1 INDEPENDENT RANDOM VARIABLES
Lecture 9
ORIE3500/5500 Summer2009 Chen
Class Today
•
Independent Random Variables
•
Conditional Distribution
1 Independent Random Variables
As the name suggests, two random variables are independent if they are not
related with each other in any way. Having informaion on one of them does
not reveal anything about the other. It is deﬁned as follows:
X
1
,...,X
n
are
said to be independent if for any sets
A
1
,...,A
n
,
P
[
X
1
∈
A
1
,...,X
n
∈
A
n
] =
P
[
X
1
∈
A
n
]
···
P
[
X
n
∈
A
n
]
.
Recall the deﬁnition of independence of events. Here, for every set
A
1
,...,A
n
,
B
1
= [
X
1
∈
A
1
]
,...,B
n
= [
X
n
∈
A
n
] are events. For any r sets
B
k
1
,...,B
k
r
chosen, we have
P
[
X
l
1
∈ <
,...,B
k
1
,...,B
k
r
,X
l
(
n

r
)
∈ <
] = 1
·
P
(
B
k
1
)
· ···
1
·
P
(
B
k
r
)
·
1
.
Here,
l
j
denotes the subscribe not equal to any
k
i
. So random variables
X
1
,...,X
n
are independent if for any collection of sets
A
1
,...,A
n
, [
X
1
∈
A
1
]
,...,
[
X
n
∈
A
n
] are independent events.
We also learned before that the joint cdf captures all information about
the the probability distribution of random vectors. So we should expect that
there will be some criterion involving cdfs which will characterize indepen
dence of random variables. Random variables
X
1
,...,X
n
are independent if
and only if we can write the joint cdf as the product of the marginal cdfs of
the random variables, that is,
F
X
1
,
···
,X
n
(
x
1
,...,x
n
) =
F
X
1
(
x
1
)
···
F
X
n
(
x
n
)
,
∀  ∞
< x
1
,...,x
n
<
∞
.
This means that given the marginal distributions of the
independent
ran
dom variables, the joint distribution of the random vector generated by those
random variables is determined.
1
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View Full Document1 INDEPENDENT RANDOM VARIABLES
Similarly, when
X
1
,...,X
n
are discrete, they are independent if and only
if their joint pmf can be written as a product of their marginal pmfs, that is,
p
X
1
,
···
,X
n
(
x
i
1
,...,x
i
n
) =
p
X
1
(
x
i
1
)
···
p
X
n
(
x
i
n
)
,
∀
x
i
1
,...,x
i
n
.
And the case is that of jointly distributed continuous random variables. In
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 Summer '08
 WEBER
 Probability theory, independent random variables, bkr

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