Stat1301B
Probability& Statistics I
Fall 20112012
P.40
Chapter II
Random
Variables
and
Probability
Distributions
§ 2.1
Random Variables
Definition
A
random variable
ℜ
→
Ω
:
X
is a numerical valued function defined on a sample
space. In other words, a number
(
)
ω
X
, providing a measure of the characteristic of
interest, is assigned to each outcome
ω
in the sample space.
Remark
Always keep in mind that
X
is a function rather than a number. The value of
X
depends
on
the
outcome.
We
write
x
X
=
to
represent
the
event
(
)
{
}

x
X
=
Ω
∈
ω
ω
and
x
X
≤
to represent the event
(
)
{
}

x
X
≤
Ω
∈
ω
ω
.
Example 2.1
Let
X
be the number of aces in a hand of three cards drawn randomly from a deck
of 52 cards. Denote A as an ace card and N as a nonace card. Then
Ω
= {AAA, AAN, ANA, ANN, NAA, NAN, NNA, NNN}
The space of
X
is {0, 1, 2, 3}. Hence
X
is discrete.
{
}
3
,
2
,
1
,
0
:
→
Ω
X
such that
(
)
3
=
AAA
X
(
)
(
)
(
)
2
=
=
=
NAA
X
ANA
X
AAN
X
(
)
(
)
(
)
1
=
=
=
NNA
X
NAN
X
ANN
X
(
)
0
=
NNN
X
Using simple probability calculations, we have
(
)
{
}
(
)
78262
.
0
3
52
3
48
0
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
=
=
NNN
P
X
P
(
)
{
}
(
)
20416
.
0
3
52
2
48
1
4
,
,
1
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
×
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
=
=
NNA
NAN
ANN
P
X
P
(
)
{
}
(
)
01303
.
0
3
52
1
48
2
4
,
,
2
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
×
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
=
=
NAA
ANA
AAN
P
X
P
(
)
{
}
(
)
00018
.
0
3
52
3
4
3
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
=
=
AAA
P
X
P
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