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Lecture 11
Sequences of IID Random
Variables
IID Random Variables
•
A great variety of applications require openended
measurements, which are random in nature
•
Such measurements may not be handled as N
dimensional vectors, but rather as a sequence (i.e.,
similar to series in calculus)
•
In a course of a large number of experiments, the
next trial does not depend on previous ones
•
In addition, the probabilistic behavior of the
experiment is invariant from trial to trial,
–
e. g. representing trials as random variables
we
get that
is independent of
.
i
X
j
X
j
i
≠
∀
}
{
i
X
±
Combining these observations yields the
following definition
Definition 1:
A sequence of random
variables
is called a sequence of
independent, identically distributed (IID)
random variables if the following
conditions are met:
1.
2. For the joint CDF of any finite
subsequence of random variables
we have
+∞
=
1
}
{
i
i
X
N
i
R
x
x
F
x
F
i
X
∈
∀
∈
∀
≡
,
),
(
)
(
L
k
n
k
X
1
}
{
=
∏
=
=
L
i
i
L
X
X
x
F
x
x
F
L
n
n
1
1
......
)
(
)
,...
(
1
Remark
•
Note:
The absence of any dependence in a
process means that the process has “no
memory”, i.e.
future is completely
independent of the present and of the past
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Sums of IID random variables
¾
Example:
In the Dow Jones game, the
total gain from an investment strategy is
the sum of the daily gains.
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 Fall '08
 Krim

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