Math 519 – B. Davis
Qualifying Examination
January, 2011
Do not do arithmetic or simplify answers.
(15) 1. Let
X
and
Y
be independent random variables each uniform on [0
,
1]. Give a
function
H
(
x,y
)=(
u
(
x,y
)
,v
(
x,y
))
such that
H
(
X,Y
) has a uniform distribution on the parallelogram with vertices
(0
,
0)
,
(0
,
2)
,
(3
,
3), and (3
,
5).
(20) 2. A pond contains
M
golden ﬁsh and
K
silver ﬁsh. The ﬁsh are removed one at a
time at random until all the ﬁsh remaining in the pond are the same color. Find
the expectation and the variance of the number of ﬁsh remaining in the pond.
(15) 3. Let
X
i
,1
≤
i
≤
10, be independent exponential random variables. Le
Y
1
and
Y
2
be independent random variables which are both discrete uniform on
{
0
,
1
,
2
}
and which are also independent of the exponentials. Find the probability that of
these twelve random variables, the
Y
j
’s are the second and third order statistics,
that is the second and third smallest of the twelve numbers.
(25) 4. Sunny, partly cloudy, and cloudy days each have probability 1/3 of occurring
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 Spring '09
 Math, Probability, Probability theory, B. Davis

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