Probability for Engineering Applications
Assignment #3, due at the beginning of class on
Thursday, February 9
th
, 2006
1.
(2 pts) (Textbook exercise 2.47) In this problem we derive the multinomial coeffi
cient.
Suppose we partition a set of
n
distinct objects into
m
subsets
B
1
, B
2
,
· · ·
, B
m
of
size
k
1
, k
2
,
· · ·
, k
m
, respectively, where
k
i
≥
0
∀
i
and
k
1
+
k
2
+
· · ·
+
k
m
=
n
.
a.
Let
N
i
denote the number of possible outcomes when the
i
th
subset is selected. Show
that
N
1
=
n
k
1
!
,
N
2
=
n

k
1
k
2
!
,
· · ·
, N
m

1
=
n

k
1
 · · · 
k
m

2
k
m

1
!
, N
m
= 1
Basically we are interested in the following: We have a total on
n
objects. From these,
we first pick
k
1
objects. How many ways of doing this (denote this by
N
1
)? From the
remaining objects, we now pick
k
2
objects. How many ways of doing this (denote this
by
N
2
)? And so on.
b.
Show that the total number of partitions is then
N
1
N
2
· · ·
N
m
=
n
!
k
1
!
k
2
!
· · ·
k
m
!
2.
(1 pt) (Textbook exercise 2.49) Show that
P
[
A

B
] satisfies the axioms of probability:
i. 0
≤
P
[
A

B
]
≤
1
ii.
P
[
S
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 Fall '08
 GELFAND
 Probability, Probability theory, Textbook exercise

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