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ECEN 303: Assignment 8
Problems:
1. If
X
is a random variable that is uniformly distributed between

1 and 1, fnd the PDF of
r

X

and the PDF of

ln

X

.
2. Find the PDF of
e
X
in terms of the PDF of
X
. Specialize the answer to the case where
X
is
uniformly distributed between 0 and 1.
3. Find the PDF of

X

1
/
3
and

X

1
/
4
in terms of the PDF of
X
.
4. Let
X
and
Y
be independent random variables, uniformly distributed in the interval [0
,
1].
Find the CDF and the PDF of

X

Y

.
5. Let
X
and
Y
be the Cartesian coordinates of a randomly chosen point (according to a uniform
PDF) in the triangle with vertices at (0
,
1), (0
,

1), and (1
,
0). Find the CDF and the PDF
of

X

Y

.
6.
Competing exponentials.
The lifetimes of two lightbulbs are modeled as independent and
exponential variables
X
and
Y
, with parameters
λ
and
μ
, respectively. The time at which a
lightbulb ±rst burns out is
Z
= min
{
X,Y
}
.
Show that
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 Fall '07
 Chamberlain

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