e statisticallyindependent i f t h e e v e n t s

{ * c -A } a n d { Y e t } a r e

statistically independent f o r

a l l A , B c - B C R ) ,

Reca l l . . .1 9 . 6T h u s w e c a n t a k e a s a n equivalent

def in i t ion . . .

Define: T w o jointly distr ibuted R V s

X a n d Y a r e statisticallyindependent i f f

£*, Ny) = fact) . f , ly).

Suppose w e h a v e 1 9 . 7

R V X d e f i n e d o n I S , , F , , P , )a n d

R V Y d e f i n e d o n 1 1 2 , # , Pz ) .

W e c a n f o r m a j o i n t experiment

( S , F , P) w i t h

S - S , X S -F = o( {AxB : A E F , and

B E Fz } )

P = a probability m e a s u r e consistent

w i t h P , a n d P z

Th e n X a n d Y c a nb e viewed a s

jo int ly d i s t r i b u t e d o n C S , F , P ) ,

theorem: I f random experiments 1 9 . 8

( S , , F , , P , ) a n d (Sa,Fz,Pz) a r e

independent experiments, t h e n t h e j - d i s t

R V s X a n d Y o n l l s , I , P ) a r e

statistically independent, where * w a s

def ined o n ( S , , F , , P , ) a n d Y w a s

def ined o n ( S z i t z , Pz ) .

O n e f u n c t i o n o f t w o R 1 9 . 9

G i ve n t w o j - d i s t R V s X and Y and a

f u n c t i o n

gc . . . ) : R - → R ,

w e c a n f o r m a n e w R V

€ - g CK, Y ) .

G i ve n f * y ' x .y ) o r F*µ i x .y) and get,y),w e w o u l d l i k e t o f i n d fzcz) o r Fz ( Z ) .

L e t D z C I R ' ( D z E ① ( I R 2 ) ) 1 9 . 1 0

D E E { ( x , y ) E 1122: g a r ,y ) E Z } ,A Z E R

÷ x

T h e e v e n t

{ # ¥ 3 = {gax.iu.az}[E¥}

= { ( X , W ) E D z } ←

= { W E S : (Kew),Yew) ) E D z } .

1 6 . 9

i . [ I z ) = P ( {FEZ} )=P( { (X ,Y )eDz3) 1 9 . 1 1

= I f f * , ex,;) dadyD z

= I f f * , imy). Izzccxay)) dxdy.1122

We c a n f i n d t h e j o i n t pdf a s

fzcz) = dFEfz¥.

I 9 . 1 2Examples: g i x .y ) = x t y

Z =

g CK,Y )= X + Y

Fz I z ) = P I { Z E E I 3 )= P ( { ( X , Y)EDz3)

where D z = {Cx,y ) e 1122 :xyt.LI#3z-ThusFzCZ)=f,yzff*pH,y3dxd/

t o z- y¥¥¥¥g¥

¥faf*, i x .y ' * DY

N o w i f X a n d Y a r e statistically 1 9 . 1 3

independent ( X 1 Y ) , t h i s becomes

Fzcz) = f f f#ext - f , ly) d x dy÷÷.= 1- ( f f * A ) d)f , ly) dy

- a s

¥ E y T•

= § F* (z-y) f.ua/1dy.

1 6 . 1 2

19.14Furthermore,

fzcz) = dfIz¥ = adz { ,§f , g ) Fate-'1)dy}= [ f , up d%{¥Y)dy

i s commutative=-f)f r 'y) f * e z - y, y,convolution

= ( fy#f*) ( z ) = (f**fµ) Cz)convolution integral

1 9 . 1 5

theorem. L e t X a n d Y b e t w o

j - d i s t ,independent R V s wi th

marginal pdfs f a s t ) a n d frey),

respectively. T h e n t h e pdf o f

t h e i r s u m € = K t Y i s

given by t h e convo lu t ion

fzcz) e ( f * # f , ) ( z )= [facxlf.plz - x ) DX

= f f , ey) facz-Y) dy.- i s

Exampled: L e t X a n d Y be t w o j-dist 1 9 . 1 6

independent exponential RVs , bothw i t h m e a n 1 . L e t

# = * + 'Y.

F i n d f z c z ) .

fzcz) = LI f , l y, f a Cz-y) dy=[ #exp ( 1 ) 'tailgate'Pft¥)%¥;'d,= It#exp(Eu ) dy =¥ze×ptE)'%¥j.

1 9 . 1 7Tw o f u n c t i o n s o f t w o R

Given t w o R Vs H and Y wi th j -pdff*µ Cx,y), and given t w o n e w R V s

Z - g ( X . Y ) ,

W - h ( X ,Y ) ,

w e w a n t t o f i n d few I Z ,w ) .

