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MA 519 Introduction to Probability
January 2000, Qualifying Examination
Instructor: Yip
•
This qualifying exam contains
ﬁve
questions.
•
By all means
simplify
your answers as much as possible.
•
It might be useful to know that for any positive integers
m
and
n
,
Z
1
0
x
m
(1

x
)
n
dx
=
m
!
n
!
(
m
+
n
+1)!
•
A normal table is provided at the end.
1. There are
n
people among whom are
A
and
B
. They stand in a row randomly. What
is the probability that there will be exactly
r
people between
A
and
B
?
What is the corresponding probability if they stand in a circular ring? (In this case,
consider only the arc going from
A
to
B
in the positive (i.e. counterclockwise) direc
tion.)
2. Consider a large collection of coins such that the probability
p
of a coin giving a head
is itself a random variable which is uniformly distributed in [0
,
1].
Let
X
be the total number of heads in
n
tossing of the coins. Find
P
(
X
=
i
)(
i
=
0
,
1
,...n
) in the following two situations:
(a) Pick a coin at random and then toss
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