11-02 Cont Random Variables (1)

# 11-02 Cont Random Variables (1) - B T R Y 4 8 S T S C I 4 8...

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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 2 3 7 S e c t i o n 4 . 1 : P r o p e r t i e s o f C u m u l a t i v e D i s t r i b u t i o n F u n c t i o n T h e C D F o f a n y r a n d o m v a r i a b l e X ( r e g a r d l e s s o f w h e t h e r i t i s d i s c r e t e o r n o t ) i s d e fi n e d a s f o l l o w s : F ( x ) = P ( X ≤ x ) , f o r a n y x ∈ R . B a s i c p r o p e r t i e s : 1 . F ( · ) i s a n o n- d e c r e a s i n g f u n c t i o n : i f a < b , t h e n F ( a ) ≤ F ( b ) 2 . l i m x → ∞ F ( x ) = 1 3 . l i m x → − ∞ F ( x ) = 4 . F ( · ) i s a r i g h t c o n t i n u o u s f u n c t i o n . F o r m a l l y : f o r a n y x ∈ R , l i m ǫ ↓ F ( x + ǫ ) = F ( x ) B T R Y 4 8 / S T S C I 4 8 F a l l 2 1 2 3 8 I m p o r t a n c e o f C D F : a n y p r o b a b i l i t y w e ’ d c a r e t o c o m p u t e b a s e d o n X c a n b e e x p r e s s e d i n t e r m s o f F ( · ) . E . g . , s u p p o s e a < b a n d c o n s i d e r P ( a < X < b ) = P ( X < b ) − P ( X ≤ a ) = ( ? ) − F ( a ) I n g e n e r a l : P ( X < b ) negationslash = F ( b ) . R e l a t i o n s h i p ? 1 . C a n w r i t e { X < b } = ∪ ∞ n = 1 D n , w h e r e D i = { X ≤ b − 1 n } . 2 . S i n c e D 1 ⊂ D 2 ⊂ D 3 ⊂ · · · , w e h a v e ∪ ∞ n = 1 D n = l i m n → ∞ D n a n d c o n t i n u i t y o f p r o b a b i l i t y t h e o r e m ( S e c 2 . 1 ) t h e n i m p l i e s P ( X < b ) = P parenleftBig l i m n → ∞ D n parenrightBig = l i m n → ∞ P ( D n ) = l i m n → ∞ F parenleftbigg b − 1 n parenrightbigg S a m e a r g u m e n t w o r k s i f b − n − 1 i s r e p l a c e d b y b − k n , w h e r e k n i s a n y d e c r e a s i n g s e q u e n c e o f n u m b e r s t h a t c o n v e r g e s t o . I n s u m m a r y : P ( a < X < b ) = l i m n → ∞ F ( b − k n ) − F ( a ) B T R Y 4 8 / S T S C I 4 8 F a l l 2 1 2 3 9 E x a m p l e : S u p p o s e X ∼ B i n o m i a l ( 6 , 2 / 3 ) . E x p r e s s t h e f o l l o w i n g p r o b a b i l i t i e s i n t e r m s o f F ( · ) , t h e C D F o f X . • P ( 2 < X ≤ 4 ) • P ( 2 < X < 4 ) • P ( 2 < X ≤ 4 . 9 9 ) • P ( 2 < X < 4 . 9 9 ) R e c a l l : f o r a n y x , F ( x ) = 6 summationdisplay i = I ( i ≤ x ) parenleftbigg 6 i parenrightbigg ( 2 / 3 ) i ( 1 / 3 ) 6 − i B T R Y 4 8 / S T S C I 4 8 F a l l 2 1 2 4 F o r a n y r a n d o m v a r i a b l e X : g i v e n a , b ∈ R a n d a d e c r e a s i n g s e q u e n c e k n t h a t c o n v e r g e s t o : • P ( a < X ≤ b ) = F ( b ) − F ( a ) • P ( a ≤ X < b ) = l i m n → ∞ F ( b − k n ) − l i m n → ∞ F ( a − k n ) • P ( a < X < b ) = l i m n → ∞ F ( b − k n ) − F ( a ) • P ( a ≤ X ≤ b ) = F ( b ) − l i m n → ∞ F ( a − k n ) L a s t f o r m u l a i m p l i e s : f o r a n y c , P ( X = c ) = F ( c ) − l i m n → ∞ F ( c − k n ) ....
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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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11-02 Cont Random Variables (1) - B T R Y 4 8 S T S C I 4 8...

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