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View Full Document 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
• 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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View Full Document• 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.
 Spring '10
 KungYao

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