Random Variable
1
Random Variable
1.1
Let
(
29
,
,
P
Ω Γ
be a probability space. Let
X
be a function from
Ω
to
R
, that is,
:
X
R
Ω →
, where
R
is
the set of real numbers.
X
is said to be a
random variable
if for every real number
a
,
{
}
Γ
∈
≤
Ω
∈
a
X
)
(
:
ϖ
ϖ
,
therefore, for a random variable
X
,
{
}
(
29
:
(
)
(
)
is defined
P
X
a
P X
a
ϖ
ϖ
∈Ω
≤
≡
≤
.
1.2
Let
:
X
R
Ω →
be a random variable. The
cumulative distribution function
(cdf) or just a distribution of
X
is defined for any real number
x
by
{
}
(
29
( )
:
(
)
(
)
X
F
x
P
X
x
P X
x
ϖ
ϖ
=
∈Ω
≤
≡
≤
.
1.3
X
is said to be a
discrete
random variable
if the range of
X
is discrete (the range of
X
contains finitely
many or countably infinitely many real numbers).
Let
:
X
R
Ω →
be a discrete random variable. Suppose
(
29
{
}
1
,
,
,
i
X
x
x
Ω =
L
L
.
Let
(
29
{
}
:
i
i
A
X
x
ϖ
ϖ
=
=
,
1,2,
i
=
L
. Observe that
(
29
{
}
:
i
i
A
X
x
ϖ
ϖ
=
=
,
1,2,
i
=
L
is a sequence of events,
which form a partition of the sample space
Ω
.
The function
( )
X
f
x
, called the
probability mass function
(pmf) , or just mass function, of
X
is defined as
follows:
(
29
{
}
(
29
(
29
(
29
{
}
{
}
1
2
1
:
=
if
,where
,
,
,
,
( )
0
if
,
,
,
,
i
i
x
i
i
i
X
i
P
X
x
P A
x
x
x
X
x
x
x
f
x
x
x
x
ϖ
ϖ
∈Ω
=
=
∈
Ω =
=
∉
L
L
L
L
L
.
( )
X
f
x
satisfies:
(a)
0
( )
1
X
f
x
≤
≤
,
(b)
{
}
1
2
,
,
( )
1
X
x
x
x
f
x
∈
=
∑
L
,
(c)
(
29
(
29
{
}
(
29
(
29
{
}
1
2
,
,
:
X
x
A
x
x
P X
A
P
X
A
f
x
ϖ
ϖ
∈ ∩
∈
≡
∈Ω
∈
=
∑
L
, where
A
R
⊂
.
The probability model of a discrete random variable
X
is a list of the distinct values
x
i
of
X
together with their
associated probabilities
(
29
{
}
(
29
(
29
(
)
:
X
i
i
i
f
x
P
X
x
P X
x
ϖ
ϖ
=
∈Ω
=
≡
=
:
(
29
(
29
(
29
(
29
1
2
3
1
2
3
X
X
X
X
X
x
x
x
f
x
f
x
f
x
f
x
L
L
Example#
If a coin is tossed twice, the number of heads, which turn up, can be 0, 1 or 2 according to the outcome of the
experiment.
Sample space
{
}
HH
HT
TH
TT
,
,
,
=
Ω
,
Γ =
P
(
29
Ω
, power set of
Ω
, and
(
29
E
n
P E
=
Ω
.
(
29
,
,
P
Ω Γ
is a probability space.
1