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Midterm 1: Stat 426.
Moulinath Banerjee
University of Michigan
October 27, 2005
Announcement:
The total number of points is 30 but the maximum you can score is 25.
(1) (a) Suppose that the radius of a circle is a random variable having the following probability
density function.
f
(
x
) =
1
8
(3
x
+ 1)
,
0
< x <
2
and 0 otherwise. Determine the probability density function of the area of the circle. (6)
(2) Let (
X,Y
) be uniformly distributed inside the ellipse given by
x
2
/a
2
+
y
2
/b
2
= 1. Thus, the
joint density of (
X,Y
) is:
f
(
x,y
) =
1
π ab
1
‰
x
2
a
2
+
y
2
b
2
<
1
±
.
where recall that 1
{
x
2
/a
2
+
y
2
/b
2
<
1
}
is the indicator function that is 1 if
x
2
/a
2
+
y
2
/b
2
<
1
and 0 otherwise.
(a) Let (
U,V
) = (
²
1
X,²
2
Y
) where
²
1
and
²
2
are either 1 or 1.
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 Winter '08
 moulib
 Statistics

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