This preview shows pages 1–5. Sign up to view the full content.
Stat1301B
Probability& Statistics I
Fall 20092010
P.39
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
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
∈
and
x
X
≤
to represent the event
{ }

x
X
≤
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
Refer to the same example from Chapter I, we have
{ }
( )
78262
.
0
0
=
=
=
NNN
P
X
P
{ }
( )
20415
.
0
3
06805
.
0
,
,
1
=
×
=
=
=
NNA
NAN
ANN
P
X
P
{ }
( )
01302
.
0
3
00434
.
0
,
,
2
=
×
=
=
=
NAA
ANA
AAN
P
X
P
{ }
( )
00018
.
0
3
=
=
=
AAA
P
X
P
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document Stat1301B
Probability& Statistics I
Fall 20092010
P.40
Example 2.2
The annual income
ω
of a randomly selected citizen has a sample space
[ )
∞
=
Ω
,
0.
Suppose the annual income is taxable if it exceeds
c
. Let
X
be the taxable income.
Then the space of
X
is also
[
)
∞
,
0
a
n
d
[ )
∞
→
Ω
,
0
:
X
such that
()
⎩
⎨
⎧
>
−
≤
=
c
c
c
X
ωω
,
,
0
.
Note
: Conventionally, we use capital letters
X
,
Y
, … to denote random variables
and small letters
x
,
y
, … the possible numerical values (or
realizations
) of
these variables.
§ 2.2
Distribution of the Discrete Type
Definition
A random variable
X
defined on the sample space
Ω
is called a
discrete random
variable
if
() ()
{ }
:
Ω
∈
=
Ω
X
X
is countable (e.g.
{ }
,...
2
,
1
,
0
:
→
Ω
X
).
§ 2.2.1
Probability Mass Function and Distribution Function
Definition
The
probability mass function
(
pmf
) of a discrete random variable
X
is defined as
() ( )
x
X
P
x
p
=
=
,
( )
Ω
∈
X
x
,
where
Ω
X
is the countable set of possible values of
X
.
Example 2.3
For the previous example of card drawing, the pmf of
X
is
78262
.
0
0
=
p
,
20415
.
0
1
=
p
,
( )
01302
.
0
2
=
p
,
00018
.
0
3
=
p
.
Stat1301B
Probability& Statistics I
Fall 20092010
P.41
Conditions for a pmf
Since pmf is defined through probability, we have the following conditions for
p
to
be a valid pmf:
1.
()
0
≥
x
p
for
Ω
∈
X
x
;
0
=
x
p
for
Ω
∉
X
x
2.
1
=
∑
Ω
∈
X
x
x
p
3.
( )
∑
∈
=
∈
A
x
x
p
A
X
P
where
( )
Ω
⊂
X
A
Example 2.4
Is
6
x
x
p
=
,
3
,
2
,
1
=
x
a valid pmf ?
( )
x
p
1/2
1/3
1/6
1
2
3
(){ }
3
,
2
,
1
=
Ω
X
1.
0
6
>
=
x
x
p
for all
3
,
2
,
1
=
x
.
2.
1
2
1
3
1
6
1
3
1
=
+
+
=
∑
=
x
x
p
3.
(
)
(
)
2
1
2
1
2
=
+
=
≤
p
p
X
P
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document Stat1301B
Probability& Statistics I
Fall 20092010
P.42
Definition
The (
cumulative
)
distribution function
(
cdf
) of the discrete random variable
X
is
defined as
() ( ) ( )
∑
≤
=
≤
=
x
t
t
p
x
X
P
x
F
,
∞
<
<
∞
−
x
.
Example 2.5
Using previous example,
()
6
x
x
p
=
,
3
,
2
,
1
=
x
, we have
6
1
1
1
1
=
=
=
≤
=
X
P
X
P
F
(
)
(
)
(
)
...
99999
.
1
566
.
1
6
1
1
5
.
1
5
.
1
=
=
=
=
=
=
≤
=
F
F
X
P
X
P
F
() ( ) () ()
2
1
2
1
2
2
=
+
=
≤
=
p
p
X
P
F
() ( ) () ( ) ( )
1
3
2
1
3
3
=
+
+
=
≤
=
p
p
p
X
P
F
As can be seen, the cdf of a discrete random variable would be a
stepfunction
with
x
p
as the size of the jumps at the possible value
x
.
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 05/04/2011 for the course STAT 1301 taught by Professor Smslee during the Spring '08 term at HKU.
 Spring '08
 SMSLee
 Statistics, Probability, The Land

Click to edit the document details