Midterm 1, Winter 06
Moulinath Banerjee
University of Michigan
February 22, 2006
Announcement:
The exam carries 30 points but the maximum you can score is 25.
(1) If
X
and
Y
are two uncorrelated random variables, are they necessarily independent? On the
other hand, if they are independent, are they necessarily uncorrelated? Justify your answers.
(6)
(2) Let
X
and
Y
be independent standard normal random variables. Set
U
= (3
X
+ 4
Y
)
/
5 and
V
= (4
X

3
Y
)
/
5.
(i) Find the joint density of (
U, V
) and also the marginal densities. (8)
(ii) Show that
U
2
+
V
2
is distributed like an exponential random variable with parameter
1
/
2. (4)
(3) Two helicopters land independently of each other on the plane.
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 Winter '08
 moulib
 Statistics, Variance, Probability theory, University of Michigan, Moulinath Banerjee, exponential random variable, uncorrelated random variables

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