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1
Lecture3
Random Variables
Let (
Ω
,
F
,
P
)
be a probability model for an experiment,
and
X
a function that maps every
to a unique
point
the set of real numbers. Since the outcome
is not certain, so is the value
Thus if
B
is some
subset of
R
, we may want to determine the probability of
“
”. To determine this probability, we can look at
the set
that contains all
that maps
into
B
under the function
X
.
,
Ω
∈
ξ
,
R
x
∈
.
)
(
x
X
=
B
X
∈
)
(
Ω
∈
=

)
(
1
B
X
A
Ω
∈
Ω
R
)
(
X
x
A
B
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Obviously, if the set
also belongs to the
associated field
F
, then it is an event and the probability of
A
is well defined; in that case we can say
However,
may not always belong to
F
for all
B
, thus
creating difficulties. The notion of random variable (r.v)
makes sure that the inverse mapping always results in an
event so that we are able to determine the probability for
any
Random Variable (r.v)
:
A finite single valued function
that maps the set of all experimental outcomes
into the
set of real numbers
R
is said to be a r.v, if the set
is an event
for every
x
in
R
.
)
(
1
B
X
A

=
)).
(
(
"
)
(
"
event
the
of
y
Probabilit
1
B
X
P
B
X

=
∈
ξ
(1)
)
(
1
B
X

.
R
B
∈
)
(
⋅
X
Ω
{}
)
(

x
X
≤
)
(
F
∈
3
Alternatively
X
is said to be a r.v, if
where
B
represents semidefinite intervals of the form
and all other sets that can be constructed from these sets by
performing the set operations of union, intersection and
negation any number of times. The Borel collection
B
of
such subsets of
R
is the smallest
σ
field of subsets of
R
that
includes all semiinfinite intervals of the above form. Thus
if
X
is a r.v, then
is an event for every
x
. What about
Are they also events ?
In fact with
since
and
are events,
is an event and
hence
is also an event.
}
{
a
x
≤
<
∞
a
b
{}
{
}
?
,
a
X
b
X
a
=
≤
<
b
X
≤
}
{
b
X
a
b
X
a
X
≤
<
=
≤
∩
a
X
a
X
c
=
≤
{
}
)
(

x
X
x
X
≤
=
≤
ξ
F
B
X
∈

)
(
1
}
{
a
X
≤
(2)
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Thus,
is an event for every
n
.
Consequently
is also an event. All events have well defined probability.
Thus the probability of the event
must
depend on
x
. Denote
The
role
of the subscript
X
in (4) is only to identify the
actual r.v.
is said to the Probability Distribution
Function (PDF) associated with the r.v
X
.
≤
<

1
a
X
n
a
±
∞
=
=
=
≤
<

1
}
{
1
n
a
X
a
X
n
a
{}
)
(

x
X
≤
ξ
.
0
)
(
)
(

≥
=
≤
x
F
x
X
P
X
(4)
)
(
x
F
X
(3)
5
Distribution Function
: Note that a distribution function
g
(
x
) is nondecreasing, rightcontinuous and satisfies
i.e., if
g
(
x
) is a distribution function, then
(i)
(ii) if
then
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This note was uploaded on 04/24/2011 for the course ECE 6303 taught by Professor Voltz during the Spring '11 term at NYU Poly.
 Spring '11
 voltz

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