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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.
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