W e w i l l s t a r t by finding FE i w

( Z ,W),

t h e joint c . d .f .

Fzulz,w ) = P ( { E E - 2 3 1 { N E W 3 ) 1 9 . 1 8

= p ( { ( X i u ) EDzw3 )where

D z w€ { (x .y) E R ? gcx.ysszaudhcx.glEw}

"Few ( Z i n ) = f f f a , I X . y) dxdy

Dewa n d

fzwcz.ws = 22FII.gg#.

19.19

÷±÷I÷÷÷÷÷:÷÷÷÷...".ru.w i t h j - p d f f * , Ix,y). L e t

Z = g CH,Y ) a n d W - h ( * i t ) , a n d

a s s u m e t h e functions gex,y) and Why)

satisfy t h e following cond i t i ons :

( 1 ) T h e equations z=gcx,y) and w e hurry) c a nb e uniquely (simultaneously) solved fo r × andyi n t e r m s o f Z a n d W .

( 2 ) Th e partial derivatives 22¥,22¥,¥z,¥w

e x i s t a n d a r e continuous.• • o

(Theorem continued)1 9 . 2 0

T h e n t h e j - p d f o f Z and W i s

fzuftiwl-fxylxlz.ws,-117W)))}{¥¥),

wh e r e t h e Jacob ian is t h e d e t e rm i n a n t

2¥ Fu'si:#¥ f- 2¥.LI - I I . 2¥.2 z

22¥

Proofs: S e e Papoulis

Example: L e t X and Y be t w o 1 9 . 2 1

z e r o - m e a n i . i .de (independent, identically

distributed) Gauss ian R V s , both wi thv a r i a n c e 0 2 ;

1×+1×41=-5*1×1 f.ua/1=z'Fozexpf-lx2ztoYzI}.

L e t R E - F t a n d Q E tan' l 'T,X)

1-a x i sF i n d ftp.olriol.""i¥¥÷÷÷....r = 1 ¥ 1 2 , Q = tarity,x )

X C r ,Q ) = r c o s t

ye r , o ) = r s i n o

c o s t - r sing

1 9 . 2 2

I:#it:÷÷÷¥.tts i n Q r c o s G

= r c o 520 t r s 1h20 = r (cos>O t s 1h20) = r ,

Ii . f , for,G )

= # ( H r , 0))-¥4150)) I r l

= ¥ . exp {'%EFFrt.IE#.I4=zr=exp( I E ) . too.is?'teriFs'

1 9 . 2 3n d

f , c r i = ftp.o.fr, o ) d o=-?-=#exp ( I o ) ' 'to?R ¥ g h ¥ f

face) = ? (exercise)

W e c a n loosen t h e constraints ( i ) 19.24

and ( z ) o f t h e l a s t theorem. . .

theorem: l e t X and Y be t w o j-dist R V sw i t h j - pdf f *µ i x .y ) , a n d l e t

Z=glX,Y) a n d W - h CK ,Y ) .

T o f i n d fe z , I z ,W), w e m u s t f i n d

a l l r e a l solutions CXn,yn) such t h a t

gcxn.tn) - Z and hen,yn) - W ,

f o r n = 112, i i . , N o

I9 . 2 5t h e n

fzwlz.ws = ¥ , ( Y 'Zin'sY i'Z ,w))) 221¥14,/+ . . . t f * , N , I z ,WI,Y , H ,w)))2%1,14 /

= E' £ , l x n ' z ,wt,Yn ' ZND' I224¥14).h = I

1 6 .2 4

Auxiliaryvariables 1 9 . 2 6

• Sometimes, w e w a n t t o f i n d t h e pdf o f

Z - g ( X i i )when f * , i x .y) i s given.

• I t ' s o f t e n e a s i e r t o u s e t h e d i rec t

technique ( H , Y ) 1 0 (EL, 1W) t o f i n d

f z ( z ) .

• B u t w e only have o n e R V € .

w h a t d o w e d o ?

1 9 . 2 7• Yo u introduce a n (arbitrary)

aux i l i a r yR1 W - h CK ,Y ) .

• T h e n you f i n d few l a w ) using t h e

d i r e c t p d f method.

• Then you integrate o v e r w t o get fzcz):

felz)=§fzµlz,w ) d w .

H ow d o yo u pick t h e a u x . R V 1W - h CK,' i )?

I . P i c k i t t o make finding few'Z ,w ) easy,2 . P i c k i t t o make t h e integration easy.