p. 611
n
n
1
,
···
,n
m
=
n
!
n
1
!
×···×
n
m
!
ways
.
Example: MISSISSIPPI
11
4
,
1
,
2
,
4
=
11!
4!1!2!4!
.
Example (Die Rolling).
Q
: If a balanced (6sided) die is rolled 12 times,
P
(each face appears twice)=??
Sample space of rolling the die once (basic experiment):
0
= {1, 2, 3, 4, 5, 6}.
The sample space for the 12 trials is
=
0
×
×
0
=
0
12
An outcome
∈
is
=(
i
1
,
i
2
, …,
i
12
), where
1
≤
i
1
, …,
i
12
≤
6.
There are 6
12
possible outcomes in
, i.e., #
= 6
12
.
Among all possible outcomes, there are
of which each face appears twice.
P
(each face appears twice) =
12
2
,
2
,
2
,
2
,
2
,
2
=
12!
(2!)
6
12!
(2!)
6
1
6
12
.
p. 612
Generalization.
Consider a basic experiment which can result in one of
m
types of outcomes. Denote its sample space as
0
= {1, 2, …,
m
}.
Let
p
i
=
P
(outcome
i
appears),
then,
(i)
p
1
, …,
p
m
≥
0, and
(ii)
p
1
+
+
p
m
= 1.
Repeat the basic experiment
n
times. Then, the sample
space for the
n
trials is
=
0
×
×
0
=
0
n
Let
X
i
= # of trials with outcome
i
,
i
=1, …,
m
,
Then,
(i)
X
1
, …,
X
m
:
→
R
, and
(ii)
X
1
+
+
X
m
=
n
.
The joint pmf of
X
1
, …,
X
m
is
p
X
(
x
1
,...,x
m
)=
P
(
X
1
=
x
1
,...,X
m
=
x
m
)
=
n
x
1
,
···
,x
m
p
x
1
1
×···×
p
x
m
m
.
for
x
1
, …,
x
m
≥
0 and
x
1
+
+
x
m
=
n
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Documentp. 613
Proof. The probability of any sequence with
x
i
i
’s is
and there are
such sequences.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '11
 ShaoWeiCheng
 Probability, Probability theory, X1, random variables X1, Xk Ak

Click to edit the document details