QUALIFYING EXAMINATION
August 2009
MA 519 (M. D. Ward)
1.
Consider
n
≥
2 random points, placed uniformly and independently, onto the edge of a
circle with circumference 1. [An “arc” denotes a path on the circle.]
1a.
(5 pts) Find the density of the length of the arc that connects the ﬁrst point to the
closest of the other points.
1b.
(3 pts) Find the density of the straight line distance (i.e.,
not
on the circle) between
the two points described in part
1a
.
1c.
(5 pts) Consider the length of the largest arc that contains none of the
n
points in its
interior. Prove that the length of this largest arc goes to 0 in probability as
n
→ ∞
.
2.
(5 pts) Consider independent random variables
X
and
Y
, with
X
uniform on (0
,
2) and
with
Y
uniform on (0
,
3). Let
M
= max(
X,Y
) and
m
= min(
X,Y
). Find
P
(
m
2
> M
).
3.
(5 pts) Ten students organize a tournament. Each student competes against each other
student exactly once; all competitions are independent. In each competition, both students
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 Spring '09
 Probability theory, pts, probability density function, M. D. Ward

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