week3_F10 - Online EE131A TA session : Week 3 TA: Ni-Chun...

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Online EE131A TA session : Week 3 TA: Ni-Chun Wang [email protected]
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Review : Independence • Events A and B are independent if P( A B )=P( A )P( B ). ( )( ). (P( A | B )=P( A ); P( B | A )=P( B )) general the events A A A e independent In general, the events A 1 , A 2 , …, A n are independent if 12 k 1 2 k ii i i i i i < i < i n P(A A A ) = P(A )P(A ) P(A ) ∩∩ "" – Ex: A , B and C are independent if P( A B C )=P( A )P( B )P( C ) k 1 i < i < i ≤≤ " P( A B )=P( A )P( B ) P( B C )=P( B )P( C ) P( A C )=P( A )P( C ) 2
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• 1. Show that if A and B are independent events, then and c A c d B and A c d c are also A and B , A and B, and A and B , are also independent. A () ( ) ( ) C PA B PA PA B ∩= −∩ B B ( )- ( ) ( ) ( )(1 ( )) ()( ) C PA PAPB PA PB APB = =− = => A and B c are indep. A c PAPB ( ) ( ) ( )- ( ) ( ) C B PB PA B PB PBPA = ( )) ( ) ( ) C PBPA = => A c and B are indep. ) [ ( ) ] CC C A B P A B = 1 ( ) 1 ( ( ) ( ) ( )) (1 ) (() ( ) ) PA PB PA B ∩∪ + 3 ( ( )) ( ( ) ( )) ( ) ( )) ( )(1 ( )) ( ) C =−− = C => A c and B c are indep.
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• 2. Suppose two dice are rolled. “Th fi t di h b A=“The first die shows an even number.” B=“The sum of the two dice is 4.” C=“The outcomes on the two dice differ at most by 2.” Are {A,B,C} independent? 31 1,3),(2,2),(3,1)} ( ) BP B = == 36 1 () 66 2 PA × × {( ,3),( , ),(3, )} 36 12 {(1,1) ~ (1,3),(2,1) ~ (2,4),(3,1) ~ (3,5) 24 2 2) ~ (4 6) (5 3) ~ (5 6) (6 4) ~ (6 6)} ( ) C C = = = (4,2) ~ (4,6),(5,3) ~ (5,6),(6,4) ~ (6,6)} 36 3 PC ⇒= 1 ( ) ()()() 36 PA B C PAPBPC ∩∩ = = 4 => A, B and C are not indep.
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This note was uploaded on 11/05/2010 for the course ELECTRICAL EE131A taught by Professor Kungyao during the Spring '10 term at UCLA.

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week3_F10 - Online EE131A TA session : Week 3 TA: Ni-Chun...

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