11-11 Joint Distr Random Variables (1)

# 11-11 Joint Distr Random Variables (1) - B T R Y 4 8 S T S...

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Unformatted text preview: B T R Y 4 8 / S T S C I 4 8 F a l l 2 1 3 8 C h a p t e r 6 : J o i n t l y D i s t r i b u t e d R a n d o m V a r i a b l e s E x a m p l e ( 6 . 1 . 1 a ) : S u p p o s e t h a t 3 b a l l s a r e r a n d o m l y s e l e c t e d f r o m a n u r n c o n t a i n i n g 3 r e d , 4 w h i t e , a n d 5 b l u e b a l l s . L e t X a n d Y d e n o t e , r e s p e c t i v e l y , t h e n u m b e r o f r e d a n d w h i t e b a l l s c h o s e n . P o s s i b l e c o m b i n a t i o n s : Y = Y = 1 Y = 2 Y = 3 X = ? ? ? ? X = 1 ? ? ? ? X = 2 ? ? ? ? X = 3 ? ? ? ? E . g . , i n c o m b i n a t i o n , “ X = a n d Y = ” i s t h e e v e n t “ 3 b l u e b a l l s ” . B T R Y 4 8 / S T S C I 4 8 F a l l 2 1 3 9 D e fi n i t i o n : G i v e n a n y t w o r a n d o m v a r i a b l e s X a n d Y , t h e i r j o i n t p r o b a b i l i t y d i s t r i b u t i o n i s c o m p l e t e l y d e t e r m i n e d b y t h e i r j o i n t C D F : F ( x , y ) = P ( X ≤ x , Y ≤ y ) , − ∞ < x , y < ∞ T h i n k o f P ( X ≤ x , Y ≤ y ) a s P ( A B ) , w h e r e A = { X ≤ x } a n d B = { Y ≤ y } . T h e m a r g i n a l d i s t r i b u t i o n o f X : F X ( x ) = P ( X ≤ x ) = P ( X ≤ x , Y ≤ ∞ ) = F ( x , ∞ ) T h e m a r g i n a l d i s t r i b u t i o n o f Y : F Y ( y ) = P ( Y ≤ y ) = P ( X ≤ ∞ , Y ≤ y ) = F ( ∞ , y ) N o t a t i o n h e r e c o v e r s t h e b i v a r i a t e c a s e ; h o w e v e r , i d e a s e x t e n d i...
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## This note was uploaded on 11/29/2010 for the course STSCI 4080 taught by Professor Strawderman during the Fall '10 term at Cornell.

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11-11 Joint Distr Random Variables (1) - B T R Y 4 8 S T S...

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